Points, orders and residues

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

A function on the curve \(v^2=f(w)\) has zeros and poles at points of the curve, and a differential has poles and residues. This lesson makes these notions precise for a polynomial \(f\) of degree three or four without repeated root, over an algebraically closed field of odd characteristic, by expanding functions in a parameter at each point. Three results are proved.

• A function without poles is constant (Theorem 4.2). So a function is determined, up to a constant factor, by the orders of its zeros and poles.
• The differential \(\omega=dw/v\) has neither zeros nor poles, and a differential without poles is a constant multiple of \(\omega\) (Proposition 5.2, Corollary 5.3).
• A logarithmic differential \(dF/F\) has at most simple poles, and its residue at a point is the order of \(F\) there (Proposition 5.4).

With the Cartier operator of The Cartier operator these are the tools for comparing the differential \(dw/((1-\lambda w)v)\), which carries the coefficient sums, with logarithmic differentials.

The lesson treats one family of curves directly. It uses formal power series and Laurent series over a field (core course Abstract Algebra II) and Sections 1, 2 and 4 of the lesson on the Cartier operator.

1. The points

Let \(k\) be an algebraically closed field of odd characteristic \(p\), and \[ \begin{aligned} f={}&f_0+f_1w+f_2w^2\\ &+f_3w^3+f_4w^4 \end{aligned} \] a polynomial in \(k[w]\) of degree three or four without repeated root. Let \(K\) be the field of functions \(A+Bv\), \(A,B\in k(w)\), \(v^2=f\), with its derivative \(D\), as in the lesson on the Cartier operator.

The points of the curve are the following.

• The affine points: the pairs \((a,b)\in k^2\) with \(b^2=f(a)\). Such a point is a branch point if \(b=0\), that is, if \(a\) is a root of \(f\), and an ordinary point otherwise. Over every \(a\in k\) there are two ordinary points, or one branch point.
• The points at infinity. In the coordinates \(u=1/w\), \(t=v/w^2\) the equation is \(t^2=f^*(u)\) with \[ \begin{aligned} f^*(u)={}&f_4+f_3u+f_2u^2\\ &+f_1u^3+f_0u^4 . \end{aligned} \] The points at infinity are the points \((0,t)\) of this curve. If \(f\) has degree four, there are two of them, \(\infty_+\) and \(\infty_-\), with \(t=\pm c\), where \(c^2=f_4\); they are ordinary points of \(t^2=f^*(u)\). If \(f\) has degree three, \(f_4=0\) and \(f_3\ne0\); there is one point at infinity, \(\infty\), with \(t=0\), a branch point of \(t^2=f^*(u)\).

2. Expansions

Let \(k((z))\) be the field of Laurent series in a variable \(z\). The order of a nonzero series is its lowest exponent; a series of order \(0\) is a unit of \(k[[z]]\).

Lemma 2.1.

1. Let \(e\in k[[z]]\) have constant term \(b^2\) with \(b\ne0\). There is exactly one \(s\in k[[z]]\) with \(s^2=e\) and constant term \(b\).
2. Let \(g\in k[[x]]\) have constant term \(g_0\ne0\). There is exactly one \(\varphi\in k[[y]]\) with constant term \(0\) such that \[ \varphi(y)\,g(\varphi(y))=y . \] It has the form \(\varphi=y/g_0+\cdots\).

Proof. (1) Write \(s=b+s_1z+\cdots\). The coefficient of \(z^n\) in \(s^2\) is \(2bs_n\) plus a polynomial in \(s_1,\dots,s_{n-1}\); since \(2b\ne0\), the \(s_n\) are determined one after another. (2) Write \(\varphi=\varphi_1y+\varphi_2y^2+\cdots\). The coefficient of \(y\) in \(\varphi\,g(\varphi)\) is \(g_0\varphi_1\), and for \(n\ge2\) the coefficient of \(y^n\) is \(g_0\varphi_n\) plus a polynomial in \(\varphi_1,\dots,\varphi_{n-1}\). So \(\varphi_1=1/g_0\), and the \(\varphi_n\) are determined one after another. \(\blacksquare\)

To every point \(P\) we attach two series \(w_P,v_P\in k((z))\) with \(v_P^2=f(w_P)\).

