The quadratic large sieve and the reflected moment

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

This lesson proves Proposition 4.1 of The coefficient family and the plan of the zero-free argument, the bound for the mean square of \(B_m(Z)\) over a box of rows. The reflected expansion of The Weyl reflection writes \(B_m(Z)\) as a bilinear form in the squarefree part \(\mathfrak d\) of the row and a squarefree column ideal \(\mathfrak c'\), coupled by the quadratic symbol \(\chi_{\mathfrak d}(c')^3\) and by a profile that decays rapidly once \(q_{\mathfrak c'}\) exceeds \(q_{\mathfrak d}^2q_{\mathfrak f}^2/Z\). A large sieve inequality for quadratic characters over number fields, due to Goldmakher and Louvel and extending Heath-Brown's theorem over \(\mathbb Q\), bounds such bilinear forms. The lesson checks that the exterior symbols form a family to which it applies, and then estimates the retained part of the expansion with it and the remaining part with the decay of the profile.

Prerequisites. S-integers and sixth-power classes, Section 3 and Lemma 4.1; Sextic reciprocity and Kummer characters, Proposition 2.1 and Lemma 3.1; The coefficient family and the plan of the zero-free argument, Section 4; Norm sheets and exact Gaussian smoothing, Corollary 5.2; A theta expression for the coefficient sums, Lemmas 1.1 and 2.1; The Weyl reflection, (2.2), Lemma 2.2, Lemma 3.1, Section 4 and Lemma 5.1; Counting ideals of bounded norm, Theorem 15.3; Quasi-characters and Hecke characters, Proposition 6.1; Hensel's lemma, squares and roots of unity in p-adic fields, Corollary 1.2; Schwartz functions and Fourier inversion, Section F4.

1. Quadratic Hecke families

Let \(k\) be a number field and \(\mathfrak c_0\) a nonzero integral ideal of \(\mathcal O_k\). Write \(I(\mathfrak c_0)\) for the set of nonzero integral ideals prime to \(\mathfrak c_0\), and \(I^{\mathrm{sf}}(\mathfrak c_0)\) for its squarefree members. The infinite type of a Hecke character \(\chi\) of modulus \(\mathfrak f\) is the character \(\chi_\infty\) of \(k\otimes_{\mathbb Q}\mathbb R\) with \(\chi((x))=\chi_\infty(x)\) for every \(x\in\mathcal O_k\) with \(x\equiv1\pmod{\mathfrak f}\).

Definition 1.1 (Goldmakher and Louvel [GL, Definition 1], for \(n=2\)). A quadratic Hecke family with respect to \(\mathfrak c_0\) is a collection \(\{\chi_{\mathfrak a}:\mathfrak a\in I^{\mathrm{sf}}(\mathfrak c_0)\}\) of primitive Hecke characters of trivial infinite type such that

  1. the order of each \(\chi_{\mathfrak a}\) divides \(2\);
  2. there are a finite group \(G\), a homomorphism \([\cdot]\) from the multiplicative monoid \(I(\mathfrak c_0)\) to \(G\), and a map \(C\colon G\times G\to\{\pm1\}\) with \(\chi_{\mathfrak a}(\mathfrak b)=\chi_{\mathfrak b}(\mathfrak a)\,C([\mathfrak a],[\mathfrak b])\) for all coprime \(\mathfrak a,\mathfrak b\) in the family;
  3. for all coprime \(\mathfrak a,\mathfrak b\) in the family with \([\mathfrak a]=[\mathfrak b]\), the character \(\chi_{\mathfrak a}\overline{\chi_{\mathfrak b}}\) is a primitive Hecke character of conductor \(\mathfrak a\mathfrak b\).

