# Modular curves over Q and the Eichler–Shimura congruence *Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Public domain (CC0).* A Hecke operator changes an elliptic curve by taking quotients by finite subgroups. Frobenius changes its coefficients by taking powers in a finite field. In characteristic \(p\), the degree-\(p\) Hecke correspondence separates into Frobenius and its transpose. This geometric identity produces the quadratic polynomials that will describe the Galois representations of weight-two forms. There are two counting issues to keep distinct. An ordinary elliptic curve in characteristic \(p\) has two subgroup schemes of rank \(p\), while the Hecke correspondence still has degree \(p+1\). Also, a space of cusp forms of dimension \(g\) supplies a Tate module of dimension \(2g\), but its Hecke trace is the Frobenius trace, without an extra factor of two. We prove both assertions, including the linear algebra needed to obtain the characteristic polynomial. ## 1. Modular curves and their good models For \(N\ge1\), let \(Y_0(N)\) classify elliptic curves together with a cyclic subgroup of order \(N\), and let \(Y_1(N)\) classify an elliptic curve together with a point of exact order \(N\). Over an algebraically closed field in which \(N\) is invertible, these descriptions mean isomorphism classes of pairs \[ (E,C),\qquad (E,P). \tag{1} \] The compactifications \(X_0(N)\) and \(X_1(N)\) add cusps. Over \(\mathbf C\), they are the compact Riemann surfaces attached to \(\Gamma_0(N)\) and \(\Gamma_1(N)\). The analytic construction is actually proved in the earlier *Modular curves and their genus*, §§1–2, and its moduli interpretation in *Complex tori, elliptic curves and the moduli interpretation of modular curves*, Theorem 4.1. The following stronger arithmetic theorem remains a required theorem; its construction is among the proof obligations identified in §7. **Model theorem.** These curves have smooth projective models over \(\mathbf Z[1/N]\). Their noncuspidal parts have the moduli interpretation (1), and \(Y_1(N)\) is a fine moduli scheme for \(N\ge4\). In particular, \(X_0(N)\), \(X_1(N)\), and their Jacobians have good reduction at every prime \(p\nmid N\). The distinction between a coarse space and a fine space matters. A point of a coarse curve over a nonclosed field is not a universal elliptic curve with its level structure. In particular, a coarse rational point records a Galois-invariant isomorphism class; twists and descent must still be considered. The fine scheme \(Y_1(N)\) represents the functor of isomorphism classes in families and carries a universal pair. **Lemma 1.0 (the interior level and tame quotient calculations).** Once a smooth elliptic family is given over a base on which \(N\) is invertible, its full-level, cyclic-level and exact-order-point functors are finite étale over that base. At an interior coarse-moduli point in residue characteristic greater than three, a smooth one-parameter elliptic deformation chart has a smooth invariant chart under its stabilizer. *Proof.* Multiplication by \(N\) has differential multiplication by \(N\) on the tangent line, at every point by translation, and is therefore étale. Its kernel is proper with finite fibres, hence finite étale. Its rank is \(N^2\) by the multiplication-degree proof in the earlier Tate-module lesson. Locally in the étale topology it is \((\mathbf Z/N)^2\). Choices of a basis, a cyclic direct summand, or an exact-order vector then form finite constant sets on this cover. Transition changes of basis permute the sets, which descend as finite étale schemes. The exact-order condition is checked prime by prime, so this includes composite \(N\). In characteristic greater than three write the fibre as \(y^2=x^3+ax+b\). An origin-preserving automorphism has the form \((x,y)\mapsto(c^2x,c^3y)\), with \(c^4a=a\) and \(c^6b=b\). This follows by comparing the unique pole spaces of orders two and three at the origin: their bases are \(1,x\) and \(1,x,y\); substituting their possible affine changes into the short cubic eliminates the extra shifts. Since \(a,b\) are not both zero, its group is cyclic of order dividing four or six, and its order is invertible in the residue field. In a strict complete one-parameter chart \(R[[t]]\), let a cyclic effective stabilizer of order \(e\) fix the origin. Its tangent action is faithful: an element of prime-to-characteristic order with tangent eigenvalue one has invariant parameter obtained by averaging its translates of \(t\); that parameter has unit linear coefficient, so fixing it makes the element the identity. Thus its tangent eigenvalue has order \(e\). Lift that eigenvalue \(\zeta\); the root-lifting proof in Lesson 01 lifts the prime-to-characteristic roots of unity to \(R\). Averaging \[ s=\frac1e\sum_{i=0}^{e-1}\zeta^{-i}\sigma^i(t) \] gives an eigenparameter, with unit linear coefficient. Formal substitution recursively inverts \(t\mapsto s\), and \(\sigma(s)=\zeta s\). Its invariant ring is exactly \(R[[s^e]]\): an invariant coefficient of \(s^n\) is zero unless \(e\mid n\), since \(1-\zeta^n\) is then a unit. This is a smooth formal chart. An ineffective stabilizer is discarded before applying the calculation. Thus tame coarse quotienting adds no singularity. This calculation proves the interior finite-level and tame-quotient steps; it does not construct the underlying elliptic deformation stack, its compactification, or the wild invariant charts in characteristics two and three. Those stronger steps of the stated model theorem are still required. ∎ The absence of automorphisms in the fine range can be checked on geometric fibers. First its geometric origin-preserving automorphism group is finite. The line bundle \(\mathcal O(3O)\) embeds it as a cubic, by the actual all-characteristic degree-three proof in *Abelian varieties*, Example 9.12. Every origin-preserving automorphism acts on its three-dimensional section space, and the group preserving the cubic and its origin is a closed subgroup of \(\operatorname{PGL}_3\), cut out by the polynomial-preservation equations. Its tangent space consists of vector fields vanishing at the origin: an infinitesimal map is locally the identity plus a derivation, and its overlaps identify those derivations as a global vector field. The elliptic tangent bundle is trivial by translation, while \(H^0(\mathcal O(-O))=0\) since a negative-degree line bundle has no section. Thus the group has zero tangent space, hence dimension zero at the identity and, by translation, everywhere. A zero-dimensional scheme of finite type has finitely many geometric points. Every such automorphism consequently has finite order. A nonidentity origin-preserving automorphism \(u\) of an elliptic curve has \(\deg(u-1)\le4\); equality four occurs only for \(u=-1\). Indeed, its finite-order characteristic polynomial has roots \(\lambda,\lambda^{-1}\) on the unit circle, so \[ \deg(u-1)=2-(\lambda+\lambda^{-1})\le4. \tag{2} \] The degree and characteristic-polynomial identity is the one established for endomorphisms in the Tate-module lesson. If equality holds, \(u+1\) is nilpotent in the endomorphism algebra and hence zero: a nonzero endomorphism of an elliptic curve is an isogeny and becomes invertible in that algebra. If \(u\) fixes a point of order \(N\), its kernel contains the \(N\) distinct points generated by that point, so \(\deg(u-1)\ge N\). This is impossible for \(N\ge5\); for \(N=4\), the only remaining candidate \(-1\) does not fix a point of order four. The representability and construction of the scheme remain part of the model theorem. ### 1.0Q. A relative cyclic quotient with a scheme-theoretic torsion marking This construction has the characteristic-zero scope needed for the cyclic Fricke correspondence. Let \(S\) be a Noetherian \(\mathbf Q\)-scheme, let \(E/S\) be an elliptic scheme with zero \(O\), and let \(P\in E[N](S)\) have exact order \(N\geq1\) on every geometric fibre. In particular \([N]P=O\) is an identity of sections; fibrewise order alone is insufficient on a nonreduced base, as proved in §1.0Q.2.3A. No reducedness, normality, connectedness or field hypothesis is imposed on \(S\). All quotients, group laws and maps below are relative scheme maps, and the constructed objects commute with arbitrary base change. The quotient and dual have the unambiguous identity \(\widehat q q=[N]\). The twice-quotiented marked object is identified with \((E,-P)\) by the dual identification. Composing that identification with \([-1]\) is the *chosen central normalization* yielding a composite \([-N]\). At \(N=1,2\) this choice is not forced by the point marking. The resulting coefficient-square convention still requires the central automorphism action in the modular descent object; the coefficient descent is still a separate required construction. #### 1.0Q.1. The precise previously written foundations The construction uses the following earlier results. 1. The finite nonnegative projective complex computing coherent cohomology after every base change is the actual proof of *Base change and the Grothendieck complex*, Lemma 3.1 and Theorem 3.2. Its finite coherent-cohomology and affine Čech prerequisites retain their earlier-programme scope. 2. Field curve Riemann–Roch, duality, positive-degree vanishing and degree-three very ampleness are proved in *Abelian varieties*, Lemma 9.11 and Example 9.12, and used explicitly below. Their projective duality, regular-local and proper coherent-finiteness foundations are retained. The earlier elliptic Tate-module lesson, Sections 0.C–0.F, proves proper coherent finiteness, formal functions and proper finite-geometric-fibre finiteness over locally Noetherian bases. Theorem 0.F.1 and the family-embedding argument in Section 0.F.4 supply exactly the finiteness and whole-base closed-immersion steps used here. No unrestricted normalization, conductor or quasi-finite factorization theorem is an input to this Noetherian construction. The basic affine sheaf, Proj, derived, curve and étale prerequisites of those local proofs remain explicit. 3. Faithfully flat module, algebra and affine-morphism descent are proved at the beginning of *Quotients and torsors*. For related affine and global quotient results, see Theorem 4.3, Corollary 4.4 and Theorem 5.2 of that lesson, and Lemma 5.4 for finite affine orbit neighbourhoods. Sections 1.0Q.3–4 below prove the special finite-constant-group quotient directly. 4. The differential criterion for finite flat étale maps, étale local-ring regularity, smoothness over a perfect field and from flatness and smooth fibres, and étale coordinates have actual proofs in *Étale morphisms and their local structure*, Lemma 1.2 and Theorems 1.3 and 5.1, and *Smooth morphisms*, Theorems 2.1, 3.1 and 4.1. Their commutative-algebra flatness and regularity prerequisites are retained. 