• Ordinary affine point \(P=(a,b)\). \(w_P=a+z\), and \(v_P\) is the square root of \(f(a+z)\) with constant term \(b\) (Lemma 2.1(1)).
• Affine branch point \(P=(a,0)\). Then \(f(a+x)=x\,g(x)\) with a polynomial \(g\), \(g(0)=f'(a)\ne0\). Let \(\varphi\) be the series of Lemma 2.1(2), so that \(f(a+\varphi(y))=y\). Put \(w_P=a+\varphi(z^2)\) and \(v_P=z\).
• \(\infty_\pm\), for \(f\) of degree four. \(w_P=1/z\) and \(v_P=z^{-2}s\), where \(s\) is the square root of \(f^*(z)\) with constant term \(\pm c\).
• \(\infty\), for \(f\) of degree three. Then \(f^*(u)=u\,g^*(u)\) with \(g^*(0)=f_3\ne0\). Let \(\varphi\) be the series with \(f^*(\varphi(y))=y\). Put \(w_P=1/\varphi(z^2)\) and \(v_P=z/\varphi(z^2)^2\).

In each case \(v_P^2=f(w_P)\); at infinity this follows from \(f(1/u)=f^*(u)/u^4\).

Lemma 2.2. For every point \(P\) there is exactly one homomorphism of fields \[ \iota_P\colon K\to k((z)) \] that is the identity on \(k\) and maps \(w\) to \(w_P\) and \(v\) to \(v_P\). It is injective.

Proof. For \(a'\in k\) the series \(w_P-a'\) is not zero. Every nonzero polynomial in \(w\) is a constant times a product of factors \(w-a'\), because \(k\) is algebraically closed; so substituting \(w_P\) for \(w\) maps nonzero polynomials to nonzero series, and extends to a homomorphism \(k(w)\to k((z))\). Since \(v_P\) is a root of \(X^2-f(w_P)\), it extends to \(K=k(w)[X]/(X^2-f)\). A homomorphism of fields is injective. It is unique because \(K\) is generated by \(k\), \(w\) and \(v\). \(\blacksquare\)

The series \(\iota_P(g)\) is the expansion of \(g\) at \(P\). For \(g\ne0\) its order is the order \(\operatorname{ord}_P(g)\) of \(g\) at \(P\). So \[ \begin{gathered} \operatorname{ord}_P(gg')=\operatorname{ord}_P(g)+\operatorname{ord}_P(g'),\\ \operatorname{ord}_P(g+g')\ge\min(\operatorname{ord}_P(g),\operatorname{ord}_P(g')). \end{gathered} \] The function \(g\) has a zero at \(P\) if \(\operatorname{ord}_P(g)>0\) and a pole if \(\operatorname{ord}_P(g)<0\). If \(\operatorname{ord}_P(g)\ge0\), \(g\) is regular at \(P\), and the constant term of its expansion is its value \(g(P)\). At an affine point \((a,b)\), \(w(P)=a\) and \(v(P)=b\). When \(f(0)=1\), the expansion at \(O=(0,1)\) is the one of Section 5 of the lesson on the Cartier operator, with \(z\) written for \(w\).

The definitions give the following list of orders.

• At an ordinary affine point over \(a\): \(\operatorname{ord}_P(w-a)=1\) and \(\operatorname{ord}_P(v)=0\).
• At an affine branch point over \(a\): \(\operatorname{ord}_P(w-a)=2\) and \(\operatorname{ord}_P(v)=1\).
• At \(\infty_\pm\), for degree four: \(\operatorname{ord}_P(w)=-1\) and \(\operatorname{ord}_P(v)=-2\).
• At \(\infty\), for degree three: \(\operatorname{ord}_P(w)=-2\) and \(\operatorname{ord}_P(v)=-3\).