Named result N3 (Goldmakher and Louvel [GL, Theorem 1.1]). Let \(\{\chi_{\mathfrak a}\}\) be a quadratic Hecke family with respect to \(\mathfrak c_0\). For every \(\epsilon>0\), all \(M,N\ge1\) and all complex numbers \(\lambda_{\mathfrak b}\), \[ \begin{aligned} &\sum_{\substack{\mathfrak a\in I^{\mathrm{sf}}(\mathfrak c_0)\\ q_{\mathfrak a}\le M}}\Bigl|\sum_{\substack{\mathfrak b\in I^{\mathrm{sf}}(\mathfrak c_0)\\ q_{\mathfrak b}\le N}}\lambda_{\mathfrak b}\,\chi_{\mathfrak b}(\mathfrak a)\Bigr|^2\\ &\quad \le C_{k,\mathfrak c_0,\epsilon}\,(MN)^\epsilon(M+N)\\ &\qquad \sum_{\substack{\mathfrak b\in I^{\mathrm{sf}}(\mathfrak c_0)\\ q_{\mathfrak b}\le N}}|\lambda_{\mathfrak b}|^2 . \end{aligned} \] It extends Heath-Brown's large sieve for quadratic characters over \(\mathbb Q\) [HB]. It is used without proof in the programme.

2. The family of exterior quadratic symbols

Keep \(F\supseteq\mu_{12}\), \(S\) and the allowed generators of Lessons 2–4. For each finite place \(v\in S\) choose \(m_v\ge1\) with \(1+\mathfrak p_v^{m_v}\subseteq F_v^{\times2}\) (possible because \(F_v^{\times2}\) is open), and let \(\mathfrak q_S=\prod_{v\in S\ \text{finite}}\mathfrak p_v^{m_v}\). Exterior ideals correspond to the ideals in \(I(\mathfrak q_S)\), with the same norms and the same multiplication.

For a squarefree exterior ideal \(\mathfrak c\) with allowed generator \(c\), let \(\Xi_{\mathfrak c}=\kappa_c^3\), with \(\kappa_c\) the Kummer character of Lemma 3.1 of Lesson 3. Let \(B_2\) be the image of \(B\) in \(F_S^\times/F_S^{\times2}\) and \(\mathrm{cl}_2(\mathfrak a)\in B_2\) the class of an allowed generator of an exterior ideal \(\mathfrak a\).

Lemma 2.1 (the exterior family [OpenAI-029, Lemma 6.4]). The characters \(\Xi_{\mathfrak c}\) form a quadratic Hecke family with respect to \(\mathfrak q_S\), with \(G=B_2\), \([\cdot]=\mathrm{cl}_2\) and \(C=1\). For every exterior ideal \(\mathfrak d\), \[ \Xi_{\mathfrak c}(\mathfrak d)=\chi_{\mathfrak d}(c)^3 , \tag{2.1} \] both sides being \(0\) when \(\mathfrak c\) and \(\mathfrak d\) share a prime.

Proof. The characters. Two allowed generators differ by an element of \(U_S^6\), whose Kummer character is trivial (Lemma 3.1(3) of Lesson 3); so \(\Xi_{\mathfrak c}\) is well defined. By Lemma 3.1(1) of Lesson 3 and \((a,b)_{6,v}^3=(a,b)_{2,v}\) (Section 3 of Lesson 2), \(\Xi_{\mathfrak c}\) is a Hecke character of order dividing \(2\) with local components \(y\mapsto(c,y)_{2,v}\). At a complex place a character of finite order is trivial, because \(\mathbb C^\times\) is connected; so the infinite type is trivial.

Conductor. At an exterior prime \(\mathfrak p\nmid\mathfrak c\), \(c\) is a unit and the symbol of two units is \(1\), so the component is unramified. At \(\mathfrak p\mid\mathfrak c\), principal units are squares (Corollary 1.2 of Hensel's lemma, squares and roots of unity in p-adic fields, as \(\mathfrak p\nmid2\)), so the conductor exponent is at most \(1\) (Proposition 6.1 of Quasi-characters and Hecke characters); and with \(c=pu_c\), a unit \(u\) gives \((c,u)_{2,\mathfrak p}=(p,u)_{2,\mathfrak p}=(u,p)_{2,\mathfrak p}^{-1}\)\(\equiv u^{-(q_{\mathfrak p}-1)/2}\), which is \(-1\) for a nonsquare unit by (3.2) and Lemma 3.1 of Lesson 2; so the exponent is exactly \(1\). At a finite \(v\in S\) the component is trivial on \(1+\mathfrak p_v^{m_v}\). So the conductor is \(\mathfrak c\mathfrak s_{\mathfrak c}\) with \(\mathfrak s_{\mathfrak c}\mid\mathfrak q_S\).