5. The field multiplication degree \(n^2\), torsion basis, dual isogeny, and every Weil-pairing property used below are the complete actual proofs in LG-GAL's elliptic Tate-module lesson, Lemmas 1.1–1.4. Its Sections 0.C–0.F prove the Noetherian proper-étale, elliptic multiplication and relative-isogeny assertions; Sections 1.4A.1–1.4A.2 prove the field cyclic quotient, constants and rigidity used by the Weil pairing. Lemma 3.0 supplies the further local lifting argument. Section 1.0Q.6 below extends the field pairing to families without determining maps on a nonreduced base merely from its fibres. Thus this construction binds to specific written algebraic and coherent foundations. It does not certify their entire transitive closure. #### 1.0Q.2. Relative projective coordinates and torsion ##### 1.0Q.2.1. The two coherent-complex consequences used here On an affine base \(\operatorname{Spec}R\), use the finite nonnegative projective complex \(K^\bullet\) of foundation 1. Locally make its terms free. A unit entry of a differential can be reduced by row and column operations to an identity block; the complex equation makes the adjoining row and column zero in the neighbouring differentials. Delete this acyclic two-term block. At a chosen prime repeat until every remaining differential has zero residue matrix. The ranks of its remaining terms then equal the fibre cohomology dimensions. Every deletion is a split chain homotopy equivalence and remains so after arbitrary tensor product. If a line bundle \(L\), flat over the base, has only \(H^0\) on every geometric fibre, with that dimension equal to \(r\), this cancellation leaves just \(R^r\) in degree zero on a neighbourhood. Consequently \(\pi_*L\) is locally free of rank \(r\), and its formation commutes with every base change. This conclusion does not assume the base is reduced. For later use, suppose instead that the base is reduced and a line bundle is trivial on every geometric fibre of a genus-one family. Its fibre dimensions are one in degrees zero and one and zero elsewhere. Near a chosen point cancellation leaves \(R\xrightarrow{a}R\) in degrees zero and one. The same fibre dimensions at every point of this neighbourhood force \(a\) to vanish in every residue field. Reducedness gives \(a=0\). Hence its \(H^0\) is an invertible module commuting with base change. Evaluation \(\pi^*\pi_*L\to L\) is an isomorphism: it is an isomorphism on every fibre and a map of invertible modules, so its local coefficient is a unit. Locally trivializing \(\pi_*L\) therefore trivializes \(L\) on the whole family. Reducedness is used here and is not silently imposed on the arbitrary \(S\). We also need universal constants on a proper flat family \(X/S\) with geometrically connected reduced fibres and a section. The constants map \(R[0]\to K^\bullet\) is split by restriction to that section. This can be represented by chain maps: choose an affine cover with the section contained in its first member locally on the base, evaluate its degree-zero Čech term there, and compose with the projective complex comparison. The constant cocycle has evaluation one. Split off \(R[0]\). The remaining nonnegative complex has zero fibre \(H^0\); cancellation at any base point therefore removes its degree-zero term. Its \(H^0\) is zero after every base change. It follows that \[ \pi_*\mathcal O_X=\mathcal O_S \] universally. This argument applies to \(E\) and its fibre products. It preserves nilpotent base parameters. ##### 1.0Q.2.2. A local short Weierstrass family, proved over the base The zero section is an effective Cartier divisor: in a smooth relative curve coordinate its ideal has one generator with nonzero relative derivative. On a geometric elliptic fibre, foundation 2 gives \[ h^0(\mathcal O(nO))=n,\qquad H^1(\mathcal O(nO))=0\quad(n>0). \] Section 1.0Q.2.1 makes all these section modules locally free with arbitrary base change. The inclusion of constants identifies \(\pi_*\mathcal O(O)\) with \(\mathcal O_S\), since it is an isomorphism on all fibres. For \(n\geq2\), the divisor exact sequence for \((n-1)O\subset nO\), together with \(H^1(\mathcal O((n-1)O))=0\), makes the successive section quotient the invertible module \(\pi_*(\mathcal O(nO)|_O)\). After shrinking the base, choose functions \(x,y\) representing generators of the rank-one pole-order quotients for \(2O,3O\). Their leading pole coefficients are units. The seven functions \(1,x,y,x^2,xy,x^3,y^2\) in the rank-six module for \(6O\) have one relation with unit coefficients of \(x^3,y^2\): their other pole orders are distinct, and reduction on every fibre gives the displayed rank. Gaussian elimination with the unit leading coefficients gives a relation \[ y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6. \] Rescale \(x,y\) together to make the two leading coefficients equal, as in the field construction; no extraction of a root of a base unit is needed. Since \(2,3\) are invertible, completing the square and translating \(x\) give \[ E:\quad y^2=x^3+ax+b,\qquad 4a^3+27b^2\in\mathcal O_S^\times. \tag{Q1} \] Here is the scheme-embedding check. The three sections of \(\mathcal O(3O)\) generate it on every fibre, hence on \(E\) by Nakayama. They define \(i:E\to\mathbf P^2_S\). This map is proper (closed graph followed by a proper projection), and every fibre map is a closed immersion by the field very-ampleness proof. Its fibres over points of \(\mathbf P^2_S\) are finite, so foundation 2 makes \(i\) finite. On an affine target chart its algebra cokernel is finite. After tensoring with every base residue field it is zero, because the corresponding fibre map is a closed immersion; finite pushforward here is just affine tensor product. Local Nakayama makes the cokernel zero, so \(i\) is a closed immersion. Its image lies in the cubic given by the relation. That cubic is smooth and flat over \(S\), by the Jacobian calculation with the discriminant unit. The ideal of \(E\) in it has zero fibre: tensoring the ideal sequence is left exact because \(E/S\) is flat. Nakayama makes the ideal zero. This proves (Q1) as a family isomorphism, including over nonreduced \(S\). Here is the relative pointed group-law construction used for the universal short cubic. For the point-class and secant arguments, see *Abelian varieties*, Lemma 9.0a and Theorem 9.0. On the square of the universal cubic the three minors of the two coordinate vectors generate the pullback of the projective diagonal ideal, hence its Cartier diagonal ideal. Dividing by a local equation gives three coefficients generating the unit ideal; they glue to the secant-or-tangent line morphism. On the pulled-back cubic this line cuts an effective relative Cartier divisor with fibre degree three, since an integral cubic has no line component. It contains the two universal point graphs as Cartier divisors, including their intersection. More explicitly, in an étale curve parameter their ideals are \(t-p,t-q\). Modulo the second ideal the first becomes \(q-p\), a local equation of the diagonal in the parameter base and a non-zero-divisor. If the line equation is \((t-p)h\) and vanishes modulo \(t-q\), this non-zero-divisor forces \(h\) to vanish there. Thus the equation is divisible by \((t-p)(t-q)\). Away from the intersection the ideals are comaximal and the same divisibility holds. The residual divisor has fibre length one; its local derivative in a curve parameter is a unit. The étale-coordinate criterion makes it étale; it is proper and thus finite, and finite étale rank one is an isomorphism to the parameter base by the unit map and Nakayama. It is therefore the graph of a morphism giving the third point. Reflection through the origin is its restriction, and composing reflection with the third-point map gives addition. These computations construct the operations over \[ B=\mathbf Q[a,b,(4a^3+27b^2)^{-1}]. \tag{Q2} \] Group identities hold there by the actual field point-class proof and reducedness of the smooth universal products; thus they hold after arbitrary pullback. They agree with the supplied elliptic group law on \(E/S\), also for nilpotent \(S\). Indeed their difference, formed using the cubic law, is a map \(E^2\to E\) whose geometric images are all the zero section. Locally the cubic has an affine neighbourhood of that section, so this map lands in that neighbourhood. Universal constants for \(E^2\), proved in §1.0Q.2.1, make it factor through \(S\); its value on the zero pair is zero. This proves the equality of group maps without a reducedness inference on \(S\). A pointed isomorphism between short families has \(x=cx'+r,\ y=dy'+sx'+t\) from the same pole filtration, with \(c,d\) units and \(d^2=c^3\). Put \(u=d/c\); then \(c=u^2,d=u^3\). Comparing the \(x'^2,x'y',y'\) coefficients, with \(2,3\) invertible, gives \(r=s=t=0\). Thus all such changes are \(x=u^2x',y=u^3y'\). This computation is over the base ring and permits nilpotents. It will make the pairing construction coordinate-independent. ##### 1.0Q.2.3. Multiplication, the constant cyclic subgroup, and torsion frames For every \(n\geq1\), the tangent group law gives \(d[n]=n\,\mathrm{id}\), at every point by translation. Across a square-zero ideal, smoothness first lifts a source point; the inverse differential corrects its prescribed image uniquely. Thus \([n]\) is formally étale and of finite presentation, hence étale by foundation 4. Its graph makes it proper. The actual relative finiteness lemma in the Tate lesson makes it finite étale. On every fibre its rank is \(n^2\), by the complete field Lemma 1.1. Hence \[ E[n]\longrightarrow S \text{ is finite étale of rank } n^2. \tag{Q3} \] No connectedness of \(S\) is needed for this fibrewise constant rank. Because \([N]P=O\) holds as sections, each \(iP\) is a section of \(E[N]\). Sections of a separated étale scheme are open and closed immersions. Distinct \(iP\), \(0\leq i0\), (Q4) says \(r^2=Nr\), hence \(r=N\). Its base change along itself is the disjoint union of \(N\) identities. Differential base change and faithful flatness give \(\Omega_{A/D}=0\); the finite-presentation, flat differential criterion makes \(A/D\) étale. Therefore \[ U\to Y_U=\operatorname{Spec}D \text{ is finite étale of degree }N,\qquad G\times U\simeq U\times_{Y_U}U. \tag{Q8} \] For completeness \(D\) is of finite type over the Noetherian \(R\). Choose finite \(D\)-module generators \(b_i\) of \(A\), including one, and finite \(R\)-algebra generators of \(A\). Form the \(R\)-subalgebra \(D_0\subset D\) generated by the coefficients expressing those generators and all products \(b_ib_j\) in the \(b_i\). Their \(D_0\)-span is a subalgebra containing all the \(R\)-algebra generators, hence is \(A\). Thus \(A\) is finite over the Noetherian finite-type \(D_0\). Its submodule \(D\) is finite over \(D_0\), so is of finite type and Noetherian. This supplies the finite-presentation assertions above. The invariants commute with *arbitrary* change of \(R\): \(e\) is an idempotent, \(D=\operatorname{im}e\), and the splitting \(A=\operatorname{im}e\oplus\ker e\) survives every tensor product. The image of the tensor idempotent is exactly the new invariants. This use of \(N^{-1}\) is valid on all \(\mathbf Q\)-bases; it is not an assertion at primes dividing \(N\). #### 1.0Q.4. Invariant relative affine charts and gluing We now produce the charts of §1.0Q.3, rather than assume an orbit quotient exists. On an affine base open, §1.0Q.2.2 embeds \(E\) in projective space. Any finite set of points of \(E\) has an affine open neighbourhood: if their homogeneous primes are \(\mathfrak p_i\), choose a positive-degree homogeneous element avoiding all of them by homogeneous prime avoidance. Explicitly discard primes contained in another; choose for each remaining prime a homogeneous \(a_i\notin\mathfrak p_i\) in the irrelevant ideal and homogeneous \(b_{ij}\in\mathfrak p_j\setminus\mathfrak p_i\). The products \(a_i\prod_{j\ne i}b_{ij}\), raised to a common degree and added, avoid all the primes, since only the designated summand survives modulo each prime. The corresponding projective standard open cuts out an affine open of the closed embedded \(E\). This is the full finite-set argument of the programme Lemma 5.4, written here for the needed projective case. Apply this to the finite orbit of a point, obtaining an affine open \(V\) containing it. Set \[ U=\bigcap_{g\in G}g(V). \tag{Q9} \] It is stable and still contains the orbit. It is affine: intersections of two affine opens in a separated scheme over an affine base are affine, since they are inverse images of its closed diagonal in the product of those affines. Finite intersection proves the assertion. Alternatively if \(V=E_s\) for a section of an ample power \(L^d\), (Q9) is the nonvanishing open of the norm section \(\bigotimes_g g^*s\) of the canonically