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Ordinary primes for $\mathrm{GL}_2$-type abelian varieties and weight $2$ modular forms

Tian Wang, Pengcheng Zhang

TL;DR

The article proves positive-density results for ordinary primes in families of GL$_2$-type abelian varieties and weight $2$ modular forms, extending known cases beyond $g=1,2$ to higher dimensions under endomorphism-field hypotheses. It develops a unified approach via compatible systems of Galois representations, leveraging prime-splitting in endomorphism fields and Chebotarev-type arguments to identify a set of primes with ordinary Frobenius data. The work yields concrete instances over $Q$ and quadratic fields, including when $ ext{End}_F(A) ensorQ$ contains a degree $g$ field $K$, and translates these results to modular forms through Eichler–Shimura and big-image phenomena. The findings bolster Serre–Ogus-type conjectures on the density of ordinary primes and have implications for $p$-adic and Iwasawa-theoretic aspects of abelian varieties and weight $2$ modular forms.

Abstract

Let $A$ be a $g$-dimensional abelian variety defined over a number field $F$. It is conjectured that the set of ordinary primes of $A$ over $F$ has positive density, and this is known to be true when $g=1, 2$, or for certain abelian varieties with extra endomorphisms. In this paper, we extend the family of abelian varieties whose sets of ordinary primes have positive density. Specifically, we show that if the endomorphism algebra of $A$ contains a number field $K$ of degree $g$, then under certain conditions on the fields $F$ and $K$, the set of ordinary primes of $A$ over $F$ has positive density. This includes $\mathrm{GL}_2$-type abelian varieties over $\mathbb{Q}$ (resp. quadratic number fields) of dimension $q$ or $2q$ (resp. $q$) for any rational prime $q$. The proof is carried out in the general setting of compatible systems of Galois representations, and as a consequence, it also implies a positive density result for the sets of ordinary primes of certain modular forms of weight $2$.

Ordinary primes for $\mathrm{GL}_2$-type abelian varieties and weight $2$ modular forms

TL;DR

The article proves positive-density results for ordinary primes in families of GL-type abelian varieties and weight modular forms, extending known cases beyond to higher dimensions under endomorphism-field hypotheses. It develops a unified approach via compatible systems of Galois representations, leveraging prime-splitting in endomorphism fields and Chebotarev-type arguments to identify a set of primes with ordinary Frobenius data. The work yields concrete instances over and quadratic fields, including when contains a degree field , and translates these results to modular forms through Eichler–Shimura and big-image phenomena. The findings bolster Serre–Ogus-type conjectures on the density of ordinary primes and have implications for -adic and Iwasawa-theoretic aspects of abelian varieties and weight modular forms.

Abstract

Let be a -dimensional abelian variety defined over a number field . It is conjectured that the set of ordinary primes of over has positive density, and this is known to be true when , or for certain abelian varieties with extra endomorphisms. In this paper, we extend the family of abelian varieties whose sets of ordinary primes have positive density. Specifically, we show that if the endomorphism algebra of contains a number field of degree , then under certain conditions on the fields and , the set of ordinary primes of over has positive density. This includes -type abelian varieties over (resp. quadratic number fields) of dimension or (resp. ) for any rational prime . The proof is carried out in the general setting of compatible systems of Galois representations, and as a consequence, it also implies a positive density result for the sets of ordinary primes of certain modular forms of weight .

Paper Structure

This paper contains 16 sections, 30 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Lemmas on primes and number fields
  3. Correspondence between places and embeddings
  4. Splitting of primes
  5. Extensions of degree $q$ or $2q$
  6. Results on Galois representations
  7. Galois representations of abelian varieties
  8. Galois representations of modular forms
  9. Proof of \ref{['inert-in-K-theorem']}
  10. Proof of \ref{['galois-rep-theorem']}
  11. A proof assuming \ref{['density-of-S(c)']}
  12. Density of $S(c)$
  13. Applications of \ref{['galois-rep-theorem']} and \ref{['inert-in-K-theorem']}
  14. Applications to abelian varieties
  15. Application to modular forms
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $A/\mathbb{Q}$ be an abelian variety of dimension $q$ (resp. $2q$) for some rational prime $q$. Assume that $\operatorname{End}_{\mathbb{Q}}(A)\otimes\mathbb{Q}$ contains a number field $K$ of degree $q$ (resp. $2q$). Then, the set of ordinary primes of $A$ over $\mathbb{Q}$ has positive density

Theorems & Definitions (65)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Lemma 2.1
  • proof
  • ...and 55 more