Values. At an exterior prime \(\mathfrak d\nmid\mathfrak c\), \(\Xi_{\mathfrak c}(\mathfrak d)=\kappa_c(\mathfrak d)^3=\chi_{\mathfrak d}(c)^3\) by Lemma 3.1(1) of Lesson 3. At \(\mathfrak d\mid\mathfrak c\) the prime divides the conductor, so the ideal character vanishes, and \(\chi_{\mathfrak d}(c)^3=0\) by the unit mask. Both sides of (2.1) are multiplicative in \(\mathfrak d\).

Reciprocity. The map \(\mathrm{cl}_2\) is well defined, since \(U_S^6\subseteq F_S^{\times2}\), and multiplicative, since products of allowed generators are allowed generators of the product (Lemma 4.1 of Lesson 2); its image lies in the finite group \(B_2\). For coprime squarefree \(\mathfrak c,\mathfrak d\), (2.2) of Lesson 3 with \(\epsilon=1\) gives \(\chi_{\mathfrak d}(c)=\chi_{\mathfrak c}(d)\), hence \(\Xi_{\mathfrak c}(\mathfrak d)=\Xi_{\mathfrak d}(\mathfrak c)\), which is property 2 with \(C=1\).

Products. Let \(\mathfrak c_1,\mathfrak c_2\) be coprime and squarefree with \(\mathrm{cl}_2(\mathfrak c_1)=\mathrm{cl}_2(\mathfrak c_2)\). Since \(\Xi\) is real, \(\Xi_{\mathfrak c_1}\overline{\Xi_{\mathfrak c_2}}\) has local components \(y\mapsto(c_1c_2,y)_{2,v}\). At \(v\in S\), \(c_1c_2=(c_1/c_2)c_2^2\) is a square in \(F_v\), so the component is trivial. At a prime of \(\mathfrak c_1\mathfrak c_2\) the valuation of \(c_1c_2\) is \(1\), and the argument above gives exponent \(1\); elsewhere it is unramified. So the conductor is exactly \(\mathfrak c_1\mathfrak c_2\), which is property 3. \(\square\)

Corollary 2.2 (quadratic large sieve for the exterior symbols). For \(C,D\ge1\), every \(\epsilon>0\), and complex \(z_{\mathfrak c}\) supported on squarefree exterior ideals with \(q_{\mathfrak c}\le C\), \[ \begin{aligned} &\sum_{\substack{\mathfrak d\ \text{squarefree}\\ q_{\mathfrak d}\le D}}\Bigl|\sum_{\mathfrak c}z_{\mathfrak c}\,\chi_{\mathfrak d}(c)^3\Bigr|^2\\ &\quad\le C_{F,S,\epsilon}\,(C+D)(CD)^\epsilon\\ &\qquad\times\sum_{\mathfrak c}|z_{\mathfrak c}|^2 .\quad(2.2) \end{aligned} \] The same bound holds after deleting any set of rows, and after multiplying each row by a factor of absolute value at most one.

Proof. N3 with \(M=D\), \(N=C\), \(\lambda_{\mathfrak c}=z_{\mathfrak c}\), and (2.1). Deleting rows or multiplying them by such factors does not increase the left side. \(\square\)

A restriction on the columns, for instance to one class \([c]\in B\), or a unimodular factor depending on the column alone, is absorbed into the coefficients \(z_{\mathfrak c}\); the family itself is applied in full.

3. Counting lemmas

Lemma 3.1. There is \(C_F\) such that, for exterior ideals:

  1. \(\#\{\mathfrak a:q_{\mathfrak a}\le t\}\le C_Ft\) for \(t\ge1\);
  2. \(\sum_{C\le q_{\mathfrak a}<2C}q_{\mathfrak a}^{-1}\le2C_F\) for \(C\ge1\);
  3. \(\sum_{q_{\mathfrak a}\le t}q_{\mathfrak a}^{-1}\le2C_F\,(1+\log_2t)\) for \(t\ge1\);
  4. \(\#\{\mathfrak h\ \text{powerful}:q_{\mathfrak h}<2P\}\)\(\le C_F\,\zeta_F(3/2)\,(2P)^{1/2}\) for \(P>0\).