translation-linearized bundle \(\bigotimes_g g^*L^d\). Thus this is an invariant relative ample-chart construction, not a fieldwise quotient formula. These \(U\)'s cover \(E\). Construct \(Y_U=\operatorname{Spec}\Gamma(U,\mathcal O)^G\) by §1.0Q.3. Each of its fibres is precisely one orbit, as the torsor identity (Q8) shows after a geometric base change. If \(W\subset U\) is invariant open, its closed invariant complement has closed image under the finite \(q_U\). Hence \[ O_W=Y_U\setminus q_U(U\setminus W),\qquad q_U^{-1}O_W=W. \tag{Q10} \] Restriction of (Q8) and of the finite faithfully flat cover represents the fppf orbit sheaf of \(W\) by this open. Indeed every test point of the target lifts after that cover, and two lifts differ by exactly one group element. For two charts \(U,V\), their intersection therefore has the same represented orbit sheaf as an open of both \(Y_U,Y_V\). The unique identifications satisfy the triple cocycle, so the \(Y_U\)'s glue to a scheme \(E'\) and the maps glue to \[ q:E\to E'=E/C. \tag{Q11} \] The map is finite étale, surjective and of degree \(N\), and \(C\times_SE\simeq E\times_{E'}E\). It represents the fppf quotient over \(S\). An invariant map into any scheme descends uniquely: first for affine targets use the invariant functions on each chart; for a general target its affine preimages are invariant opens, descend by (Q10), and glue those affine maps. This supplies the scheme-valued universal property used below, not merely an orbit-set property. All the statements commute with arbitrary \(T\to S\). The torsor relation survives base change; the same lifting calculation identifies \(E'_T\) with the quotient of \(E_T\). Equivalently on affine base-change charts the Reynolds splitting proves the invariant-ring compatibility directly. The schemes over different affine opens of \(S\) identify by the same quotient universal property, so this is a global relative construction even when \(S\) has no global Weierstrass coordinates. #### 1.0Q.5. The elliptic group quotient and its dual On the quotient sheaf set \(q(x)+q(y)=q(x+y)\), inversion \(q(x)\mapsto q(-x)\), and zero \(q(O)\). Addition is invariant under the product action \(C^2\) on \(E^2\); the product torsor gives its scheme descent. The unit, inverse, commutativity and associativity identities can be checked after the appropriate finite faithfully flat product covers, so are actual scheme identities. The kernel of \(q\) is \(C\), by its torsor relation. The quotient is of finite presentation over \(S\) by the affine finite-type argument in §1.0Q.3 and Noetherianity. Its diagonal is closed: its pullback by \(q\times q\) is the closed union of translation graphs, and faithfully flat ideal descent, proved in the opening of *Quotients and torsors*, descends that closed immersion. It is universally closed over \(S\): a closed subset of \(E'_T\) and its inverse image in \(E_T\) have the same image in \(T\), because \(q_T\) is surjective; the latter image is closed by properness of \(E/S\). Hence \(E'/S\) is proper. Flatness descends explicitly. For an injection of base modules, tensor its pullback to a quotient chart with the faithfully flat \(D\)-algebra \(A\). It becomes injective since \(E/S\) is flat. Faithfulness gives injectivity already after tensoring with \(D\), so \(D\) is flat over the base. On a geometric fibre the étale cover \(E_{\bar s}\to E'_{\bar s}\) descends regularity of local rings, by the actual étale-local-ring proof in foundation 4. Over characteristic zero regular finite-type curves are smooth. The flat, finite-presentation fibre criterion therefore makes \(E'/S\) smooth. Its fibres are geometrically connected, since they are surjective images of the connected elliptic fibres, and have dimension one by the finite map. Their genus is one. On each smooth fibre, translation trivializes the rank-one differential bundle of this proper one-dimensional group: at a point translate its tangent space back to zero, algebraically by the group maps. Thus its canonical degree is zero. The curve duality formula \(2g-2=\deg\Omega^1\) gives \(g=1\). Together with its zero section this proves \(E'/S\) is an elliptic scheme. In particular §1.0Q.2.2 also supplies its relative projective model; projectivity of the quotient was not presumed in its construction. The scheme-section identity \([N]P=O\) gives \([N](x+iP)=[N]x\) as scheme maps, so the quotient universal property uniquely produces \[ \widehat q:E'\to E,\qquad \widehat q q=[N]_E. \tag{Q12} \] It is a group homomorphism by faithful descent of that equality. The other identity follows after precomposing with \(q\), a faithfully flat cover: \[ q\widehat q=[N]_{E'}. \tag{Q13} \] The differential of \(\widehat q\) is an isomorphism, by (Q12), \(dq\) invertible and \(N\) invertible. The square-zero lifting argument in §1.0Q.2.3 makes it étale. Its closed graph makes it proper, and the actual relative finiteness lemma makes it finite étale. It is surjective because (Q12) is. Degrees on fibres multiply; \(N\deg\widehat q=N^2\), so its degree is \(N\). It is the dual isogeny: on every field fibre the actual Lemma 1.3 characterizes the dual by (Q12), and (Q12) itself uniquely characterizes the relative map without a fibrewise equality inference. This map and both identities commute with all base changes by their construction and uniqueness. As fppf sheaves, \[ \ker\widehat q=q(E[N])\simeq E[N]/C. \tag{Q14} \] For a section of the kernel, lift locally through the cover \(q\); (Q12) says the lift is \(N\)-torsion. Two such lifts differ by \(C\). This proves (Q14) over all test schemes. #### 1.0Q.6. A relative Weil pairing, with nilpotent bases retained We construct the pairing over the short universal family (Q2) and then descend through its coordinate changes. This step is necessary: the field pairing alone does not specify a morphism over an arbitrary nonreduced \(S\). The finite étale algebra of \(T=E_B[n]^2\) is normal and reduced. Here is the algebra needed for that assertion without an unproved normality import. If \(R\) is an integrally closed Noetherian domain and \(A/R\) finite étale, flatness injects \(A\) into \(A_K\), a product of separable fields. The trace pairing is perfect: on every residue field, separable field factors have perfect trace (a primitive-element Vandermonde calculation), and Nakayama applied to the determinant gives invertibility locally over \(R\). If \(z\in A_K\) is integral over \(A\), it is integral over \(R\). For every \(a\in A\), \(\operatorname{Tr}(za)\) lies in \(K\) and is integral over \(R\), hence in \(R\). The perfect trace pairing puts \(z\) in \(A\). Thus \(A\) is integrally closed in its product of fraction fields and is reduced. The field-factor idempotents are integral and extend to \(A\), so it is a product of normal domains. Apply this with the localization of the polynomial domain \(B\). The same assertion applies to every finite étale cover of \(B\) used below. Let \(U,V\) be the universal torsion sections on \(E_T\), and put \[ D_V=(V)-(O),\qquad M=[n]^*\mathcal O(D_V). \] Every geometric fibre of \(M\) is trivial by the field point-class and dual-multiplication proof in the Tate lesson: its class is \([n]V=0\). The second coherent consequence in §1.0Q.2.1 applies because \(T\) is reduced. Locally on \(T\) choose a trivialization of \(M\), giving a relative rational function \(g_V\) with Cartier divisor \([n]^*D_V\). These divisors restrict correctly to fibres: \([n]\) is finite étale and the point divisors are relative Cartier divisors. Translation by \(U\) preserves the displayed divisor since \([n]\circ t_U=[n]\). Therefore \[ e_n(U,V)=g_V(X)/g_V(X+U) \tag{Q15} \] is a regular unit on the whole relative curve: the quotient of two rational sections with the same Cartier divisor has unit coefficient in each local divisor trivialization. Universal constants from §1.0Q.2.1 make it a unit on \(T\). A different chosen trivialization of \(M\) differs by a base unit, which cancels in (Q15). Its \(n\)-th power is one by translating \(n\) times, so it defines a morphism \(T\to\mu_n\). It agrees on every field fibre with the actual field formula (G9). Bilinearity and alternation are identities of functions on finite étale covers of \(B\); their field values are exactly the full field Lemma 1.4. Those covers are reduced by the trace argument, so the identities hold as scheme identities. Perfection can be checked after a torsion-frame and roots-of-unity cover: the alternating pairing has matrix \(\left(\begin{smallmatrix}0&a\\-a&0\end{smallmatrix}\right)\) over \(\mathbf Z/n\), and \(a\) is a unit on every field fibre by that lemma. This is a perfect map between constant finite étale groups, hence an isomorphism. Composite \(n\) and even \(n\) are covered; actual alternation was proved in the field lesson rather than inferred merely from skew symmetry. Coordinate independence is a universal identity, including for nilpotent bases. The changes \(x=u^2x',y=u^3y'\) are parametrized by the normal reduced ring \(B[u,u^{-1}]\), with the corresponding change of \(a,b\). Its torsion covers are finite étale and reduced. On all field fibres the two pairings agree by the field pairing's invariance under pointed isomorphism, proved by Lemma 1.4's adjoint calculation for degree one. Hence they agree as functions on these universal covers. Pulling this identity to any base ring, including one with nilpotents, proves the overlap identity. The local pairings therefore glue on arbitrary \(E/S\), and their construction is compatible with arbitrary base change: \[ e_n:E[n]\times_SE[n]\longrightarrow\mu_{n,S} \text{ is perfect, bilinear and alternating}. \tag{Q16} \] The relative adjoint formula for this cyclic quotient also holds as a universal identity. Parametrize a primitive \(N\)-torsion section by the open-and-closed primitive part \(B_N\) of \(E_B[N]\). It is finite étale over \(B\), hence reduced and normal. Sections 1.0Q.3–1.0Q.5 construct the universal quotient and its dual over \(B_N\). On affine opens choose short coordinates for that quotient by §1.0Q.2.2. The torsion product of source and quotient over these reduced opens is finite étale and reduced. The actual field adjoint proof gives equality of the two pairing functions on every field fibre; reducedness proves the universal identity. Every \((E/S,P)\) is locally the pullback of this marked short family, and the quotient commutes with that pullback. Thus, also on nonreduced \(S\), \[ e_n^{E'}(qA,B)=e_n^E(A,\widehat qB),\qquad A\in E[n],\ B\in E'[n]. \tag{Q17} \] In this last passage no equality was inferred just from the fibres of the arbitrary \(S\). It was first proved on an actually reduced universal parameter scheme and then pulled back as an identity. This also proves arithmetic automorphism covariance of (Q16), since the whole construction is over \(\mathbf Q\). #### 1.0Q.7. The marked quotient point and arithmetic covariance Pass to \(S_\zeta=S\times_{\mathbf Q}\operatorname{Spec}\mathbf Q(\zeta_N)\) if necessary, choosing a primitive section \(\zeta_N\) of \(\mu_N\). For \(N=1\), this means \(\zeta_1=1\). By (Q16), \[ a_P:E[N]\to\mu_N,\qquad Q\mapsto e_N(P,Q) \] is surjective with kernel \(C\). To check the exact ranks, use a torsion frame. A vector of order \(N\) in \((\mathbf Z/N)^2\) is primitive: its two coordinates and \(N\) have greatest common divisor one. Bezout extends it to a basis. A perfect alternating form sends it to a surjective character; its kernel has \(N\) elements and contains its own generated subgroup, hence is exactly that subgroup. These are identities of constant groups on the étale cover, so descend as scheme identities. The fibre \(T_\zeta=a_P^{-1}(\zeta_N)\) is therefore a finite étale \(C\)-torsor, of rank \(N\), and surjective over \(S_\zeta\). Its universal lift \(Q\) need not exist on \(S_\zeta\) itself. On this cover the point \(q(Q)\) is unchanged by replacing \(Q\) with \(Q+iP\); on the double overlap every pair of lifts differs uniquely by such a multiple. Descent of the morphism \(T_\zeta\to E'\), or the scheme-valued sheaf property proved in §1.0Q.4, gives a unique section \[ P'_\zeta=q(Q)\in E'(S_\zeta). \tag{Q18} \] This proves lift independence over families, not merely at geometric points. The selected lift lies in \(E[N]\), so \([N]q(Q)=q([N]Q)=O\) is a section identity after descent. Its order is exactly \(N\): if \(r q(Q)=0\), then \(rQ\in C\), so \(1=e_N(P,rQ)=\zeta_N^r\). Fibrewise this forces \(N\mid r\), and those exact orders make it a primitive section of the finite étale torsion scheme. Formula (Q14) and the torsion-frame calculation show \[ \ker\widehat q=\langle P'_\zeta\rangle. \tag{Q19} \] Consequently \(\mathcal F_\zeta(E,P)=(E',P'_\zeta)\) is a relative marked Fricke construction, functorial for isomorphisms and base change. It retains every \(N\), without a character restriction. For \(u\in(\mathbf Z/N)^*\), \(uQ\) pairs with \(P\) to \(\zeta_N^u\). Hence \[ P'_{\zeta^u}=uP'_\zeta. \tag{Q20} \] A field or base automorphism \(\sigma\) transports all these scheme maps and the pairing, giving \(\sigma\mathcal F_\zeta\sigma^{-1}=\mathcal F_{\sigma\zeta}\). If \(\sigma\zeta=\zeta^u\), this is \(D_u\mathcal F_\zeta\), where \(D_u(E',P')=(E',uP')\). Replacing the input marking by \(dP\) replaces the selected lift by \(d^{-1}Q\), so \[ \mathcal F_\zeta D_d=D_{d^{-1}}\mathcal F_\zeta. \tag{Q21} \] If the actual modular coefficient object and its pullbacks have been descended, pullback reverses compositions and gives precisely Lesson 12's \(\sigma w\sigma^{-1}=wD_u^*\). No quotient-model existence or nebentypus projection follows just from this conditional last interpretation. #### 1.0Q.8. The Fricke square and the central sign Let \(q_2:E'\to E''=E'/\langle P'_\zeta\rangle\) be the second relative quotient, and \(P''\) its marked point selected by the same \(\zeta_N\). Formula (Q19) identifies its kernel with the kernel of \(\widehat q\). The latter finite étale surjective homomorphism is itself the quotient torsor under that kernel: the difference of two lifts gives the unique kernel element in its fibre relation. Thus the quotient universal property gives a unique isomorphism \[ v:E''\xrightarrow{\sim}E,\qquad vq_2=\widehat q,\qquad vq_2q=[N]. \tag{Q22} \] For an explicit second marking, lift \(P\) locally to \(R\in E[N^2]\) with \([N]R=P\). Such lifts exist on a finite étale cover by (Q3) and the field torsion transition, or its constant-frame calculation. Then \(qR\in E'[N]\). Formula (Q17) gives \[ e_N^{E'}(qQ,qR)=e_N^E(Q,[N]R)=e_N^E(Q,P)=\zeta_N^{-1}. \] So \(-qR\) is an allowed second lift, and \[ P''=q_2(-qR),\qquad vP''=-P. \tag{Q23} \] Descent makes these identities global. The twice-quotiented marked object is therefore canonically identified by the **dual identification \(v\)** with \((E,-P)\). Now choose the marked identification \[ j=[-1]_E\circ v:(E'',P'')\longrightarrow(E,P). \] It has the exactly computed composite \[ jq_2q=[-N]. \tag{Q24} \] Equations (Q22) and (Q24) distinguish the universal dual statement from its central normalization. At \(N=1\), all points are zero and the quotient is the identity; \(v\) and \(j=-v\) are different automorphisms with the same marking. At \(N=2\), \(P=-P\), so again both identifications preserve the marking. The marking never forces this sign at these levels. Even at other levels the notation records the selected dual-derived identification, not an assertion of uniqueness among all automorphisms of a special elliptic curve. For the coefficient object in Lesson 12, the specified identification \(j\) makes the double universal isogeny \([-N]\), whose pullback on \(R^1\) is multiplication by \(-N\), and on \(\operatorname{Sym}^{m}R^1\) by \((-N)^m\). The actual Kummer comparison proof is the Tate-module lesson, Section 4, its Kummer-comparison paragraph: finite degree-one cohomology is Picard torsion, its multiplication pullback is multiplication on point classes by the dual identity, and the surjective transition maps kill the degree-one inverse-limit correction. Consequently multiplication by \(-N\) acts by \(-N\) on every geometric degree-one stalk; equality of maps of the étale sheaf is detected by these stalks. This is not a choice of an integral self-duality. Thus the convention gives \(w^2=(-N)^m\) **provided the modular coefficient descent carries this central identification and automorphism action**. On a characteristic-zero low-level invariant object, \([-1]\) acts on the coefficient fibre by \((-1)^m\); an odd \(m\) invariant under that central stabilizer is zero. Likewise \(\chi(-1)=(-1)^k\) is the parity constraint. None of these coefficient assertions permits deleting the stabilizer and declaring a coarse-scheme marking to have forced the sign. This zero-invariant assertion uses that 2 is invertible in the coefficient field and does not assert vanishing for a characteristic-two residual coefficient object. #### 1.0Q.9. What is now constructed, and what remains For every supplied elliptic scheme over every Noetherian \(\mathbf Q\)-base with \(P\in E[N](S)\) of exact geometric-fibre order, the relative cyclic quotient, its elliptic structure, base-change compatibility, finite étale quotient and dual, relative Weil pairing, selected quotient point, cyclotomic covariance and dual/central square distinction have now been constructed from the explicit written foundations above. The construction is independent of nebentypus; its diamond covariance is what later permits arbitrary-character coefficient descent. The coefficient application in the higher-weight lesson still needs its separately stated arithmetic and descent proofs. The next construction supplies the characteristic-zero fine full-level scheme, its universal family and cyclotomic-component actions. Small-level modular coefficient descent, compactification at the cusps, models over \(\mathbf Z[1/n]\), specialization at every good prime, and the supersingular degree-\(p\) correspondence with coefficient trace remain separate construction obligations. In particular \(N^{-1}\), \(2^{-1}\) and \(3^{-1}\) were available here on the \(\mathbf Q\)-base. The construction of integral models and specialization at every good prime remains necessary. The quotient construction and its marked central return are illustrated below. The maps in the figure are the exact scheme maps just proved; no sampled geometry or numerical test replaces the argument. #### 1.0Q.10. Primary reading and evidence boundary The named earlier programme proofs supply the module-descent, coherent-complex, curve, étale and smoothness interfaces used above. Their transitive foundations remain precisely those recorded in §1.0Q.1. In particular the field pairing retains the finite translation-quotient route through *Abelian varieties*, Proposition 6.3, and *Quotients and torsors*, Theorem 5.2, and the pointed-law compatibility route through rigidity Corollary 2.3. These are explicit programme obligations until their complete supporting proofs are bound. - [Official Stacks finite quotient proof, Tag 07S5](https://stacks.math.columbia.edu/tag/07S5), including its proof and affine/orbit prerequisites, and [Tag 03C8](https://stacks.math.columbia.edu/tag/03C8), give related proofs. The special constant-group algebra (Q4)–(Q8) is written here explicitly, so the theorem citation is not its proof provider. - [Official Stacks Tag 03GW](https://stacks.math.columbia.edu/tag/03GW), [Tag 02LN](https://stacks.math.columbia.edu/tag/02LN), [Tag 02LR](https://stacks.math.columbia.edu/tag/02LR), and [Tag 02LS](https://stacks.math.columbia.edu/tag/02LS) give the corresponding proofs. The argument requires the subsidiary results specified with those proofs. - [Milne's freely accessible author notes](https://www.jmilne.org/math/CourseNotes/AV.pdf), I §13, printed pp.57–58, actually supply the reciprocal-translation convention. Lemma 13.1 and Proposition 13.2 there refer elsewhere for their proofs; those references are not imported. The actual local Tate-module Lemma 1.4 supplies the complete elliptic proofs, and §1.0Q.6 here supplies the relative passage. - [Deligne–Rapoport's freely accessible original author-hosted paper](https://publications.ias.edu/sites/default/files/Number22.pdf), I §§2 and 6 and IV §3.20–4.4, discusses the relative pairing and quotient conventions. Its relative pairing and quotient remarks are source assertions with explicit imports; they do not replace §§1.0Q.3–1.0Q.8. In particular IV §3.20 calls its relative pairing identification known, rather than writing the proof needed here. These references give conventions and further comparisons for the preceding arguments. ![Exact relative cyclic quotient, its dual, marked covariance and the chosen central return](../assets/relative-cyclic-quotient-proof-map.png) [Open Figure 1 at full size](../assets/relative-cyclic-quotient-proof-map.png) *The bases, finite étale lifts and unit indices are part of the displayed maps. Equations (Q1)–(Q24) and §1.0Q.2.3A prove the identities and the nilpotent counterexample.* ### 1.0F. An affine fine full-level moduli scheme in characteristic zero This construction supplies an actual scheme and universal elliptic family for every integer \(M\geq3\). Its test bases are all Noetherian \(\mathbf Q\)-schemes, including nonreduced bases. A full level structure means an **isomorphism of group schemes** \[ \alpha:(\mathbf Z/M\mathbf Z)^2_S\xrightarrow{\sim}E[M], \qquad P=\alpha(1,0),\quad Q=\alpha(0,1). \tag{F1} \] In particular \([M]P=[M]Q=O\) are identities of sections. Exact orders on geometric fibres alone would not be a permissible replacement. The relative scheme constructions in §1.0Q supply the short-coordinate and torsion inputs. This full-level construction remains in characteristic zero; integral models, cusp compactifications, supersingular correspondences and specialization at good primes still require their separate proofs. #### 1.0F.1. The precise starting constructions Write \[ B=\mathbf Q[a,b,\Delta^{-1}],\qquad \Delta=4a^3+27b^2, \qquad \mathscr E_B:\ y^2=x^3+ax+b, \quad O=[0:1:0]. \tag{F2} \] The projective cubic, its relative group law, multiplication maps, and their scheme identities have been constructed in §§1.0Q.2.1–1.0Q.2.3. Those sections prove the following inputs, over the base ring rather than by inferring a nilpotent-base equality from geometric fibres. * Every elliptic scheme over a Noetherian \(\mathbf Q\)-scheme has short coordinates (F2) Zariski locally on its base. * Every pointed isomorphism between two such short models is \((x,y)\mapsto(u^2x,u^3y)\), with \(u\) a base unit; the target coefficients are \((u^4a,u^6b)\). This includes nilpotent units. * Multiplication \([n]\) is finite étale of degree \(n^2\), and \(\mathscr E_B[n]\) is its actual scheme-theoretic kernel. Field torsion is \((\mathbf Z/n)^2\). * Section 1.0Q.6 constructs a base-change compatible, perfect alternating relative Weil pairing. Its coordinate-invariance identity is proved first over the reduced universal parameter ring and then pulled back to arbitrary bases. The actual section identities in (F1), rather than fibrewise order alone, are used throughout; §1.0Q.2.3A proves why this matters. The coherent, curve, finiteness, pairing and étale foundations behind these inputs remain the explicit prerequisites in §1.0Q.1 and §1.0F.10. #### 1.0F.2. A finite affine parameter scheme for actual torsion frames Let \(T_M=\mathscr E_B[M]\times_B\mathscr E_B[M]\), with its universal sections \(P,Q\). For each nonzero \((i,j)\in(\mathbf Z/M)^2\), the equalizer \([i]P+[j]Q=O\) is open and closed in \(T_M\): it is the inverse image of the open-and-closed zero section of the