For the branch points this uses \(\varphi(z^2)=z^2/g_0+\cdots\).

3. Functions of w

Let \(e_P=2\) if \(P\) is an affine branch point or the point \(\infty\) of a cubic, and \(e_P=1\) for the other points. For a nonzero \(r\in k(w)\) and \(a\in k\) let \(n_a(r)\) be the exponent of \(w-a\) in the factorization of \(r\), and \(\deg r\) the degree of the numerator minus that of the denominator.

Lemma 3.1. Let \(r\in k(w)\), \(r\ne0\).

1. If \(P\) is an affine point over \(a\), then \(\operatorname{ord}_P(r)=e_P\,n_a(r)\).
2. If \(P\) is a point at infinity, then \(\operatorname{ord}_P(r)=-e_P\deg r\).
3. If \(r\) is regular at all affine points, it is a polynomial. If \(r\) is regular at all points, it is constant.

Proof. Write \(r=c\prod_i(w-a_i)^{n_i}\) with distinct \(a_i\). At an affine point over \(a\), \(w_P-a'\) has the nonzero constant term \(a-a'\) for \(a'\ne a\), so \(\operatorname{ord}_P(w-a')=0\), and \(\operatorname{ord}_P(w-a)=e_P\) by the list of orders. At infinity, \(\operatorname{ord}_P(w-a')=\operatorname{ord}_P(w)=-e_P\) for every \(a'\). This gives (1) and (2). For (3): over every \(a\in k\) there is a point, because \(f(a)\) has a square root in \(k\). So regularity at all affine points gives \(n_a(r)\ge0\) for all \(a\), and \(r\) is a polynomial; regularity at infinity gives \(\deg r\le0\). \(\blacksquare\)

4. Functions without poles

Let \(\tau\) be the automorphism of \(K\) with \(\tau(A+Bv)=A-Bv\). For a point \(P\) let \(\bar P\) be the point \((a,-b)\) if \(P=(a,b)\) is affine, \(\infty_\mp\) if \(P=\infty_\pm\), and \(\infty\) if \(P=\infty\).

Lemma 4.1. \(\operatorname{ord}_{\bar P}(\tau g)=\operatorname{ord}_P(g)\) for every point \(P\) and every \(g\ne0\).

Proof. If \(P\) is an ordinary affine point or one of \(\infty_\pm\), then \(w_{\bar P}=w_P\) and \(v_{\bar P}=-v_P\). So \(\iota_{\bar P}\) and \(\iota_P\circ\tau\) agree on \(w\) and \(v\), hence are equal, and \(\iota_{\bar P}(\tau g)=\iota_P(g)\). If \(P\) is a branch point or the point \(\infty\) of a cubic, then \(\bar P=P\); the series \(w_P\) contains only even powers of \(z\) and \(v_P\) only odd powers. Let \(\epsilon\) be the automorphism \(z\mapsto-z\) of \(k((z))\). Then \(\iota_P\circ\tau\) and \(\epsilon\circ\iota_P\) agree on \(w\) and \(v\), hence are equal, and \(\epsilon\) preserves orders. \(\blacksquare\)

Theorem 4.2. A function \(g\in K\) that is regular at all points is constant: \(g\in k\).

Proof. Let \(g=A+Bv\). By Lemma 4.1, \(\tau g=A-Bv\) is regular at all points. So \(g+\tau g=2A\) and \(g\cdot\tau g=A^2-B^2f\) are elements of \(k(w)\) that are regular at all points, and they are constants by Lemma 3.1(3). Hence \(A\in k\) and \(B^2f=c\) with \(c\in k\). If \(B\ne0\), then \(c\ne0\) and \(f=cB^{-2}\); at a root \(a\) of \(f\) this gives \(1=n_a(f)=-2n_a(B)\), which is impossible. So \(B=0\) and \(g=A\in k\). \(\blacksquare\)

Corollary 4.3. If two nonzero functions have the same order at every point, their quotient is a constant.