Proof. (1) Exterior ideals correspond to integral ideals of \(\mathcal O_F\) of the same norm, and (16) of Theorem 15.3 of Counting ideals of bounded norm bounds their number. (2) The sum is at most \(C^{-1}\) times the number of ideals of norm less than \(2C\). (3) Split \([1,t]\) into the intervals \([2^k,2^{k+1})\), \(0\le k\le\log_2t\), and apply (2). (4) A powerful ideal has a unique expression \(\mathfrak h=\mathfrak r^2\mathfrak s^3\) with \(\mathfrak s\) squarefree: at a prime of exponent \(e\ge2\), take exponent \(0\) in \(\mathfrak s\) and \(e/2\) in \(\mathfrak r\) if \(e\) is even, and \(1\) in \(\mathfrak s\) and \((e-3)/2\) in \(\mathfrak r\) if \(e\) is odd. For each \(\mathfrak s\), the number of \(\mathfrak r\) with \(q_{\mathfrak r}<(2P/q_{\mathfrak s}^3)^{1/2}\) is at most \(C_F(2P)^{1/2}q_{\mathfrak s}^{-3/2}\) by (1), and it is \(0\) when the bound is below \(1\). Summing over \(\mathfrak s\) gives at most \(C_F(2P)^{1/2}\zeta_F(3/2)\). \(\square\)

Lemma 3.2 ([OpenAI-029, (6.34)]). For every \(\delta_1>0\) there is \(C_{\delta_1}\) with \[ \begin{aligned} &\sum_{\mathfrak f\mid\operatorname{rad}(\mathfrak h)}\ \prod_{\mathfrak p\mid\mathfrak f}E_{\mathfrak p}(\sigma)\\ &\quad =\prod_{\mathfrak p\mid\mathfrak h}\bigl(1+E_{\mathfrak p}(\sigma)\bigr)\le C_{\delta_1}\,q_{\mathfrak h}^{\delta_1} \end{aligned} \] for all \(\sigma\ge0\) and all exterior ideals \(\mathfrak h\), with \(E_{\mathfrak p}\) as in (2.5) of Lesson 9.

Proof. The identity is the expansion of the product. Each factor is at most \(4+\sqrt2\) by Lemma 2.2 of Lesson 9. The primes with \(q_{\mathfrak p}<(4+\sqrt2)^{1/\delta_1}\) are finitely many and contribute at most a constant; every other factor is at most \(q_{\mathfrak p}^{\delta_1}\), and \(\prod_{\mathfrak p\mid\mathfrak h}q_{\mathfrak p}\le q_{\mathfrak h}\). \(\square\)

4. Proof of the reflected second moment

We prove Proposition 4.1 of Lesson 5: for a compact \(\mathscr K\subset F_S^\times\), \(\epsilon>0\), \(Z\ge2\) and \(P,D\ge1\), \[ \begin{aligned} &\sum_{\substack{m\in\mathscr B_Z(\mathscr K)\\ P\le q_{\mathfrak h(m)}<2P\\ D\le q_{\mathfrak d(m)}<2D}}|B_m(Z)|^2\\ &\quad \le C_{\mathscr K,\epsilon}\,Z^\epsilon P^{1/2}\Bigl(D+\frac{D^2P}Z\Bigr).\quad(4.1) \end{aligned} \] The deduction of the total bound (4.4) of Lesson 5 from (4.1) is given there. All constants below may depend on \(F\), \(S\), \(\eta\), \(\mathscr K\), \(\epsilon\) and the fixed theta data.

Step 1: from rows to ideals. By (2.2) of Lesson 8, \(B_m(Z)\) depends on \(m\) only through \(\mathfrak m=mR\) and \(\upsilon_m\in U_6\). For an exterior ideal \(\mathfrak m=\mathfrak h\mathfrak d\), with \(\mathfrak h\) powerful and \(\mathfrak d\) squarefree and prime to \(\mathfrak h\), and for \(\upsilon\in U_6\), let \(B_{\mathfrak m,\upsilon}(Z)\) be the common value, given by the expansion (5.1) of Lesson 9. By Lemma 1.1(2) of Lesson 8, at most \(C_{\mathscr K}=|U_S\cap\mathscr K\mathscr K^{-1}|\) rows generate one ideal. Hence the left side of (4.1) is at most \[ C_{\mathscr K}\sum_{\upsilon\in U_6}\ \sum_{\substack{\mathfrak h\ \text{powerful}\\ P\le q_{\mathfrak h}<2P}}\ \sum_{\substack{\mathfrak d\ \text{squarefree}\\ (\mathfrak d,\mathfrak h)=1\\ D\le q_{\mathfrak d}<2D}}|B_{\mathfrak h\mathfrak d,\upsilon}(Z)|^2 . \] If the dyads contain a row, then \(q_{\mathfrak h}q_{\mathfrak d}=|m|_S\) is at most a constant times \(Z^{M_0}\), so \[ \begin{gathered} P,D\ge1,\\ PD\le C_{\mathscr K}'Z^{M_0};\quad(4.2) \end{gathered} \] otherwise the left side of (4.1) is \(0\). Fix \(\upsilon\).