separated finite étale torsion scheme. Remove the union of these finitely many equalizers and call the resulting open-and-closed subscheme \(F_M\). Thus \[ F_M=\operatorname{Spec}R_M\longrightarrow\operatorname{Spec}B \quad\text{is finite étale and surjective}. \tag{F3} \] Its geometric fibres consist of the ordered bases of \((\mathbf Z/M)^2\), so its degree is \(|\operatorname{GL}_2(\mathbf Z/M)|\). On \(F_M\), the map \((i,j)\mapsto[i]P+[j]Q\) from the constant group to the torsion scheme is an isomorphism: split both finite étale schemes on an étale cover and check the bijection of finite constant sets. This proves the scheme identity (F1), including after every base change. This is an explicit affine construction from (F2): take the kernel of the already constructed multiplication map, its square, and the complements of the displayed finite list of equalizers. It does not use a modular representability theorem. Both sections avoid \(O\) on every geometric fibre, hence factor through the affine cubic. Moreover \(F_M\) is closed in the product of two affine cubics, since it is open and closed in the closed torsion square. Consequently \(R_M\) is a quotient of \(B[x_P,y_P,x_Q,y_Q]\); its defining closed component is the one specified by the multiplication kernels and equalizers above. Two units will remove the coordinate ambiguity: \[ d=x_P-x_Q\in R_M^\times,\qquad c_0=y_P\in R_M^\times. \tag{F4} \] For the first, in any algebraically closed characteristic-zero fibre equal \(x\)-coordinates give \(Q=P\) or \(Q=-P\), since their \(y\)-coordinates have equal squares. Either relation contradicts a torsion basis. For the second, \(y_P=0\) would give \(P=-P\), hence \(2P=O\), whereas \(P\) has order \(M\geq3\). These prove **invertibility**, not an equality inferred from fibres: a nonunit in any local ring belongs to its maximal ideal and therefore vanishes in the corresponding residue field. This argument permits every nilpotent base parameter. The coordinate scaling defines an actual \(\mathbf G_m\)-action \[ R_u(a,b,x_P,y_P,x_Q,y_Q) =(u^4a,u^6b,u^2x_P,u^3y_P,u^2x_Q,u^3y_Q). \tag{F5} \] It preserves the kernel and frame conditions by its pointed group isomorphism. In particular \(d\) has weight two and \(c_0\) weight three. The stronger observation \[ w=c_0/d\in R_M^\times,\qquad w(R_u z)=u\,w(z) \tag{F6} \] provides a unit of weight one. It makes global normalization possible without choosing a square root. #### 1.0F.3. An affine slice and its explicit sign quotient Define \[ W_M=\operatorname{Spec}C_M,\qquad C_M=R_M/(w-1). \tag{F7} \] The following are inverse morphisms of schemes: \[ W_M\times\mathbf G_m\longrightarrow F_M,\quad(z,u)\mapsto R_u z, \qquad F_M\longrightarrow W_M\times\mathbf G_m,\quad z\mapsto\bigl(R_{w(z)^{-1}}z,w(z)\bigr). \tag{F8} \] All coefficients in the second map are regular because \(w\) is a unit; (F6) proves the two composites are the identities as morphisms. Hence this is an affine scheme quotient of the scaling ambiguity, with the actual invariant algebra \(R_M^{\mathbf G_m}=C_M\): in \(R_M=C_M[u,u^{-1}]\), invariance under \(u\mapsto tu\) forces every nonzero Laurent weight coefficient to vanish. This coefficient calculation works over nonreduced rings too. Here is also another explicit normalized sign quotient. Put \[ Z_M=\operatorname{Spec}A_M,\quad A_M=R_M/(d-1), \quad \tau=R_{-1}|_{Z_M},\quad D_M=A_M^{\langle\tau\rangle}. \tag{F9} \] The group \(\mu_2\) over \(\mathbf Q\) is the constant group \(\{1,-1\}\). On \(Z_M\), put \(c=y_P\); it is a unit and \(\tau(c)=-c\). The fixed locus of the nonidentity element is empty as a scheme: its defining ideal contains \(2c\), a unit. Thus the action is schematically free, even on arbitrary test rings with nilpotents. There is no assumed finite-quotient existence assertion here. For every \(h\in A_M\), write \[ h=h_++c(h_-/c),\qquad h_\pm=(h\pm\tau h)/2. \] Both \(h_+\) and \(h_-/c\) are invariant. Uniqueness follows by applying \(\tau\) and subtracting; \(2c\) is a unit. Therefore, with \(v=c^2\in D_M^\times\), \[ A_M=D_M\oplus D_Mc \simeq D_M[T]/(T^2-v). \tag{F10} \] This finite free rank-two algebra is étale, since \(2T\) is a unit. After tensoring with \(A_M\), its polynomial factors as \((T-c)(T+c)\) with comaximal factors. The Chinese remainder theorem consequently gives \[ A_M\otimes_{D_M}A_M\xrightarrow{\sim}A_M\times A_M, \quad a\otimes b\mapsto(ab,a\tau(b)). \tag{F11} \] Thus \(Z_M\to\operatorname{Spec}D_M\) is faithfully flat finite étale, and (F11) is precisely its constant-group torsor relation. Every base change retains (F10)–(F11); the invariant projection \((1+\tau)/2\) also proves invariants commute with arbitrary scalar extension. For an explicit finite set of generators, write \(p=x_P\) and \(r=y_Q/c\) on \(Z_M\). Then \[ x_Q=p-1,\quad v=p^3+ap+b\in D_M^\times, \quad D_M=\mathbf Q[a,b,p,r,\Delta^{-1},v^{-1}]\ \text{as its image in }A_M. \tag{F12} \] Indeed \(A_M\) is generated by these invariants and \(c\), with \(c^2=v\) and \(y_Q=cr\); (F10) then identifies its invariant part with the displayed image. The inherited torsion and basis equations are imposed by the ring \(A_M\) in (F9), so (F12) does not drop them. The quotient also has the scheme-valued universal property. An invariant map from \(Z_M\) to an affine target descends by the invariant ring. For a general scheme target, the preimage of an affine open is invariant. Its complement has closed image under the finite quotient map; the torsor relation identifies its fibres with the two-element orbits, so its saturated image is an open of \(\operatorname{Spec}D_M\). On affine subopens of that image descend the ring maps, then glue by faithful flat uniqueness. The same calculation says that every quotient test point has a lift on the finite étale cover and that two lifts differ by a unique element of \(\{1,-1\}\). Hence the affine scheme represents the fppf orbit sheaf, not merely the set of geometric orbits. Finally (F8) identifies the two constructions directly. On \(W_M\) set \(d_W=x_P-x_Q=y_P\), a unit. Under (F8), the equation \(d=1\) becomes \(u^2d_W=1\), and \(\tau\) sends \(u\) to \(-u\). Thus \[ A_M=C_M[u]/(u^2-d_W^{-1}),\quad c=u, \qquad D_M\simeq C_M. \tag{F13} \] Equivalently, normalizing \(d\) by an étale-local square root leaves exactly the free sign ambiguity just quotiented. The weight-one unit removes that ambiguity globally. We henceforth call this affine scheme \(Y_M=W_M=\operatorname{Spec}D_M\). #### 1.0F.4. The universal elliptic family, zero, group and entire frame Restrict (F2) and its universal frame to \(W_M\). This already gives a projective smooth elliptic scheme \(\mathscr E_M/Y_M\), its zero, group operations and the actual isomorphism \(\alpha_M:(\mathbf Z/M)^2_{Y_M}\simeq\mathscr E_M[M]\). Here is an explicit description over the constant-quotient ring that verifies effectivity of the proposed sign descent as well. Using (F12), define the projective cubic \[ \mathscr E_D:\quad vY^2Z=X^3+aXZ^2+bZ^3, \quad O=[0:1:0],\quad P_D=(p,1),\quad Q_D=(p-1,r). \tag{F14} \] It is an elliptic scheme: the change \(x'=vX,y'=v^2Y\) gives the short equation with coefficients \(av^2,bv^3\), whose discriminant factor is \(v^6\Delta\), a unit. Its short universal group law therefore constructs all the operations in (F14). After the faithfully flat extension (F10), \[ [X:Y:Z]\longmapsto[X:cY:Z] \tag{F15} \] identifies it with the short family over \(Z_M\), carrying the zero and both marked points to the original ones. This is a group isomorphism, by the pointed-short-coordinate calculation of §1.0Q.2.2. The lifted involution on that family is \((z,x,y)\mapsto(\tau z,x,-y)\); (F15) is invariant because both \(c\) and \(y\) change sign. Thus it descends the family, zero and group maps with their entire scheme structure. The points in (F14) satisfy the actual identities \([M]P_D=[M]Q_D=O\), since these identities hold after the faithfully flat extension (F15). The frame map from the constant group is an isomorphism for the same reason; isomorphism of finite affine schemes is detected after faithful scalar extension. This proves descent of the full frame, including nilpotent bases. The direct short model on \(W_M\) is obtained from (F14) by \[ x_*=X/v,\qquad y_*=Y/v, \quad a_*=a/v^2,\quad b_*=b/v^3, \quad P_*=(p/v,1/v),\quad Q_*=((p-1)/v,r/v). \tag{F16} \] On \(Z_M\) this is exactly normalization by \(u=c^{-1}\), and (F13) identifies its base with \(W_M\). This explicitly identifies the two universal families; an abstract effectivity theorem for descent of arbitrary proper schemes is unnecessary. The parameter \(F_M\) is smooth of dimension two over \(\mathbf Q\), being finite étale over the smooth affine plane open \(B\). The scaling derivation satisfies \(\delta(d)=2d\). The differential of \(d\) in the cotangent bundle of \(F_M\), restricted to \(Z_M\), is consequently unimodular; the Jacobian criterion in smooth coordinates makes \(Z_M\) a smooth curve. The finite étale cover (F10) descends regularity of local rings, and over the perfect field \(\mathbf Q\) this makes \(Y_M\) a smooth affine curve. Its finite presentation follows also directly from (F7). No normality or reducedness is imposed on its test bases. #### 1.0F.5. The representing property over arbitrary Noetherian bases Let \(\mathcal L_M(S)\) be isomorphism classes of pairs \((E/S,\alpha)\) as in (F1), with pointed group isomorphisms respecting \(\alpha\). We prove the natural bijection \[ \operatorname{Hom}_{\mathbf Q}(S,Y_M)\xrightarrow{\sim} \mathcal L_M(S),\qquad f\mapsto f^*(\mathscr E_M,\alpha_M). \tag{F17} \] Cover \(S\) by opens with the short coordinates proved in §1.0Q.2.2. The supplied frame gives a map from each open to \(F_M\). On that open \(w=y(P)/d\) is a unit by (F4), so rescale the coordinates by \(w^{-1}\). This gives an actual map to \(W_M\) and an isomorphism from the supplied family to the pullback of its universal family. On an overlap, the two original short coordinate systems differ by (F5), say by a base unit \(t\). Their values of \(w\) differ by \(t\), by (F6). For example \[ \frac{t^4a}{(tw)^4}=\frac a{w^4},\qquad \frac{t^2x_P}{(tw)^2}=\frac{x_P}{w^2},\qquad \frac{t^3y_P}{(tw)^3}=\frac{y_P}{w^3}. \tag{F18} \] The same identity holds for both curve coordinate functions and every other coefficient. Thus the maps to the closed affine slice and the family isomorphisms agree as scheme maps on every overlap. They glue to \(f\) and an isomorphism \((E,\alpha)\simeq f^*(\mathscr E_M,\alpha_M)\). No equality in this gluing argument is inferred just from fibres. Any pointed marked isomorphism between normalized short models is (F5). Both normalized values of \(w\) are one, so (F6) forces \(t=1\) in the base ring itself. Their coefficients and all marked coordinates consequently agree. In particular every automorphism of a full-level marked family is the identity, and the global marked identification just constructed is unique. Conversely a pullback of the universal short family already has \(w=1\), so the normalization procedure returns the original morphism \(f\). These prove both inverses in (F17), not just a geometric classification. All kernels, equalizers, unit identities, ring quotients, polynomial changes, and universal scheme maps above commute with base change. Pulling (F18) back gives the same normalized maps and family isomorphisms. Thus (F17) is natural for every morphism between Noetherian \(\mathbf Q\)-bases, without flatness, reduction, normality or connectedness assumptions. The explicit universal family and its identities also pull back along arbitrary \(\mathbf Q\)-scheme morphisms. #### 1.0F.6. Every matrix action and its action on the universal family Use a row frame and a right action. For \(g=\left(\begin{smallmatrix}A&B\\C&D\end{smallmatrix}\right) \in\operatorname{GL}_2(\mathbf Z/M)\), set \[ (P_g,Q_g)=(AP+CQ,BP+DQ),\qquad \alpha_g=\alpha\circ g. \tag{F19} \] The raw transformation \(r_g\) of \(F_M\) is a morphism: it uses the already constructed group operations, and both resulting sections are nonzero torsion sections. It commutes with (F5), and \(r_h\circ r_g=r_{gh}\). The following formula provides the descended morphism on the affine slice without an unspecified quotient action. On \(W_M\), put \[ w_g=\frac{y(P_g)}{x(P_g)-x(Q_g)}\in\mathcal O(W_M)^\times. \] Both numerator and denominator are units by the proof of (F4) applied to the new basis. Define \[ T_g=R_{w_g^{-1}}\circ r_g:W_M\longrightarrow W_M. \tag{F20} \] Explicitly its coefficient and marking functions are \[ (a,b,P,Q)\longmapsto \left(\frac a{w_g^4},\frac b{w_g^6}, \left(\frac{x(P_g)}{w_g^2},\frac{y(P_g)}{w_g^3}\right), \left(\frac{x(Q_g)}{w_g^2},\frac{y(Q_g)}{w_g^3}\right)\right). \tag{F21} \] These are regular functions on the whole slice. If \(w_{gh}^{\rm raw}=w(r_{gh}z)\), then \(w_h(T_gz)=w_g(z)^{-1}w_{gh}^{\rm raw}(z)\). Substitution in (F20) gives \(T_h\circ T_g=T_{gh}\) as morphisms. In particular \(T_{g^{-1}}\) is its inverse. This constructs all matrices, for composite and even \(M\) as well as prime \(M\). The lift to the universal family, covering \(T_g\), is \[ L_g:(z,x,y)\longmapsto \left(T_gz,x/w_g(z)^2,y/w_g(z)^3\right). \tag{F22} \] It is a pointed group isomorphism to the target family; its projective coordinate formula extends it across \(O\). Its precise frame covariance is \[ L_g\circ\alpha_z=\alpha_{T_gz}\circ g^{-1}. \tag{F23} \] Indeed it sends \(\alpha_zg\) to the target normalized frame. The same unit calculation gives \(L_hL_g=L_{gh}\) over \(T_hT_g\). For the central matrix \(-I\), \(w_g=-1\). Hence \[ T_{-I}=\mathrm{id}_{Y_M},\qquad L_{-I}=[-1]_{\mathscr E_M}. \tag{F24} \] The central subgroup \(\{I,-I\}\) acts trivially on the base, while the universal family retains the central sign. This does not contradict fineness: \([-1]\) is an isomorphism from the object with frame \(-\alpha\) to the object with frame \(\alpha\), and is not an automorphism respecting the same frame for \(M\geq3\). Formula (F24) is needed before any coefficient descent. #### 1.0F.7. Weil components and determinant covariance Restrict the actual relative pairing of §1.0Q.6 to the universal frame. Its value \(\epsilon_M=e_M(P,Q)\) defines a morphism to \(\mu_M\). It is primitive on every geometric fibre: if its order were a proper divisor of \(M\), the alternating pairing matrix in this basis could not be perfect. The primitive part of the finite étale scheme \(\mu_M\) is open and closed, so the morphism factors through it as a scheme morphism: \[ h_M:Y_M\longrightarrow K_M=\operatorname{Spec} \mathbf Q[t]/\Phi_M(t). \tag{F25} \] Here \(\Phi_M\) is the cyclotomic polynomial. Monic division in \(t^M-1=\prod_{d\mid M}\Phi_d(t)\) constructs its integer coefficients; its roots are exactly the primitive roots, all distinct in characteristic zero. For clarity, the elementary irreducibility input can be supplied directly. Let a monic irreducible rational factor \(f\) contain a primitive root \(\zeta\), and write \(\Phi_M=fg\) with integer monic polynomials by Gauss's lemma. If \(p\nmid M\) and \(\zeta^p\) were a root of \(g\), then \(f\mid g(t^p)\). Reduction modulo \(p\) gives \(\bar f\mid\bar g(t)^p\), so \(\bar f\) and \(\bar g\) have a common factor. This would give a repeated factor in \(t^M-1\), contrary to its coprimality with \(Mt^{M-1}\). Thus \(\zeta^p\) is a root of \(f\). Repeat for primes not dividing \(M\); every integer prime to \(M\) is a product of such primes. The factor contains all primitive roots and equals \(\Phi_M\). Hence \(K_M=\operatorname{Spec}\mathbf Q(\zeta_M)\). Bilinearity and actual alternation of the relative pairing give the identity over the entire parameter scheme \[ e_M(P_g,Q_g)=e_M(P,Q)^{AD-BC}. \] Coordinate invariance under the pointed isomorphism (F22) then proves \[ h_M\circ T_g=[\det g]\circ h_M, \qquad [u](\zeta)=\zeta^u. \tag{F26} \] Over \(\mathbf Q(\zeta_M)\), or any larger field containing all roots, \(Y_M\) decomposes after scalar extension into the open-and-closed loci \(Y_{M,\zeta}\) defined by \(e_M(P,Q)=\zeta\), for primitive \(\zeta\). Each represents precisely that fixed-pairing functor, by (F17). Formula (F26) sends \(Y_{M,\zeta}\) to \(Y_{M,\zeta^{\det g}}\). The determinant-one subgroup preserves each. #### 1.0F.8. These loci are the geometric components This additional characteristic-zero assertion can be proved without importing the modular representability claim of an analytic reference. The earlier LG-MF lesson *Complex tori, elliptic curves and the moduli interpretation of modular curves*, Theorems 2.1–2.3 and 3.1, actually constructs the Weierstrass parametrization of \(\mathbf C/(\mathbf Z+\tau\mathbf Z)\) and proves that every complex elliptic cubic arises in this way. Its Theorem 1.1 and the existence calculation in Theorem 4.1 compute the effect of changing an oriented integral lattice basis. The elementary surjectivity \(\operatorname{SL}_2(\mathbf Z)\to \operatorname{SL}_2(\mathbf Z/M)\) is proved in the earlier congruence subgroups lesson, Lemma 1.1. Those results apply to this step. The convergent Weierstrass series, evaluated at \(1/M,\tau/M\), and then normalized by the unit (F6), give a holomorphic map \(\mathfrak H\to Y_M(\mathbf C)\). Normal convergence locally in \(\tau\) follows from a uniform lattice bound on compact subsets of \(\mathfrak H\); none of these torsion evaluations is at a lattice point. The pairing value of this standard frame is a fixed primitive root \(\eta\): the map to the finite discrete root set is continuous and \(\mathfrak H\) is connected. No unverified analytic sign convention for \(\eta\) is needed. Fix \(r\in(\mathbf Z/M)^\times\) and a matrix \(g_r\) of determinant \(r\). Apply its raw frame change and normalization; this gives a continuous map \(\mathfrak H\to Y_{M,\eta^r}(\mathbf C)\). It is surjective. Indeed uniformize any marked complex elliptic curve. Its frame is the standard lattice frame times a matrix \(g\), necessarily of determinant \(r\) by the pairing identity. Write \(g=h g_r\) with \(h\in\operatorname{SL}_2 (\mathbf Z/M)\). Lift \(h\) to an integral determinant-one change of the oriented lattice basis. The lattice calculation in the earlier Theorem 1.1 absorbs \(h\) into a new parameter \(\tau\), leaving exactly the chosen frame \(g_r\). The representing property (F17) identifies its normalized point with the given point of our constructed scheme. The continuous image of connected \(\mathfrak H\) is connected. A nontrivial algebraic open-and-closed decomposition of \(Y_{M,\eta^r,\mathbf C}\) would give such a decomposition of its analytic points; both pieces have complex points because they are nonempty schemes of finite type over an algebraically closed field. Therefore this algebraic curve is connected. It is smooth; distinct irreducible components of a smooth curve cannot meet, since its regular local rings are domains. Thus it is irreducible. A decomposition over \(\overline{\mathbf Q}\) would remain a decomposition after extending to \(\mathbf C\), proving geometric connectedness and irreducibility over \(\mathbf Q(\zeta_M)\). These prove that the primitive pairing loci are exactly the \(\varphi(M)\) geometric components, not merely unproved labels for unions of components. In particular (F25) has geometrically connected fibres and is surjective. #### 1.0F.9. Arithmetic transport of the components All parameter equations, normalization maps, group laws, matrix maps and the relative pairing were constructed over \(\mathbf Q\). A field automorphism \(\sigma\) therefore transports their actual scheme maps and universal families. If \(\sigma\zeta_M=\zeta_M^u\), then it sends the geometric component with value \(\zeta_M^r\) to the component with value \(\zeta_M^{ur}\). Thus the arithmetic component action is the cyclotomic action, while the matrix component action is its determinant (F26). A fixed component carries the determinant-one matrix action; general matrices give the stated isomorphisms between components. If a semilinear arithmetic action on the component with value \(\zeta_M\) is wanted, compose its transport by \(\sigma\) with \(T_{\operatorname{diag}(u^{-1},1)}\). This explicitly brings the value back to \(\zeta_M\); it does not pretend that arbitrary determinants preserve that component. The lifts (F22), with the covariance (F23) and central sign (F24), accompany every such transport. #### 1.0F.10. Reading, retained foundations, and the exact completion boundary The construction uses the following written earlier programme proofs, with their exact foundational interfaces still retained: * *Base change and the Grothendieck complex*, Lemma 3.1 and Theorem 3.2, supply the finite projective coherent complex used in the relative short coordinates and universal constants of §1.0Q. * *Abelian varieties*, Lemma 9.0a, Theorem 9.0, Lemma 9.11 and Example 9.12, give the written cubic group law, point-class argument, curve duality and degree-three embedding. The earlier Tate-module lesson, Theorem 0.F.1 and Section 0.F.4, supplies Noetherian finite-fibre finiteness and whole-base closed immersions; its Theorem 0.C.10 supplies proper coherent finiteness. * *Quotients and torsors*, its opening module and affine-morphism descent, gives the faithful descent interface. The special constant quotient here is proved explicitly in (F10)–(F13), rather than imported from its general quotient theorem. * *Étale morphisms and their local structure*, Lemma 1.2, Theorem 1.3 and Theorem 5.1, and *Smooth morphisms*, Theorems 2.1, 3.1 and 4.1, supply the written differential, local regularity and smooth coordinate criteria. * The current LG-GAL Tate-module lesson, its complete multiplication, torsion, dual and Weil-pairing Lemmas 1.1–1.4, and the relative construction in §1.0Q supply the precise elliptic and pairing inputs. The central convention is proved locally in §1.0Q.8; no later lesson is a proof provider for this construction. * The LG-MF uniformization and reduction proofs named in §1.0F.8, together with the valence lesson's complete Theorem 3.1, supply the additional component argument. Only the already written analytic constructions are used. The free original [Deligne–Rapoport paper, IV §5.1–5.4](https://publications.ias.edu/sites/default/files/Number22.pdf) (PDF pages 84–85) describes the analytic frame quotient and its determinant components; its recollection of uniformization and reduction is not substituted for the complete earlier proofs used in §1.0F.8. The official [Stacks finite quotient proof, Tag 07S5](https://stacks.math.columbia.edu/tag/07S5) gives a related comparison. Its general theorem does not replace our algebra. The Noetherian finiteness and family-embedding interface is now the written local proof in the earlier Tate-module lesson, Sections 0.C–0.F. Those proofs apply on nonreduced bases and keep the full coherent and infinitesimal data. Retained transitive foundations are the affine sheaf and Čech calculation, the coherent-complex base-change construction; finite flat/projective module algebra; projective Cohen–Macaulay duality and regular-local/DVR algebra; and the algebraic standard smooth and formally étale criteria; and, for §1.0F.8 alone, elementary complex analysis, the earlier valence/Eisenstein identities, and the elementary finite-type algebraically closed field point theorem. Gauss's lemma and elementary integer arithmetic enter the displayed cyclotomic proof. These specific inputs retain their existing programme scope; recursive closure of every supporting theorem is not claimed by this construction. The new result is (F17), an affine fine full-level moduli scheme with an actual universal elliptic family over arbitrary Noetherian \(\mathbf Q\)-bases, all \(\operatorname{GL}_2(\mathbf Z/M)\) maps and universal lifts, the central sign, and the cyclotomic geometric components with determinant and arithmetic transport. The construction works for every \(M\geq3\). It does not yet construct low-level coefficient invariants, descent from this fine cover to the particular higher-weight modular coefficient object, cusps, models over integer rings, specialization at any good prime, or the supersingular degree-\(p\) correspondence and its coefficient trace. Those remain separately required constructions with no level or prime deleted. The following figure records the exact normalized parameter schemes, universal family maps and component transport. The full argument above, including the unique marked isomorphism, accompanies the figure. The earlier Tate-module lesson now contains the needed Noetherian finite-fibre proof and specialized field quotient, constants and rigidity arguments. Their basic sheaf, curve, algebra and étale prerequisites remain explicit. This affine moduli construction does not certify their entire recursive closure. ![Exact fine full-level parameter, universal elliptic family and matrix lifts](../assets/fine-full-level-moduli-proof-map.png) [Open Figure 2 at full size](../assets/fine-full-level-moduli-proof-map.png) *Equations (F1)–(F26) and §§1.0F.8–1.0F.10 prove the six panels, including geometric connectivity, arithmetic transport and the precise foundational scope. The base map for −I is identity, while its universal family lift is [−1]; the frame and family action must both be retained in later coefficient descent.