Proof. The quotient and its inverse have order \(0\) at every point. \(\blacksquare\)

5. Differentials

For a Laurent series \(\ell=\sum_nb_nz^n\) let \(\ell'=\sum_nnb_nz^{n-1}\). The expansion at \(P\) of a differential \(\eta=g\,dw\) is \[ \iota_P(\eta)=\iota_P(g)\,w_P'\;dz . \] The series \(w_P'\) is not zero: it is \(1\) at an ordinary affine point, \(2z\varphi'(z^2)\) at an affine branch point, \(-z^{-2}\) at \(\infty_\pm\), and \(-2z\varphi'(z^2)/\varphi(z^2)^2\) at the point \(\infty\) of a cubic; here \(\varphi'\) has a nonzero constant term and \(2\ne0\) in \(k\). The order of the series \(\iota_P(g)w_P'\) is the order \(\operatorname{ord}_P(\eta)\), and its coefficient of \(z^{-1}\) is the residue \(\operatorname{res}_P(\eta)\). The differential has a pole at \(P\) if \(\operatorname{ord}_P(\eta)<0\), a simple pole if the order is \(-1\). For a function \(g\), \[ \operatorname{ord}_P(g\eta)=\operatorname{ord}_P(g)+\operatorname{ord}_P(\eta). \]

Lemma 5.1. For every function \(g\) and every point \(P\), \[ \iota_P(dg)=\iota_P(g)'\,dz . \]

Proof. We have to show \(\iota_P(D(g))\,w_P'=\iota_P(g)'\). Let \(\delta_1(g)=\iota_P(D(g))\) and \(\delta_2(g)=\iota_P(g)'/w_P'\). Both maps \(\delta\colon K\to k((z))\) are additive, vanish on \(k\), have \(\delta(w)=1\), and satisfy \[ \delta(gg')=\delta(g)\iota_P(g')+\iota_P(g)\delta(g'). \] Two such maps agree on polynomials in \(w\), by the product rule and induction on the degree, then on \(k(w)\), since \(\delta(A)=\delta(B\cdot A/B)\) determines \(\delta(A/B)\). They agree on \(v\), because \(2\iota_P(v)\delta(v)=\delta(f)\) and \(2\iota_P(v)\ne0\). So they agree on all \(A+Bv\). \(\blacksquare\)

Proposition 5.2. The differential \(\omega=dw/v\) has order \(0\) at every point.

Proof. \(\iota_P(\omega)=(w_P'/v_P)\,dz\), and \(w_P'/v_P\) is

• \(1/v_P\) at an ordinary affine point, where \(v_P\) is a unit;
• \(2z\varphi'(z^2)/z=2\varphi'(z^2)\) at an affine branch point;
• \(-z^{-2}/(z^{-2}s)=-1/s\) at \(\infty_\pm\), where \(s\) is a unit;
• \(-2\varphi'(z^2)\) at the point \(\infty\) of a cubic.

Each of these is a unit of \(k[[z]]\). \(\blacksquare\)

Corollary 5.3. A differential that has no pole is \(c\,\omega\) with \(c\in k\).

Proof. Let \(\eta=g\,dw=(gv)\,\omega\). By Proposition 5.2, \(\operatorname{ord}_P(gv)=\operatorname{ord}_P(\eta)\ge0\) for all \(P\). By Theorem 4.2, \(gv\in k\). \(\blacksquare\)

So the differentials without poles form a vector space of dimension one over \(k\). This is the statement that the curve has genus one.

Proposition 5.4. Let \(F\in K\), \(F\ne0\), and let \(P\) be a point with \(n=\operatorname{ord}_P(F)\). Then \(\operatorname{ord}_P(dF/F)\ge-1\), and \[ \operatorname{res}_P(dF/F)=n\quad\text{in }k . \] In particular \(dF/F\) has no pole at \(P\) if \(p\) divides \(n\).