Step 2: retained and discarded terms. Fix \(\delta>0\), to be chosen small, and put \[ \begin{gathered} Q=q_{\mathfrak a'_{\mathfrak f}}q_{\mathfrak c'}q_{\mathfrak n'}^3,\\ t_{\mathfrak d}=\frac{q_{\mathfrak d}^2q_{\mathfrak f}^2}Z,\\ \mathcal L=\frac{8D^2P}Z,\\ R_0=Z^\delta\mathcal L . \end{gathered} \] The profile in (5.1) of Lesson 9 is \(\Phi_{\upsilon,\kappa}(Q/t_{\mathfrak d})\), and \(t_{\mathfrak d}<\mathcal L\) because \(q_{\mathfrak d}<2D\) and \(q_{\mathfrak f}^2\le q_{\mathfrak h}<2P\) ((2.2) of Lesson 9). Split each term of (5.1) of Lesson 9 according to \(Q\le R_0\) or \(Q>R_0\), and write \(B_{\mathfrak h\mathfrak d,\upsilon}(Z)=\operatorname{Ret}_{\mathfrak h,\mathfrak d}+\operatorname{Tail}_{\mathfrak h,\mathfrak d}\) accordingly; for fixed \(\mathfrak f\) and \(\mathbf s=(u,z_d,z_c)\in U_6\times B^2\) write \(\operatorname{Ret}_{\mathfrak f,\mathbf s,\mathfrak d}\) and \(\operatorname{Tail}_{\mathfrak f,\mathbf s,\mathfrak d}\) for the corresponding parts, which still contain the sums over \(\mathfrak a'_{\mathfrak f},\mathfrak n',\mathfrak c'\). The threshold \(R_0\) depends only on the dyads. If \(R_0<1\), every term is discarded, because \(Q\ge1\).

Step 3: the discarded terms. Put \(\mathcal J=\max(Z^\delta,\mathcal L^{-1})\) and fix \(N>\sigma>1/2\). If \(Q>R_0\), then \(Q/t_{\mathfrak d}>R_0/\mathcal L=Z^\delta\) and \(Q/t_{\mathfrak d}\ge1/t_{\mathfrak d}>\mathcal L^{-1}\), so \(Q/t_{\mathfrak d}\ge\mathcal J\ge1\). By Lemma 3.1 of Lesson 9, \(|\Phi_{\upsilon,\kappa}(x)|\le C_Nx^{-N}\le C_N\mathcal J^{-(N-\sigma)}x^{-\sigma}\) for \(x\ge\mathcal J\). With (4.5), (5.7) of Lesson 9 and \(t_{\mathfrak d}<\mathcal L\), \[ \begin{aligned} &|\operatorname{Tail}_{\mathfrak f,\mathbf s,\mathfrak d}|\\ &\quad \le C_N\mathcal J^{-(N-\sigma)}t_{\mathfrak d}^\sigma\mathcal D_{\mathfrak f}(\sigma)\\ &\quad \le C_{N,\sigma}\,\mathcal L^\sigma\mathcal J^{-(N-\sigma)}\prod_{\mathfrak p\mid\mathfrak f}E_{\mathfrak p}(\sigma).\quad(4.3) \end{aligned} \] Summing over \(\mathfrak f\in\mathcal F(\mathfrak h)\) with Lemma 3.2, and over the \(|U_6||B|^2\) choices of \(\mathbf s\), gives \(|\operatorname{Tail}_{\mathfrak h,\mathfrak d}|\le C\,q_{\mathfrak h}^{\delta_1}\mathcal L^\sigma\mathcal J^{-(N-\sigma)}\). Squaring, and summing over the at most \(2C_FD\) rows \(\mathfrak d\) and the at most \(CP^{1/2}\) powerful \(\mathfrak h\) (Lemma 3.1), \[ \begin{aligned} &\sum_{\mathfrak h,\mathfrak d}|\operatorname{Tail}_{\mathfrak h,\mathfrak d}|^2\\ &\quad \le C\,P^{1/2+2\delta_1}D\,\mathcal L^{2\sigma}\mathcal J^{-2(N-\sigma)} .\quad(4.4) \end{aligned} \] By (4.2), \(P\le C'Z^{M_0}\) and \(\mathcal L\le8(PD)^2/Z\le CZ^{2M_0-1}\), while \(\mathcal J\ge Z^\delta\). Given \(A_0>0\), choose \(N\) with \[ \begin{aligned} 2\delta(N-\sigma)&>A_0+2M_0\delta_1\\ &\quad+2\sigma(2M_0-1); \end{aligned} \] then (4.4) is at most \(CZ^{-A_0}P^{1/2}D\). This covers every dyad, including those with \(R_0<1\).

