Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Frobenius sign separation for abelian varieties

Alina Bucur, Francesc Fité, Kiran S. Kedlaya

TL;DR

This work extends Frobenius-sign separation results from elliptic curves to higher-dimensional abelian varieties by exploiting Sato–Tate and Mumford–Tate frameworks alongside a Bach kernel-based analytic approach. Under the generalized Riemann hypothesis for a finite set of motivic $L$-functions and the assumption $\text{Hom}(A,A')=0$, the authors establish the existence of a prime of good reduction $\mathfrak p$ not dividing $NN'$ for which the Frobenius traces $a_{\mathfrak p}(A)$ and $a_{\mathfrak p}(A')$ are nonzero and have opposite signs, with an explicit bound on the norm: $\mathrm{Nm}(\mathfrak p)=O([k:\mathbb Q]^2 \log(2|\Delta_k|)^2 g^4 (g')^4 (g+g')^2 \log(2NN')^2)$. The method generalizes the elliptic-curve technique of Chen–Park–Swaminathan by adapting Bach’s kernel to the product of Sato–Tate groups, enabling a concrete, effectively computable constant dependence on $k$, $g$, and $g'$, and reducing prior error terms. The result broadens the scope of effective Frobenius-sign separation by reducing assumptions and improving the bound, thereby providing a practical criterion for detecting sign differences in Frobenius traces in higher dimensions. This has potential implications for understanding the arithmetic of abelian varieties and their Galois representations, especially in contexts where nonisogenous factors interact nontrivially.

Abstract

Let A and A' be nonzero abelian varieties defined over a number field k such that Hom(A,A')=0. Under the Generalized Riemann hypothesis for motivic L-functions attached to A and A', we show that there exists a prime p of k of good reduction for A and A' at which the Frobenius traces of A and A' are nonzero and differ by sign, and such that the norm of p is O_{k,g,g'}(log(2NN')^2), where N and N' respectively denote the absolute conductors of A and A'. We also make the dependence of the big-O constant on k and the dimensions g,g' of A,A' explicit up to an effectively computable absolute constant. Our method extends that of Chen, Park, and Swaminathan who considered the case in which A and A' are elliptic curves.

Frobenius sign separation for abelian varieties

TL;DR

This work extends Frobenius-sign separation results from elliptic curves to higher-dimensional abelian varieties by exploiting Sato–Tate and Mumford–Tate frameworks alongside a Bach kernel-based analytic approach. Under the generalized Riemann hypothesis for a finite set of motivic -functions and the assumption , the authors establish the existence of a prime of good reduction not dividing for which the Frobenius traces and are nonzero and have opposite signs, with an explicit bound on the norm: . The method generalizes the elliptic-curve technique of Chen–Park–Swaminathan by adapting Bach’s kernel to the product of Sato–Tate groups, enabling a concrete, effectively computable constant dependence on , , and , and reducing prior error terms. The result broadens the scope of effective Frobenius-sign separation by reducing assumptions and improving the bound, thereby providing a practical criterion for detecting sign differences in Frobenius traces in higher dimensions. This has potential implications for understanding the arithmetic of abelian varieties and their Galois representations, especially in contexts where nonisogenous factors interact nontrivially.

Abstract

Let A and A' be nonzero abelian varieties defined over a number field k such that Hom(A,A')=0. Under the Generalized Riemann hypothesis for motivic L-functions attached to A and A', we show that there exists a prime p of k of good reduction for A and A' at which the Frobenius traces of A and A' are nonzero and differ by sign, and such that the norm of p is O_{k,g,g'}(log(2NN')^2), where N and N' respectively denote the absolute conductors of A and A'. We also make the dependence of the big-O constant on k and the dimensions g,g' of A,A' explicit up to an effectively computable absolute constant. Our method extends that of Chen, Park, and Swaminathan who considered the case in which A and A' are elliptic curves.
Paper Structure (4 sections, 5 theorems, 72 equations)

This paper contains 4 sections, 5 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Conjectural framework
  3. Use of the Bach kernel
  4. Proof of the main theorem

Key Result

Theorem 1.1

Let $A$ (resp. $A'$) be an abelian variety defined over $k$ of dimension $g$ (resp. $g'$), and with absolute conductor $N$ (resp. $N'$). Suppose that the Mumford--Tate conjecture holds for $A$ and $A'$, and that the generalized Riemann hypothesis holds for $L(s,\chi)$, where $\chi$ is an irreducible such that $a_\mathfrak p(A)$ and $a_\mathfrak p(A')$ are nonzero and of opposite sign. The implied

Theorems & Definitions (13)

  • Theorem 1.1
  • Conjecture 2.1: Generalized Riemann hypothesis
  • Remark 3.1
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • proof
  • Remark 3.5
  • ...and 3 more