Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Adjoint $L$-functions, congruence ideals, and Selmer groups over $\mathrm{GL}_n$

Ho Leung Fong

TL;DR

The paper develops a precise bridge between special values of adjoint L-functions on GL$_n$ and congruence phenomena in automorphic cohomology. By embedding automorphic representations into cohomology and exploiting cup-product pairings, the authors show that, under natural hypotheses, the cohomological congruence ideal is governed by a normalized adjoint L-value $L^{alg}(1,\pi,\mathrm{Ad}^0,\epsilon)$. This yields concrete congruence criteria for automorphic representations and, in CM fields, a lower bound for certain Selmer groups via Galois deformation theory, relating arithmetic invariants to adjoint L-values. The work generalizes prior GL$_2$ results to GL$_n$, integrates Betti-Whittaker periods, and provides a framework connecting automorphic congruences, period data, and Selmer groups with potential implications for Bloch–Kato-type conjectures.

Abstract

In this paper, we relate $L(1,π,\mathrm{Ad}^\circ)$ to the congruence ideals for cohomological cuspidal automorphic representations $π$ of $\mathrm{GL}_n$ over any number field. We then use this result to deduce relationships between the congruences of automorphic forms and adjoint $L$-functions. For CM fields, we apply the result to obtain a lower bound on the cardinality of certain Selmer groups in terms of $L(1,π,\mathrm{Ad}^\circ)$.

Adjoint $L$-functions, congruence ideals, and Selmer groups over $\mathrm{GL}_n$

TL;DR

The paper develops a precise bridge between special values of adjoint L-functions on GL and congruence phenomena in automorphic cohomology. By embedding automorphic representations into cohomology and exploiting cup-product pairings, the authors show that, under natural hypotheses, the cohomological congruence ideal is governed by a normalized adjoint L-value . This yields concrete congruence criteria for automorphic representations and, in CM fields, a lower bound for certain Selmer groups via Galois deformation theory, relating arithmetic invariants to adjoint L-values. The work generalizes prior GL results to GL, integrates Betti-Whittaker periods, and provides a framework connecting automorphic congruences, period data, and Selmer groups with potential implications for Bloch–Kato-type conjectures.

Abstract

In this paper, we relate to the congruence ideals for cohomological cuspidal automorphic representations of over any number field. We then use this result to deduce relationships between the congruences of automorphic forms and adjoint -functions. For CM fields, we apply the result to obtain a lower bound on the cardinality of certain Selmer groups in terms of .
Paper Structure (14 sections, 24 theorems, 139 equations)

This paper contains 14 sections, 24 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Strategy
  3. Notation
  4. Cohomology
  5. Cohomology and Hecke operators
  6. Regular algebraic automorphic representations
  7. More on cohomology
  8. Abstract congruence ideals
  9. L(1,pi,Ad 0) and congruences for automorphic representations
  10. Hecke eigensystems
  11. Congruence ideals for automorphic representations
  12. Betti-Whittaker periods
  13. L(1,pi,Ad 0) and congruence ideals
  14. Selmer groups

Key Result

Proposition 2.2

KT If $U$ is a neatIn KT, this is stated for good subgroup, but the proof actually works for all neat subgroups. subgroup, then where and $C_{{\mathbb A},\bullet}$ is the group of singular chains on $X\times G(\mathbb A_F^\infty)$ with $\mathbb Z$ coefficients.

Theorems & Definitions (74)

  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Lemma 2.8
  • proof
  • Lemma 2.9
  • ...and 64 more