Step 4: the retained terms. Let \(R_0\ge1\). Fix \(\mathfrak f\), \(\mathbf s\), \(\mathfrak a'_{\mathfrak f}\) and \(\mathfrak n'\); then \(\kappa=\kappa_6(\mathfrak t;z_d,z_c)\) is fixed, and \(\rho=\rho_{\upsilon,\mathfrak t,z_d}\), \(\lambda=\lambda_{\upsilon,\mathfrak t,z_c}\) are functions of the row and of the column alone. Split the columns into dyads \(C\le q_{\mathfrak c'}<2C\) with \(C\ge1\) a power of \(2\), and put \[ \begin{gathered} v_0=\frac{Zq_{\mathfrak a'_{\mathfrak f}}q_{\mathfrak n'}^3C}{D^2q_{\mathfrak f}^2},\\ \begin{aligned} x&=\log\frac{q_{\mathfrak c'}}C-2\log\frac{q_{\mathfrak d}}D\\ &\in[-2\log2,\log2]. \end{aligned} \end{gathered} \] Then \(Q/t_{\mathfrak d}=v_0e^x\). Fix \(\phi\in C_c^\infty(\mathbb R)\) equal to \(1\) on \([-2\log2,\log2]\) and put \(g(x)=\phi(x)\Phi_{\upsilon,\kappa}(v_0e^x)\). By Corollary 5.2 of Lesson 7, applied to the profiles of (3.1) of Lesson 9, \[ \begin{aligned} &\int_{\mathbb R}|\widehat g(\tau)|\,d\tau\\ &\quad \le C_N\min(v_0^\alpha,v_0^{-N})\le C .\quad(4.5) \end{aligned} \] Fourier inversion for the smooth compactly supported \(g\) (Section F4 of Schwartz functions and Fourier inversion) gives \(\Phi_{\upsilon,\kappa}(Q/t_{\mathfrak d})=\frac1{2\pi}\int\widehat g(\tau)\)\(\,(q_{\mathfrak c'}/C)^{i\tau}(q_{\mathfrak d}/D)^{-2i\tau}\,d\tau\), which separates the row and column norms. For each \(\tau\) put \[ \begin{aligned} z_{\mathfrak c'}(\tau)&=\lambda(\mathfrak c')\,q_{\mathfrak c'}^{-1/2}(q_{\mathfrak c'}/C)^{i\tau}\\ &\quad \,\mathbf 1_{C\le q_{\mathfrak c'}<2C}\,\mathbf 1_{Q\le R_0}, \end{aligned} \] a column coefficient once \(\mathfrak a'_{\mathfrak f}\) and \(\mathfrak n'\) are fixed, with \(\sum|z_{\mathfrak c'}(\tau)|^2\le\sum_{C\le q<2C}q^{-1}\le2C_F\) by (4.5) of Lesson 9 and Lemma 3.1. The row factors \(\rho(\mathfrak d)(q_{\mathfrak d}/D)^{-2i\tau}\) have absolute value at most one, and the restrictions \((\mathfrak d,\mathfrak h\mathfrak n')=1\) and \([d]=z_d\) delete rows. By Minkowski's inequality for the \(\tau\)-integral and Corollary 2.2, the \(\ell^2\) norm over \(\mathfrak d\) of the contribution of one column dyad is at most \[ \begin{gathered} C_{\delta_2}\,(D+C)^{1/2}(DC)^{\delta_2/2}\\ (\delta_2>0).\quad(4.6) \end{gathered} \] Shared primes of \(\mathfrak d\) and \(\mathfrak c'\) are already handled by the zero of \(\chi_{\mathfrak d}(c')^3\), so no condition coupling rows and columns has been dropped.