* At the cusp \(\infty\) of \(X_0(N)\), the Tate curve with parameter \(q\) and subgroup \(\mu_N\) gives the local expansion. At a general cusp of width \(w\), the analytic coordinate is \(q_c=e^{2\pi iz/w}\). The formal Tate-curve description supplies the corresponding algebraic coordinate, with roots of \(q\) and roots of unity as needed by the level structure. This is the geometric meaning of a \(q\)-expansion, rather than a new Galois representation at a cusp. Write \[ X=X_0(N),\qquad J=J_0(N)=\operatorname{Pic}^0(X),\qquad g=\dim S_2(\Gamma_0(N)). \tag{3} \] The cusp \(\infty\) is rational, so \(x\mapsto[x-\infty]\) defines the Abel–Jacobi map. The equality of the analytic genus with the dimension of holomorphic differentials is proved in *Dimension formulas for congruence subgroups*, Appendix A.7. The construction and dimension of the algebraic Jacobian are proved in *The Picard functor and the Picard scheme of a curve*, §§4–7, especially Theorems 6.2 and 7.2. Lemma 1.2 below identifies the analytic and algebraic genera. Thus both dimensions are \(g\). **Lemma 1.1 (weight two and differentials).** The map \(f\mapsto f(z)\,dz\) identifies \(S_2(\Gamma_0(N))\) with the holomorphic differentials on \(X(\mathbf C)\). *Proof.* The weight-two transformation law and \(d(\gamma z)=(cz+d)^{-2}dz\) make this differential invariant. At a cusp, \[ f(z)\,dz=\frac{w}{2\pi i}\,f(q_c)\frac{dq_c}{q_c}. \tag{4} \] It is holomorphic exactly when the constant term of the cusp expansion vanishes. There is no additional pole at an elliptic point. In a local coordinate \(u\) where its stabilizer acts by \(u\mapsto\zeta u\) of order \(e\), invariance of \(a(u)\,du\) forces \(a(u)=u^{e-1}b(u^e)\); putting \(t=u^e\) gives \(a(u)\,du=b(t)\,dt/e\). Thus it descends holomorphically. Conversely, pulling a holomorphic differential back to the upper half-plane gives a weight-two form, and (4) makes it cuspidal. These constructions are inverse. ∎ The decomposition needed in degree one is \[ H^1(X(\mathbf C),\mathbf C) \simeq S_2(\Gamma_0(N))\oplus\overline{S_2(\Gamma_0(N))}. \tag{5} \] The bar denotes the antiholomorphic summand. It supplies the second copy of the Hecke spectrum; it does not double the eventual Frobenius trace. **Lemma 1.2 (the curve realization and comparison).** For a smooth projective connected complex algebraic curve, analytification identifies algebraic and analytic line bundles and meromorphic functions, and identifies finite étale covers with finite analytic unramified covers. Consequently finite-coefficient degree-one étale cohomology equals degree-one cohomology of the analytic surface, naturally for finite correspondences. For the modular curve the decomposition (5) is Hecke equivariant. Passing to the coefficient limit gives Betti–étale comparison in degree one. *Proof.* Here are the particular algebraization arguments needed, so that a general comparison theorem is not an unproved premise. The analytic Riemann–Roch and meromorphic-section proofs for arbitrary compact curves and arbitrary line bundles are written in *Dimension formulas for congruence subgroups*, Appendix A.1–A.7. Algebraic curve Riemann–Roch and canonical degree are proved in *Abelian varieties*, Lemma 9.11. The two genera agree: choose a separating rational function on the algebraic curve. Local parameters give the ramification orders of its derivative, hence the algebraic canonical degree is \(-2d+\sum(e_x-1)\). The same local maps, analytically \(u\mapsto u^{e_x}\), give the same number by the polygon proof in the earlier analytic Appendix A.7. The identities \(\deg K=2g-2\) on both sides prove equality of genera. For an effective divisor \(D\), algebraic sections of \(\mathcal O(D)\) inject into analytic sections. If \(\deg D>2g-2\), the two Riemann–Roch formulas give the same dimension, so that injection is an isomorphism. Every meromorphic function has a finite pole divisor and therefore belongs to one such space after enlarging the divisor. Thus every analytic meromorphic function is rational. Every analytic line bundle has a nonzero meromorphic section by the earlier Appendix A.6, and is \(\mathcal O(D)\) for its finite divisor. All points of a complex algebraic curve are algebraic closed points. Its analytic line bundle consequently algebraizes. Meromorphic sections of the Hom bundle show that morphisms also algebraize: an analytic holomorphic morphism is a rational section with the prescribed nonnegative local orders, hence an algebraic morphism. This proves precisely the curve line-bundle comparison required here. Let \(Z\to X^{\mathrm{an}}\) be a finite unramified analytic covering of degree \(d\), first with connected \(Z\). It is a compact Riemann surface. Analytic Riemann–Roch, after adding a sufficiently large pole divisor away from a regular fibre, permits arbitrary prescribed values at that fibre: apply its negative-degree dual correction to the evaluation exact sequence. Choose a meromorphic function \(t\) with pairwise distinct values on the \(d\) points. Symmetric functions of its values on the sheets are meromorphic functions on \(X^{\mathrm{an}}\). Near a pole, bounded pole orders on the finitely many sheets bound their elementary symmetric functions, so they extend meromorphically there. They are rational by the preceding paragraph. The monic polynomial \[ \prod_{z\mapsto x}(T-t(z)) \] therefore has coefficients in \(\mathbf C(X)\). The sheets are distinguished generically. Analytic continuation on connected \(Z\) is transitive on them: lift paths between any two points of a fibre and project the paths. A factor of this polynomial over \(\mathbf C(X)\) would give a continuation-invariant proper collection of sheets. Hence it is irreducible and defines an extension of degree \(d\). The normalization needed here is finite by the following algebra argument. On a smooth affine chart let \(A\) be its normal Noetherian coordinate domain, \(K\) its fraction field, and \(L/K\) this finite extension. Characteristic zero makes it separable. Choose a \(K\)-basis \(e_i\) integral over \(A\), by clearing denominators in their monic equations. The trace pairing is nondegenerate: a primitive element has distinct conjugates, and the Vandermonde matrix of their powers is invertible. Such an element exists by avoiding the finitely many linear equalities between distinct embeddings in the infinite field \(K\). For an integral \(b\in L\), every \(\operatorname{Tr}(be_i)\) is integral and lies in \(K\), hence in \(A\). Consequently the integral closure is contained in the free finite trace-dual lattice of \(\sum A e_i\). Its submodule is finite by Noetherianity. Integral closure commutes with localization: multiplying a monic equation over a localization by a suitable power of its denominator makes a multiple of the element integral before localization. These affine normalizations therefore glue. Their closed local rings are DVRs by the actual proof in *Discrete valuation rings, normal rings and Serre's criterion*, Lemma 1.1 and Theorem 1.2. This supplies both finiteness and the local description, rather than an unlocated normalization premise; freely accessible comparison materials are Stacks Tags 032L and 00PD. Normalize the algebraic curve in \(L\) by this construction. The analytification of the resulting curve is \(Z\). Indeed, on a small disk every sheet is a disk over the same parameter; the integral closure of its convergent local ring in those local meromorphic sheets is the product of the disk rings, since each disk ring is a DVR. An element satisfying a monic equation with holomorphic coefficients has no pole, which also verifies this description directly. The generic identification given by \(t\) extends uniquely across its poles, by those discrete valuation rings. The algebraic map has ramification index one at every point, hence is étale. Conversely analytification of an étale map is a local biholomorphism by the local inverse theorem. An analytic map between two such covers algebraizes: its pullback on meromorphic functions is a map of the two function-field extensions, and the local valuation condition extends it on the normal curves. Disconnected covers are the disjoint union of this construction. Cover actions and equivariant maps are included by full faithfulness. This proves the finite-cover assertion. Degree-one cohomology with a finite constant abelian group classifies torsors with that group. Algebraic and analytic transition cocycles correspond under the just-proved cover equivalence; their group laws and connecting maps are preserved. Thus the degree-one comparison is canonical and additive. Kummer is exact in both topologies, and the line-bundle comparison identifies its degree-two class with the winding number of a transition function. In the algebraic theory the actual proof in *Poincaré duality for curves*, §2, identifies \(H^2(X,\mu_n)\) with \(\mathbf Z/n\) by divisor degree; on the surface its oriented fundamental class does the same. A positive local parameter has winding one, so these generators and their traces agree. Cup products are formed from the same transition maps and therefore correspond. Pullback is natural; the trace of a finite map is the sum over its sheets, extended at a ramification point with its local length. Thus traces and correspondences correspond too. For modular curves the period proof is actually written in *Group cohomology of \(\Gamma\) and the Eichler–Shimura isomorphism*, §§3–5, Theorem 5.3, including the weight-two dimension calculation. Its parabolic cocycles are exactly the period functionals on the compact surface: cusp loops are killed, and an elliptic loop has finite order and is killed with characteristic-zero constant coefficients. The handle presentation there identifies this space with \(H^1(X^{\mathrm{an}},\mathbf C)\). The integral double-coset cocycle and its period equivariance are proved in *Modular symbols and the algebraicity of Hecke eigenvalues*, §1, equations (1.3) and (1.8). This proves (5) with its Hecke action. Finally the compact surface's handle cell complex is a finite free integral complex. Reducing it modulo \(\ell^r\), taking the inverse limit, and tensoring with \(\mathbf Q_\ell\) gives its rational cohomology tensored with \(\mathbf Q_\ell\); its first integral cohomology is free. The finite comparison maps commute with reduction. Finiteness makes these towers Mittag–Leffler, so the inverse-limit comparison has no extra \(\varprojlim^1\) term. This is degree-one Betti–étale comparison, rather than an application of a finite-coefficient theorem directly to \(\mathbf Q_\ell\). ∎ **Lemma 1.3 (Tate modules, pairings and good specialization).