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The asymptotic distribution of Elkies primes for reductions of abelian varieties is Gaussian

Alexandre Benoist, Jean Kieffer

TL;DR

This work generalizes Elkies primes from elliptic curves to abelian varieties with real multiplication and proves that, under GRH and a large adelic Galois image, the count of Elkies primes in specified ranges for reductions of A distributes as a Gaussian in the limit. The authors develop a two-pronged approach: (i) a density-theoretic analysis via an explicit Chebotarev framework applied to torsion fields, and (ii) a combinatorial analysis of split matrices in $ ext{GSp}_{2h}$ to quantify Elkies-conditions through characteristic polynomials. They show convergence of all moments to those of the standard Gaussian, derive explicit asymptotics for the even moments, and obtain an asymptotic Gaussian behavior for the Elkies-prime counts on average over reductions, with implications for the SEA algorithm on abelian surfaces. The results extend prior work on elliptic curves to higher dimensions, providing new insights into the distribution of Elkies primes and confirming the practical viability of SEA-like point counting in the RM setting. The numerical experiments corroborate the Gaussian pattern and motivate further exploration of the constants $oldsymbol{}_h$ across RM families.

Abstract

We generalize the notion of Elkies primes for elliptic curves to the setting of abelian varieties with real multiplication (RM), and prove the following. Let $A$ be an abelian variety with RM over a number field whose attached Galois representation has large image. Then the number of Elkies primes (in a suitable range) for reductions of $A$ modulo primes converges weakly to a Gaussian distribution around its expected value. This refines and generalizes results obtained by Shparlinski and Sutherland in the case of non-CM elliptic curves, and has implications for the complexity of the SEA point counting algorithm for abelian surfaces over finite fields.

The asymptotic distribution of Elkies primes for reductions of abelian varieties is Gaussian

TL;DR

This work generalizes Elkies primes from elliptic curves to abelian varieties with real multiplication and proves that, under GRH and a large adelic Galois image, the count of Elkies primes in specified ranges for reductions of A distributes as a Gaussian in the limit. The authors develop a two-pronged approach: (i) a density-theoretic analysis via an explicit Chebotarev framework applied to torsion fields, and (ii) a combinatorial analysis of split matrices in to quantify Elkies-conditions through characteristic polynomials. They show convergence of all moments to those of the standard Gaussian, derive explicit asymptotics for the even moments, and obtain an asymptotic Gaussian behavior for the Elkies-prime counts on average over reductions, with implications for the SEA algorithm on abelian surfaces. The results extend prior work on elliptic curves to higher dimensions, providing new insights into the distribution of Elkies primes and confirming the practical viability of SEA-like point counting in the RM setting. The numerical experiments corroborate the Gaussian pattern and motivate further exploration of the constants across RM families.

Abstract

We generalize the notion of Elkies primes for elliptic curves to the setting of abelian varieties with real multiplication (RM), and prove the following. Let be an abelian variety with RM over a number field whose attached Galois representation has large image. Then the number of Elkies primes (in a suitable range) for reductions of modulo primes converges weakly to a Gaussian distribution around its expected value. This refines and generalizes results obtained by Shparlinski and Sutherland in the case of non-CM elliptic curves, and has implications for the complexity of the SEA point counting algorithm for abelian surfaces over finite fields.

Paper Structure

This paper contains 21 sections, 26 theorems, 113 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Setup
  3. Main results
  4. Organization
  5. Acknowledgments
  6. Statement on competing interests
  7. Data availability statement
  8. Galois representations and Elkies primes
  9. Torsion subgroups of abelian varieties with RM
  10. Elkies primes for abelian varieties with RM
  11. Elkies primes and the action of Frobenius
  12. Large Galois images
  13. Counting split matrices in GSp
  14. Characteristic polynomials and Elkies primes
  15. Estimating the size of S2h
  16. ...and 6 more sections

Key Result

Theorem 1.1

Assume the generalized Riemann hypothesis (GRH). Let $\mathcal{O}$ be an order in a totally real number field $K$ of degree $d$, and let $A$ be a polarized abelian variety of dimension $g\geq 1$ defined over a number field $F$ with RM by $\mathcal{O}$ with large Galois image. For a real number $L$, where $\Sigma_h$ denotes the set of unordered partitions of the integer $h = g/d$. Then, as $L,P\to

Figures (3)

  • Figure 1: Moment of order $2$ for $P \in [10^3, 5 \cdot 10^6]$
  • Figure 2: Distribution with $L=250$ and $P=10^7$
  • Figure 3: Evolution of the moment of order 2, with $P = 10^5$ and $L \in [20,500]$

Theorems & Definitions (54)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • ...and 44 more