In the retained range \(C\le q_{\mathfrak c'}\le R_0\) and \(q_{\mathfrak n'}\le R_0^{1/3}\). There are at most \(1+\log_2R_0\) column dyads, and \(\sum_{q_{\mathfrak n'}\le R_0^{1/3}}q_{\mathfrak n'}^{-1}\le C\log(2R_0)\) by Lemma 3.1. By (4.5) of Lesson 9, \(|\omega_{\upsilon,\mathfrak t}|\le\beta_{\mathfrak f}(\mathfrak a'_{\mathfrak f})\), and \(\sum_{\mathfrak a'_{\mathfrak f}\in\mathcal A(\mathfrak f)}\beta_{\mathfrak f}(\mathfrak a'_{\mathfrak f})=\prod_{\mathfrak p\mid\mathfrak f}E_{\mathfrak p}(0)\). Minkowski's inequality over the column dyads, \(\mathfrak n'\) and \(\mathfrak a'_{\mathfrak f}\) bounds the \(\ell^2\) norm over \(\mathfrak d\) of \(\operatorname{Ret}_{\mathfrak f,\mathbf s,\mathfrak d}\) by \[ \begin{aligned} &C_{\delta_2}\prod_{\mathfrak p\mid\mathfrak f}E_{\mathfrak p}(0)\,\log^2(2R_0)\\ &\quad \,(D+R_0)^{1/2}(DR_0)^{\delta_2/2}. \end{aligned} \] Summing over \(\mathfrak f\) with Lemma 3.2 at \(\sigma=0\), and over \(\mathbf s\), bounds the \(\ell^2\) norm of \(\operatorname{Ret}_{\mathfrak h,\cdot}\) by \(C\,q_{\mathfrak h}^{\delta_1}\log^2(2R_0)(D+R_0)^{1/2}(DR_0)^{\delta_2/2}\). Square this for each \(\mathfrak h\) and sum over the at most \(CP^{1/2}\) powerful \(\mathfrak h\); every row has one powerful part, so no inequality across different \(\mathfrak h\) is needed: \[ \begin{aligned} &\sum_{\mathfrak h,\mathfrak d}|\operatorname{Ret}_{\mathfrak h,\mathfrak d}|^2\\ &\quad\le C\,P^{1/2+2\delta_1}\log^4(2R_0)\\ &\qquad\times(D+R_0)(DR_0)^{\delta_2}.\quad(4.7) \end{aligned} \]

Step 5: the exponents. By (4.2), \(D+R_0\le8Z^\delta(D+D^2P/Z)\), \(DR_0=8Z^\delta D^3P/Z\le CZ^{3M_0-1+\delta}\), and \(1\le R_0\le CZ^{2M_0-1+\delta}\), so \(\log^4(2R_0)\le C_{\delta_3}Z^{\delta_3}\) for every \(\delta_3>0\). Choose \(\delta,\delta_1,\delta_2,\delta_3>0\) with \[ \delta+2M_0\delta_1+(3M_0-1+\delta)\delta_2+\delta_3<\epsilon . \] Then (4.7) is at most \(CZ^\epsilon P^{1/2}(D+D^2P/Z)\), and (4.4), with \(A_0=1\), is smaller. Since \(|B|^2\le2|\operatorname{Ret}|^2+2|\operatorname{Tail}|^2\), summing over the finitely many \(\upsilon\) and restoring the factor \(C_{\mathscr K}\) proves (4.1).