** Put \(J=\operatorname{Pic}^0(X)\). There are natural identifications \[ H^1_{\mathrm{\acute et}}(X,\mathbf Q_\ell(1))=V_\ell J, \qquad H^1_{\mathrm{\acute et}}(X,\mathbf Q_\ell) \simeq(V_\ell J)^\vee. \] The rational Tate module has a nondegenerate alternating pairing into \(\mathbf Q_\ell(1)\). A symmetric curve correspondence is self-adjoint. If \(J\) extends to an abelian scheme at \(p\ne\ell\), specialization identifies its generic and special Tate modules, and arithmetic Frobenius is the special-fibre Frobenius endomorphism. *Proof.* On a projective geometrically connected curve, every global unit over the algebraic closure is constant, and constants have \(\ell^r\)-th roots. The Kummer exact sequence therefore identifies \(H^1(X,\mu_{\ell^r})\) with \(\operatorname{Pic}(X)[\ell^r]=J[\ell^r]\). The exact sequence and its maps are written in *Cohomology of curves*, §6, equations (6.1)–(6.4); its former stated Picard input is supplied by the actual Picard construction just bound above and the multiplication/torsion proofs in *Abelian varieties*, Theorems 6.2 and 7.2. Under reduction of coefficients the Kummer transition is multiplication by \(\ell\) on torsion. Taking the inverse limit gives the first identification. Curve cup duality is proved for every invertible finite coefficient ring in *Poincaré duality for curves*, §§8–11, particularly Theorem 10.1, Corollary 10.2 and Theorem 11.1. Its §12 constructs the perfect cup pairing, without needing its separately stated comparison with a particular Weil-pairing convention. On the rational coefficient limit, graded commutativity gives alternation, since two is invertible even when \(\ell=2\). Twisting its trace once gives a perfect alternating form on \(V_\ell J\) into \(\mathbf Q_\ell(1)\). That form identifies \(V_\ell J(-1)\) with its dual and proves the second displayed identification. The projection formula for finite trace proves that pushforward is adjoint to pullback; the explicit trace and its compatibility with cup are in the same earlier lesson, §§2 and 7. Thus transposing a correspondence takes its adjoint. For good specialization one can work directly on torsion. Multiplication by \(\ell^r\) on a smooth group has invertible differential, so its kernel on an abelian scheme is finite étale. Over a strictly henselian trait every finite étale scheme is a disjoint union of sections: lift each special point by the henselian root-lifting property, use the étale diagonal for uniqueness, and observe that their union has all the fibres and is the whole finite scheme. This argument is written in *Néron models*, Lemma 7.1. It identifies the torsion points compatibly for every \(r\), and inertia acts trivially. On a point of the special fibre, the \(p\)-power Frobenius endomorphism raises its coordinates to the \(p\)-th powers, exactly the action of arithmetic residue Frobenius. Taking the limit proves the claim. The cyclotomic character is unramified at \(p\ne\ell\) and has arithmetic Frobenius value \(p\), by Lesson 01, Lemma 0D.1, Proposition 0F.2 and Proposition 3.1. Galois equivariance of cup consequently gives multiplier \(p\) on the Tate-module pairing. ∎ **Lemma 1.4 (algebraically closed field extension in the degrees used here).** Let \(K\subset K'\) be algebraically closed fields, let \(X/K\) be a smooth projective connected curve, and let \(n\) be invertible in \(K\). Pullback gives an isomorphism \[ H^1_{\mathrm{\acute et}}(X,\mu_n) \xrightarrow{\ \sim\ } H^1_{\mathrm{\acute et}}(X_{K'},\mu_n). \] It preserves the degree-normalized trace in degree two, cup products, and finite curve correspondences. After compatible coefficient choices and passage to the limit it gives the degree-one \(\mathbf Q_\ell\) comparison needed to extend a geometric curve from \(\overline{\mathbf Q}\) to \(\mathbf C\). *Proof.* Choose \(x\in X(K)\). The actual Picard construction in *The Picard functor and the Picard scheme of a curve*, Theorem 2.1, §§4–6 and Theorem 7.2, represents degree-zero line bundles normalized along \(x\), on every test scheme. Put \(J=\operatorname{Pic}^0_{X/K}\). Its base change represents that same functor for \(X_{K'}\): for any \(K'\)-scheme \(T\), a map \(T\to J_{K'}\) is precisely a \(K\)-map \(T\to J\), and \[ X_{K'}\times_{K'}T=X\times_KT. \] The bundle, its normalization along the section and its fibre degree in these two descriptions are identical. The functorial identifications commute with pullback, and therefore identify the representing schemes and their group laws. This proves the needed Jacobian base change directly from the written all-test-scheme Picard construction. Global units on either projective connected curve are constants and have \(n\)-th roots in its algebraically closed field. The Kummer proof of Lemma 1.3 consequently gives the natural diagram \[ \begin{array}{ccc} H^1_{\mathrm{\acute et}}(X,\mu_n)&\simeq&J[n](K)\\ \downarrow&&\downarrow\\ H^1_{\mathrm{\acute et}}(X_{K'},\mu_n)&\simeq&J_{K'}[n](K'). \end{array} \] The scheme \(J[n]\) is finite étale by the multiplication proof used in Lemma 1.3. A finite étale algebra over an algebraically closed field is a product of copies of that field: its finite separable residue extensions are the field itself, and étaleness makes the local factors reduced. Tensoring this product with \(K'\) changes none of its component labels or group-law maps. Thus the right vertical arrow is a bijection and a group isomorphism, which proves the first assertion. In degree two, *Poincaré duality for curves*, §2, identifies \(H^2(X,\mu_n)\) with \(\mathbf Z/n\) by divisor degree, and identifies the class of the point \(x\) with one. Its pullback is the point \(x_{K'}\), again of degree one. Naturality of the Kummer divisor class therefore makes pullback the identity under the two trace identifications. Pullback of cochains preserves cup products, so it also preserves their perfect degree-one pairing, whose proof was bound in Lemma 1.3. For a finite generically étale map of smooth projective curves, pullback commutes with the field extension. Pushforward is its adjoint for this perfect cup pairing, by the finite trace projection formula proved in *Poincaré duality for curves*, §§2 and 7, as used in Lemma 1.3. An adjoint is unique: equality of its pairing against every vector determines the vector by perfection. The preserved pairings therefore make this pushforward commute with field extension. Here is the additional argument for inseparable maps. A finite curve map in characteristic \(p\) factors into an iterated relative Frobenius \(h\), of degree \(d=p^r\), followed by a generically étale map. To check the factorization, a function field \(L\) of a smooth curve over the perfect field \(K\) has a separating parameter \(t\). Comparing the degrees over \(K(t^p)\) before and after the Frobenius field isomorphism gives \([L:L^p]=p\). Hence \(1,t,\ldots,t^{p-1}\) is a basis over \(L^p\), and differentiation shows \(\ker(d:L\to\Omega_{L/K})=L^p\). If the differential of a curve map vanishes, its pulled-back function field lies in \(L^p\), so its rational map factors through relative Frobenius. The rational factor extends at every point by the valuation rings of the smooth projective curves, as in Lemma 1.2. Each such factor removes a factor \(p\) from the finite degree, so this process terminates. The remaining map has nonzero differential; the differential exact sequence makes its relative differentials zero, and the minimal-polynomial derivative criterion makes its function-field extension separable. The finite radicial map \(h\) is an integral universal homeomorphism. Its pullback induces an equivalence of étale topoi and therefore an isomorphism on these cohomology groups, by the actual strict-local proof in *Pushforward, pullback and finite morphisms*, Lemma 6.1 and Theorem 6.2. Define its degree-normalized pushforward by \[ h_*=d\,(h^*)^{-1}. \] This is the usual divisor normalization: the pullback of any point has length \(d\), so \(h^*\) multiplies the degree-two divisor class by \(d\), whereas the point's pushforward has degree one. The length assertion follows because a finite map of smooth curves is flat (a finite torsion-free module over a DVR is free), and a radicial fibre has one geometric point and total length its rank \(d\). Cup naturality then gives \[ \operatorname{tr}_X(h^*a\cup b) =d\,\operatorname{tr}_Y(a\cup(h^*)^{-1}b) =\operatorname{tr}_Y(a\cup h_*b). \] Thus this pushforward is also the required adjoint. Its formula commutes with algebraically closed field extension: pullback does and the finite flat degree \(d\) is preserved. Compose the radicial and generically étale pushforwards to obtain the adjoint, trace normalization and base-change compatibility for every finite curve map. Apply this to both projections of a finite correspondence; disconnected parameter curves are treated component by component. The correspondence action is consequently preserved. Choose compatible primitive \(\ell^r\)-th roots in \(K\), also viewed in \(K'\), to identify constant coefficients with the corresponding twists. The finite comparison diagrams commute with coefficient reduction because Kummer does. Their inverse limits are isomorphic term by term; tensoring with \(\mathbf Q_\ell\) gives the rational degree-one identification. For \(K=\overline{\mathbf Q}\subset\mathbf C=K'\), combine this isomorphism with Lemma 1.2. This supplies the arithmetic application of the complex curve comparison, including the Hecke action and pairing used below. ∎ ## 2. The Hecke correspondence over \(\mathbf Q\) Fix a prime \(p\nmid N\). The correspondence has a parameter curve whose noncuspidal points are triples \((E,C,D)\), where \(D\subset E[p]\) is cyclic of order \(p\). Its two maps are \[ \pi_1(E,C,D)=(E,C),\qquad \pi_2(E,C,D)=\bigl(E/D,(C+D)/D\bigr). \tag{6} \] They and their compactifications are defined over \(\mathbf Q\). Since \(p\nmid N\), the quotient map is injective on \(C\). In characteristic zero, \(E[p]\simeq(\mathbf Z/p)^2\), whose lines number \[ \frac{p^2-1}{p-1}=p+1. \tag{7} \] Thus both projections have generic degree \(p+1\). On divisor classes and the Jacobian, our convention is \[ T_p=(\pi_2)_*(\pi_1)^*. \tag{8} \] Pullback includes the lengths of the fibers; pushforward then sends each divisor to its image. Replacing an isogeny by its dual exchanges the two projections: the composite is multiplication by \(p\), and multiplication by \(p\) preserves the cyclic subgroup \(C\). Consequently this correspondence on \(X_0(N)\) equals its transpose. The induced Hecke action on differentials is the classical action, with \[ (T_pf)(z)=p\,f(pz)+\frac1p\sum_{j=0}^{p-1} f\!\left(\frac{z+j}{p}\right). \tag{9} \] For \(f=\sum_{n\ge1}b_nq^n\), the finite sum kills terms whose index is not divisible by \(p\), so the coefficient of \(q^n\) is \[ b_{pn}+p\,b_{n/p}, \qquad b_{n/p}=0\ \text{if }p\nmid n. \tag{10} \] In particular, a normalized eigenform \(f=q+\cdots\) has \(T_p\)-eigenvalue \(b_p\). **Lemma 2.1 (normalization and the actual Hecke prerequisites).** The differential action of (8) is (9). The good-prime operators commute, are self-adjoint for the Petersson inner product, and have algebraic integer eigenvalues. They admit a common eigenbasis. *Proof.* Present a complex elliptic curve by \(\mathbf C/(\mathbf Z+\mathbf Z z)\). Its order-\(p\) kernels are generated by \(1/p\), and by \((z+j)/p\), \(0\le j