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$p$-adic Asai and twisted triple product $L$-functions for finite slope families

Ananyo Kazi, David Loeffler

Abstract

We define a two-variable $p$-adic Asai $L$-function for a finite-slope family of Hilbert modular forms over a real quadratic field (with one component of the weight, and the cyclotomic twist variable, varying independently); and a two-variable ``twisted triple product'' $L$-function, interpolating the central $L$-value of the tensor product of such a family with a family of elliptic modular forms. The former construction generalizes a construction due to Grossi, Zerbes and the second author for ordinary families; the latter is a counterpart of the twisted triple product $L$-function of arXiv:2401.13230, but differs in that it interpolates classical $L$-values in a different range of weights, in which the dominant weight comes from the Hilbert modular form. Our construction relies on a ``nearly-overconvergent'' version of higher Coleman theory for Hilbert modular surfaces.

$p$-adic Asai and twisted triple product $L$-functions for finite slope families

Abstract

We define a two-variable -adic Asai -function for a finite-slope family of Hilbert modular forms over a real quadratic field (with one component of the weight, and the cyclotomic twist variable, varying independently); and a two-variable ``twisted triple product'' -function, interpolating the central -value of the tensor product of such a family with a family of elliptic modular forms. The former construction generalizes a construction due to Grossi, Zerbes and the second author for ordinary families; the latter is a counterpart of the twisted triple product -function of arXiv:2401.13230, but differs in that it interpolates classical -values in a different range of weights, in which the dominant weight comes from the Hilbert modular form. Our construction relies on a ``nearly-overconvergent'' version of higher Coleman theory for Hilbert modular surfaces.

Paper Structure

This paper contains 31 sections, 30 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. General conventions
  3. $\mathfrak{p}_1$-overconvergent cohomology at prime-to-$p$ level
  4. Setup: Shimura varieties and models
  5. Shimura varieties over $\mathbb{Q}$
  6. Moduli spaces
  7. Automorphic sheaves
  8. Descent data
  9. Hodge height and Hasse invariants on $X$
  10. Hasse invariants
  11. Partial canonical subgroups
  12. Integral structures on modular and de Rham sheaves
  13. Interpolation sheaves for overconvergent and nearly overconvergent forms
  14. An explicit model of the sheaves
  15. Comparison with classical sheaves
  16. ...and 16 more sections

Key Result

Theorem 3.3

Let $n \geqslant 1$. If $v < \frac{1}{p^{n+1}}$, the $p$-divisible group $\mathcal{A}[\mathfrak{p}_1^{\infty}]$ over $\mathfrak{X}_v$ admits a canonical subgroup $H^{\mathop{\mathrm{can}}\nolimits}_n$ of level $n$ that lifts $\ker F_{\mathfrak{p}_1}^n := \ker F^n \cap \mathcal{A}[\mathfrak{p}_1^n]$

Theorems & Definitions (74)

  • Remark 3.1
  • Definition 3.2
  • Theorem 3.3: see Andreatta2018leHS
  • Remark 3.4
  • Theorem 3.5
  • proof
  • Corollary 3.6
  • Corollary 3.7
  • Lemma 3.8
  • proof
  • ...and 64 more