Uniformity. Replacing \(\mathscr K\) by \(c\mathscr K\) with \(c\in F_S^1\) changes neither \(\mathscr K\mathscr K^{-1}\) nor the norms \(|y|_S\) of its elements; so \(C_{\mathscr K}\) and the constant in (4.2) are unchanged, and the bound is uniform in \(c\). \(\square\)

5. Exercises

5.1. Show that the decomposition \(\mathfrak h=\mathfrak r^2\mathfrak s^3\) with \(\mathfrak s\) squarefree exists exactly for powerful \(\mathfrak h\), and that it is unique.

5.2. Let \(p\equiv1\pmod{12}\) be a rational prime and \(\chi\) a character of order \(6\) of \(\mathbb F_p^\times\). Show that \(\chi^3\) is the Legendre symbol modulo \(p\). Explain how this is the special case of (2.1) at a prime of degree one.

5.3. Let \(v\in S\) be a finite place not above \(2\), and let \(a\in F_v^\times\) be a unit that is not a square. Show that \(y\mapsto(a,y)_{2,v}\) is a nontrivial unramified character of \(F_v^\times\). Conclude that, at such a place, a difference between the local classes of \(c_1\) and \(c_2\) need not enlarge the conductor of \(\Xi_{\mathfrak c_1}\overline{\Xi_{\mathfrak c_2}}\), while equal classes always give a trivial local component there.

5.4. In Step 3, verify the inequality \(x^{-N}\le\mathcal J^{-(N-\sigma)}x^{-\sigma}\) for \(x\ge\mathcal J\ge1\), and explain why the tail bound needs no separate treatment of the dyads with \(R_0<1\).

5.5. Show that \(D^3P\le(PD)^3\) for \(P\ge1\), and deduce \(DR_0\le CZ^{3M_0-1+\delta}\) from (4.2).

6. Solutions

5.1. If \(\mathfrak h=\mathfrak r^2\mathfrak s^3\), every exponent is \(2r+3s\) with \(r\ge0\), \(s\in\{0,1\}\), which is \(0\) or at least \(2\); so \(\mathfrak h\) is powerful. Conversely the proof of Lemma 3.1(4) constructs the decomposition. It is unique because \(s\equiv e\pmod2\) determines \(s\in\{0,1\}\) and then \(r=(e-3s)/2\).

5.2. \(\chi^3\) is a character of order \(2\) of the cyclic group \(\mathbb F_p^\times\), and there is only one such character, the Legendre symbol. At a prime \(\mathfrak d\) of degree one, \(k_{\mathfrak d}=\mathbb F_p\), \(\chi_{\mathfrak d}(c)\equiv c^{(p-1)/6}\), and \(\chi_{\mathfrak d}(c)^3\equiv c^{(p-1)/2}\) is the quadratic residue symbol of \(c\), the value of the character of \(F(\sqrt c)/F\) at \(\mathfrak d\).

5.3. For a unit \(u\), \((a,u)_{2,v}=1\), the symbol of two units at a place not above \(2\) (Section 3 of Lesson 2); so the character is unramified. For a uniformizer \(\varpi\), (3.2) of Lesson 2 gives \((a,\varpi)_{2,v}\equiv a^{(q_v-1)/2}\pmod v\). A unit is a square exactly when its residue is a square (Hensel's lemma, as \(v\nmid2\)); so the residue of \(a\) is not a square, and Euler's criterion gives \(a^{(q_v-1)/2}\equiv-1\). Hence the character is nontrivial. If the local classes of \(c_1\) and \(c_2\) differ by such a unit, the component \((c_1c_2,\cdot)_{2,v}\) is nontrivial but unramified, and contributes nothing to the conductor. If the classes agree, \(c_1c_2\) is a square at \(v\) and the component is trivial, as in the proof of Lemma 2.1.

5.4. For \(x\ge\mathcal J\), \(x^{-(N-\sigma)}\le\mathcal J^{-(N-\sigma)}\) since \(N>\sigma\). If \(R_0<1\), every term is in the tail, and the bound (4.4) holds for the whole sum because \(\mathcal J=\max(Z^\delta,\mathcal L^{-1})\) was chosen so that \(Q/t_{\mathfrak d}\ge\mathcal J\) for every \(Q\ge1\).

5.5. \(D^3P=(PD)^3/P^2\le(PD)^3\). Then \(DR_0=8Z^\delta D^3P/Z\)\(\le8Z^{\delta-1}(PD)^3\le CZ^{3M_0-1+\delta}\).

Sources