Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Euler characteristic of $\ell$-adic local systems on $\mathcal{A}_n$

Olivier Taïbi

TL;DR

This work analyzes the Euler characteristics of $oldsymbol{ ext{ell-adic}}$ local systems on the moduli stack $oldsymbol{\mathcal{A}_n}$ of principally polarized abelian varieties, linking them to $oldsymbol{ ext{GSpin}}$-valued Galois representations attached to level-one Siegel modular forms. It advances the Ihara–Langlands–Kottwitz program by unconditionally reformulating Kottwitz’s conjecture in the Siegel level-one setting, employing Morel’s intersection-cohomology framework and Franke’s spectral sequence to relate IH to compactly supported cohomology, and using endoscopy to stabilize spectral expansions. A major contribution is the unconditional construction of $ ext{GSpin}$-valued Galois representations for level-one Siegel cusp forms (both odd and even spin cases), with precise local-global compatibility statements and explicit tensor decompositions of IH in terms of spin factors. The paper also delivers genus-by-genus point-count formulas over finite fields for small $n$ and proves a higher-genus analogue of Bergström–Faber–van der Geer’s conjecture at the level of $oldsymbol{ ext{l-adic}}$ realizations, while clarifying the limitations of expressing all exterior motives in terms of lower-dimensional data for large $n$. These results deepen the interplay between automorphic forms, Galois representations, and the arithmetic of Siegel modular varieties, with potential motivic interpretations and consequences for explicit arithmetic geometry.

Abstract

We study the Euler characteristic of $\ell$-adic local systems on the moduli stack $\mathcal{a}_n$ of principally polarized abelian varieties of dimension $n$ associated to algebraic representations of $\mathbf{GSp}_{2n}$, as virtual representations of the absolute Galois group of $\mathbb{Q}$ and the unramified Hecke algebra of $\mathbf{GSp}_{2n}$. To this end we take the last steps of the Ihara-Langlands-Kottwitz method to compute the intersection cohomology of minimal compactifications of Siegel modular varieties in level one, following work of Kottwitz and Morel, proving an unconditional reformulation of Kottwitz' conjecture in this case. This entails proving the existence of $\mathrm{GSpin}$-valued Galois representations associated to certain level one automorphic representations for $\mathbf{PGSp}_{2n}$ and $\mathbf{SO}_{4n}$. As a consequence we prove the existence of $\mathrm{GSpin}$-valued Galois representations associated to level one Siegel eigenforms, a higher genus analogue of theorems of Deligne (genus one) and Weissauer (genus two). Using Morel's work and Franke's spectral sequence we derive explicit formulas expressing the Euler characteristic of compactly supported cohomology of automorphic $\ell$-adic local systems on Siegel modular varieties in terms of intersection cohomology. Specializing to genus three and level one, we prove an explicit conjectural formula of Bergström, Faber and van der Geer for the compactly supported Euler characteristic in terms of spin Galois representations associated to level one Siegel cusp forms. Specializing to trivial local systems we give explicit formulas for the number of points of $\mathcal{A}_n$ over finite fields for all $n \leq 7$.

The Euler characteristic of $\ell$-adic local systems on $\mathcal{A}_n$

TL;DR

This work analyzes the Euler characteristics of local systems on the moduli stack of principally polarized abelian varieties, linking them to -valued Galois representations attached to level-one Siegel modular forms. It advances the Ihara–Langlands–Kottwitz program by unconditionally reformulating Kottwitz’s conjecture in the Siegel level-one setting, employing Morel’s intersection-cohomology framework and Franke’s spectral sequence to relate IH to compactly supported cohomology, and using endoscopy to stabilize spectral expansions. A major contribution is the unconditional construction of -valued Galois representations for level-one Siegel cusp forms (both odd and even spin cases), with precise local-global compatibility statements and explicit tensor decompositions of IH in terms of spin factors. The paper also delivers genus-by-genus point-count formulas over finite fields for small and proves a higher-genus analogue of Bergström–Faber–van der Geer’s conjecture at the level of realizations, while clarifying the limitations of expressing all exterior motives in terms of lower-dimensional data for large . These results deepen the interplay between automorphic forms, Galois representations, and the arithmetic of Siegel modular varieties, with potential motivic interpretations and consequences for explicit arithmetic geometry.

