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P-adic Asai L-functions for quadratic Hilbert eigenforms

Giada Grossi, David Loeffler, Sarah Livia Zerbes

TL;DR

This work constructs $p$-adic Asai $L$-functions for real-quadratic base fields, employing a partially ordinary version of higher Hida theory on Hilbert modular surfaces with Iwahori level at a single prime above $p$. It develops both one- and two-variable $p$-adic $L$-functions via interpolations of coherent cohomology classes, Igusa towers, and automorphic Eisenstein data, and it provides a detailed analysis of local zeta integrals to realize the Euler-factor corrections. The main contributions include a robust framework for interpolating eigenfamilies through a two-variable eigenvariety, explicit $p$-adic interpolation of zeta-integrals through Eisenstein measures, and a precise matching of local and global factors that yields interpolated critical values of Asai $L$-functions. The results advance the $p$-adic Langlands program for $ ext{GL}_2$ over real quadratic fields, with potential applications to explicit reciprocity laws and Iwasawa theory for Asai motives. The constructions combine higher Hida theory with geometric and automorphic input to interpolate critical data across $p$-adic families, enabling explicit $p$-adic measures that encode Asai $L$-values in families and across varying weights and levels.

Abstract

We construct p-adic Asai L-functions for cuspidal automorphic representations of GL2 / F, where F is a real quadratic field in which p splits. Our method relies on higher Hida theory for Hilbert modular surfaces with Iwahori level at one prime above p.

P-adic Asai L-functions for quadratic Hilbert eigenforms

TL;DR

This work constructs -adic Asai -functions for real-quadratic base fields, employing a partially ordinary version of higher Hida theory on Hilbert modular surfaces with Iwahori level at a single prime above . It develops both one- and two-variable -adic -functions via interpolations of coherent cohomology classes, Igusa towers, and automorphic Eisenstein data, and it provides a detailed analysis of local zeta integrals to realize the Euler-factor corrections. The main contributions include a robust framework for interpolating eigenfamilies through a two-variable eigenvariety, explicit -adic interpolation of zeta-integrals through Eisenstein measures, and a precise matching of local and global factors that yields interpolated critical values of Asai -functions. The results advance the -adic Langlands program for over real quadratic fields, with potential applications to explicit reciprocity laws and Iwasawa theory for Asai motives. The constructions combine higher Hida theory with geometric and automorphic input to interpolate critical data across -adic families, enabling explicit -adic measures that encode Asai -values in families and across varying weights and levels.

Abstract

We construct p-adic Asai L-functions for cuspidal automorphic representations of GL2 / F, where F is a real quadratic field in which p splits. Our method relies on higher Hida theory for Hilbert modular surfaces with Iwahori level at one prime above p.
Paper Structure (66 sections, 32 theorems, 130 equations)

This paper contains 66 sections, 32 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Higher Hida theory for $\mathfrak{p}_1$-ordinary cohomology
  3. One-variable $p$-adic $L$-functions
  4. Two-variable $p$-adic $L$-functions
  5. Relation to other work
  6. Higher Hida theory for p1-ordinary cohomology
  7. Preliminaries
  8. Hilbert modular surfaces
  9. Moduli spaces
  10. Integral models
  11. Automorphic sheaves
  12. Geometry of the special fibre for $n=0$
  13. Geometry of the special fibre for $n=1$
  14. Classicality at spherical level
  15. The $T_{\mathfrak{p}_1}$-operator
  16. ...and 51 more sections

Key Result

Theorem A

The exists a perfect complex $M_G^\bullet(k_1 + 2\kappa, k_2, w)$ of modules over $\Lambda = \mathbb{Z}_p[[\mathbb{Z}_p^\times]]$, with an action of prime-to-$p$ Hecke operators, such that for all integers $a \in \mathbb{Z}$ with $k_1 + 2a \geqslant 2$, we have a Hecke-equivariant isomorphism where $\kappa$ denotes the universal character of $\Lambda$. There is also an analogous complex $M_G^\bul

Theorems & Definitions (81)

  • Theorem A
  • Remark 1
  • Theorem B
  • Remark 2
  • Definition 1.2.2
  • Theorem C
  • Theorem D
  • Remark 1.3.1
  • Remark 2.0.1
  • Remark 2.1.1
  • ...and 71 more