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Uniform irreducibility of Galois action on the $\ell$-primary part of Abelian $3$-folds of Picard type

Mladen Dimitrov, Dinakar Ramakrishnan

TL;DR

This work extends Manin's uniform boundedness from elliptic curves to certain two-dimensional families of Picard-type abelian 3-folds with imaginary quadratic multiplication, establishing a uniform bound on the $ ext{ell}$-primary Galois action to preclude a full $ ext{k}$-rational flag in $A[ ext{ell}^r]$ for non-CM cases. The authors combine a detailed local analysis of endoscopic $ ext{U}(3)$ representations—via explicit invariant computations for $K''$, Iwahori, and hyperspecial subgroups—with a global automorphic/geometry framework on Picard modular surfaces, exploiting irregularity and root-number formulas. Central to the approach are the Shimura-surface moduli for Picard type, the Mumford–Tate framework for abelian 3-folds, and the use of Bombieri–Lang type finiteness results together with Cadoret–Tamagawa uniformity for abelian families to obtain uniform bounds independent of the particular member of the 2D family. The results deliver a genuine two-dimensional base analogue of uniform Galois-image irreducibility, with implications for uniform control of $ ext{ell}$-power torsion in Picard-type families and insights into the arithmetic of endoscopic phenomena on unitary groups.

Abstract

Half a century ago Manin showed that given a number field $k$ and a rational prime $\ell$, there exists a uniform bound for the order of cyclic $\ell$-power isogenies between two non-CM elliptic curves over $k$. We generalize this to certain $2$-dimensional families of abelian $3$-folds with multiplication by an imaginary quadratic field.

Uniform irreducibility of Galois action on the $\ell$-primary part of Abelian $3$-folds of Picard type

TL;DR

This work extends Manin's uniform boundedness from elliptic curves to certain two-dimensional families of Picard-type abelian 3-folds with imaginary quadratic multiplication, establishing a uniform bound on the -primary Galois action to preclude a full -rational flag in for non-CM cases. The authors combine a detailed local analysis of endoscopic representations—via explicit invariant computations for , Iwahori, and hyperspecial subgroups—with a global automorphic/geometry framework on Picard modular surfaces, exploiting irregularity and root-number formulas. Central to the approach are the Shimura-surface moduli for Picard type, the Mumford–Tate framework for abelian 3-folds, and the use of Bombieri–Lang type finiteness results together with Cadoret–Tamagawa uniformity for abelian families to obtain uniform bounds independent of the particular member of the 2D family. The results deliver a genuine two-dimensional base analogue of uniform Galois-image irreducibility, with implications for uniform control of -power torsion in Picard-type families and insights into the arithmetic of endoscopic phenomena on unitary groups.

Abstract

Half a century ago Manin showed that given a number field and a rational prime , there exists a uniform bound for the order of cyclic -power isogenies between two non-CM elliptic curves over . We generalize this to certain -dimensional families of abelian -folds with multiplication by an imaginary quadratic field.

Paper Structure

This paper contains 21 sections, 25 theorems, 71 equations.

Table of Contents

  1. Levels for endoscopic non-tempered representations of $\mathrm{U}(3)$
  2. The Bruhat-Tits tree of $\mathop{\mathrm{U}}\nolimits_3$
  3. Review of $L$-parameters and $A$-packets
  4. The Gross subgroup $K"$
  5. Higher Iwahori invariants via matrix coefficients
  6. Intertwining and an exponential sum
  7. Galois representations for $3$-folds of Picard type
  8. Abelian $3$-folds of Picard type and Tate modules
  9. Shimura surfaces and families of abelian threefolds of Picard type
  10. Étale fundamental groups and Mumford--Tate groups
  11. Galois stable lattices and rationality
  12. Lie images of Galois representations
  13. Irregularity for Picard modular surfaces
  14. A lemma on surfaces with isolated singularities
  15. A formula for the irregularity
  16. ...and 6 more sections

Key Result

Theorem A

Given a number field $k$, a prime number $\ell$ inert in $M$ and a finite set $S$ of places of $M$, there exists an integer $r=r(\ell,k,S)$ such that for any non-CM abelian $3$-fold $A$ over $k$ of Picard type which is semi-stable outside $S$, $A[\ell^r]$ does not contain a full $k$-rational flag.

Theorems & Definitions (46)

  • Theorem A
  • Theorem B
  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Definition 1.3
  • Proposition 1.4
  • proof
  • Lemma 1.5
  • ...and 36 more