Abstract

We study the Euler characteristic of -adic local systems on the moduli stack of principally polarized abelian varieties of dimension associated to algebraic representations of , as virtual representations of the absolute Galois group of and the unramified Hecke algebra of . To this end we take the last steps of the Ihara-Langlands-Kottwitz method to compute the intersection cohomology of minimal compactifications of Siegel modular varieties in level one, following work of Kottwitz and Morel, proving an unconditional reformulation of Kottwitz' conjecture in this case. This entails proving the existence of -valued Galois representations associated to certain level one automorphic representations for and . As a consequence we prove the existence of -valued Galois representations associated to level one Siegel eigenforms, a higher genus analogue of theorems of Deligne (genus one) and Weissauer (genus two). Using Morel's work and Franke's spectral sequence we derive explicit formulas expressing the Euler characteristic of compactly supported cohomology of automorphic -adic local systems on Siegel modular varieties in terms of intersection cohomology. Specializing to genus three and level one, we prove an explicit conjectural formula of Bergström, Faber and van der Geer for the compactly supported Euler characteristic in terms of spin Galois representations associated to level one Siegel cusp forms. Specializing to trivial local systems we give explicit formulas for the number of points of over finite fields for all .

Paper Structure

This paper contains 93 sections, 132 theorems, 572 equations.

Table of Contents

  1. Introduction
  2. Spin Galois representations for level one Siegel modular forms
  3. A conjecture of Bergström, Faber and van der Geer
  4. Intersection cohomology of local systems on the minimal compactification of $\mathcal{A}_n$ and $\mathrm{GSpin}$-valued Galois representations
  5. Kottwitz' conjecture in general
  6. The Ihara-Langlands-Kottwitz method
  7. The case of Siegel modular varieties
  8. Adding the level one assumption
  9. $\mathrm{GSpin}$-valued Galois representations
  10. Tensor product decompositions in intersection cohomology
  11. Compactly supported cohomology
  12. Acknowledgments
  13. Notations and conventions
  14. Class field theory
  15. Reductive groups and root data
  16. ...and 78 more sections

Key Result

Theorem 1

For $1 \leq n \leq 6$ and any prime number $\ell$ the virtual representation $e_c(\mathcal{A}_{n,\overline{\mathbb{Q}}}, \mathbb{Q}_{\ell})$ of $\mathrm{Gal}_{\mathbb{Q}}$ (forgetting the Hecke action) is Tate, equivalently there exists a polynomial $P_n \in \mathbb{Z}[X]$ such that for any prime po For $n=7$ this Euler characteristic is not Tate, more precisely for any prime power $q=p^m$ we have

Theorems & Definitions (322)

  • Theorem 1
  • Theorem 2: Corollary \ref{['cor:GSpin_Gal_rep_Siegel']}
  • Theorem 3: Theorem \ref{['thm:BFG']}
  • Conjecture 1: Kottwitz' conjecture Kottwitz_AA, in our simplified setting
  • Theorem 4: weaker, vague version of Theorem \ref{['theo:IH_explicit_crude']}
  • Theorem 5: Theorem \ref{['theo:odd_spin_rep']}
  • Theorem 6: Theorems \ref{['thm:loc_glob_param_SO_even']} and \ref{['theo:even_GSpin_Gal_rep']}
  • Theorem 7: Corollary \ref{['coro:Euler_ord_GL']}
  • Theorem 1.4.1: Theorem \ref{['theo:Hc_from_IH']}
  • Definition 3.1.1
  • ...and 312 more