Proof. By Lemma 5.1, \(\iota_P(dF/F)=(\ell'/\ell)\,dz\) with \(\ell=\iota_P(F)=z^ne\), \(e\) a unit. Then \(\ell'/\ell=n/z+e'/e\), and \(e'/e\in k[[z]]\). \(\blacksquare\)

The residue of \(dF/F\) is the integer \(n\) reduced modulo \(p\). A zero or pole of \(F\) whose order is divisible by \(p\) is not seen by \(dF/F\).

6. Exercises

6.1. Let \(P=(a,b)\) be an ordinary affine point. Determine the order of the function \(w-a\) at every point of the curve, for \(f\) of degree three and of degree four, and check that the sum of the orders over all points is \(0\).

6.2. Determine the order of \(v\) at every point, and check that the sum of the orders is \(0\).

6.3. Determine \(\operatorname{ord}_P(dw)\) at every point, and check that the sum is \(0\).

6.4. Show that a function that is regular at all affine points is \(A+Bv\) with polynomials \(A,B\in k[w]\).

6.5. Let \(f(0)=1\), \(O=(0,1)\) and \(\bar O=(0,-1)\). Show that \(dw/(wv)\) has simple poles at \(O\) and \(\bar O\) with residues \(1\) and \(-1\), and no other pole.

7. Solutions

6.1. By Lemma 3.1 the order is \(1\) at \(P\) and at \(\bar P=(a,-b)\), and \(0\) at the other affine points. At infinity it is \(-1\) at each of \(\infty_+,\infty_-\) for degree four, and \(-2\) at \(\infty\) for degree three. The sum is \(1+1-2=0\) in both cases.

6.2. At an affine point \((a,b)\) the value of \(v\) is \(b\); so the order is \(0\) at ordinary points and \(1\) at the branch points (the list of orders in Section 2). There are four branch points and order \(-2\) at each of \(\infty_\pm\) for degree four: \(4-4=0\). There are three branch points and order \(-3\) at \(\infty\) for degree three: \(3-3=0\).

6.3. The order of \(dw\) is the order of \(w_P'\): \(0\) at ordinary affine points, \(1\) at affine branch points, \(-2\) at \(\infty_\pm\), and \(1-4=-3\) at the point \(\infty\) of a cubic. The sums are \(4-2-2=0\) and \(3-3=0\). This agrees with Exercise 6.2 and Proposition 5.2, since \(dw=v\,\omega\).

6.4. Let \(g=A+Bv\) be regular at all affine points. By Lemma 4.1 so is \(\tau g\), hence so are \(2A\) and \(A^2-B^2f\). By Lemma 3.1(3), \(A\) and \(B^2f\) are polynomials. If \(B\) had a pole at some \(a\), then \(n_a(B)\le-1\) and \(n_a(f)\le1\), because \(f\) has no repeated root; so \(n_a(B^2f)\le-1\), and \(B^2f\) would not be a polynomial. So \(B\) is a polynomial.

6.5. \(dw/(wv)=w^{-1}\omega\). By Proposition 5.2 its order at \(P\) is \(-\operatorname{ord}_P(w)\): \(-1\) at the two points \(O,\bar O\) over \(w=0\), which are ordinary because \(f(0)\ne0\); \(0\) at the other affine points; and positive at infinity. At \(O\) the expansion is \(dz/(z\,v_O)\) with \(v_O=1+\cdots\), so the residue is \(1\); at \(\bar O\), \(v_{\bar O}=-1+\cdots\) and the residue is \(-1\).

References

• [Jordan] D. Jordan, Cartier–Miller evaluation of genus-one coefficient sums, manuscript, revised 5 October 2026, the section "Cartier evaluation and the regular identity". https://github.com/bbrhuft/Cartier-Miller-evaluation-of-genus-one-coefficient-sums