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Twisted Triple Product $p$-adic $L$-function for Finite Slope Families of Hilbert Modular Forms

Ananyo Kazi

TL;DR

This work constructs a twisted triple product $p$-adic $L$-function for finite slope families of Hilbert modular forms with $p$ unramified in a totally real field $L$. Central to the approach is a robust $p$-adic interpolation framework built from overconvergent modular forms and de Rham data, organized by VBMS with marked sections, and guided by a refined Gauss--Manin iteration that converges across multiple embeddings. The authors develop interpolation sheaves $oldsymbol{w}_k^0$ and $oldsymbol{W}_k^0$, extend them to analytic families, and define a $p$-adic iteration $ abla^s$ compatible with Marked Splittings, partial Igusa towers, and Hecke actions. They prove an interpolation formula that expresses special values of the twisted triple product $L$-function in terms of $p$-adic periods and Petersson-type pairings, recovering previous Hida-family results in compatible limits. The construction broadens the scope of $p$-adic automorphic $L$-functions to finite slope Hilbert modular forms and paves the way for p-adic Gross–Zagier-type formulas with potential arithmetic applications, including links to generalized Abel–Jacobi maps and BSD-type conjectures in the totally real setting.

Abstract

Let $L$ be a totally real field, and $p$ be a rational prime that is unramified in $L$. We construct overconvergent families of classes of relative de Rham cohomology of the universal abelian scheme over Hilbert modular varieties associated to $L$. We show that these classes come equipped with Gauss-Manin connection. We prove convergence for $p$-adic iteration of this connection, improving upon a technique due to Andreatta-Iovita. We use this to construct a $p$-adic twisted triple product $L$-function associated to finite slope families of Hilbert modular forms, extending work of Blanco-Chacon-Fornea for Hida families.

Twisted Triple Product $p$-adic $L$-function for Finite Slope Families of Hilbert Modular Forms

TL;DR

This work constructs a twisted triple product -adic -function for finite slope families of Hilbert modular forms with unramified in a totally real field . Central to the approach is a robust -adic interpolation framework built from overconvergent modular forms and de Rham data, organized by VBMS with marked sections, and guided by a refined Gauss--Manin iteration that converges across multiple embeddings. The authors develop interpolation sheaves and , extend them to analytic families, and define a -adic iteration compatible with Marked Splittings, partial Igusa towers, and Hecke actions. They prove an interpolation formula that expresses special values of the twisted triple product -function in terms of -adic periods and Petersson-type pairings, recovering previous Hida-family results in compatible limits. The construction broadens the scope of -adic automorphic -functions to finite slope Hilbert modular forms and paves the way for p-adic Gross–Zagier-type formulas with potential arithmetic applications, including links to generalized Abel–Jacobi maps and BSD-type conjectures in the totally real setting.

Abstract

Let be a totally real field, and be a rational prime that is unramified in . We construct overconvergent families of classes of relative de Rham cohomology of the universal abelian scheme over Hilbert modular varieties associated to . We show that these classes come equipped with Gauss-Manin connection. We prove convergence for -adic iteration of this connection, improving upon a technique due to Andreatta-Iovita. We use this to construct a -adic twisted triple product -function associated to finite slope families of Hilbert modular forms, extending work of Blanco-Chacon-Fornea for Hida families.
Paper Structure (36 sections, 102 theorems, 191 equations)

This paper contains 36 sections, 102 theorems, 191 equations.

Table of Contents

  1. Introduction
  2. Application to construction of p-adic L-function
  3. Acknowledgements
  4. The setup
  5. The weight space
  6. Analyticity of the universal character.
  7. The weight space for the group G
  8. The Hilbert modular variety
  9. The Shimura variety associated to $G$
  10. Hilbert modular forms for the group G
  11. Adelic Hilbert modular forms
  12. Atkin--Lehner involution.
  13. The partial Igusa tower
  14. Splitting of de Rham sheaf
  15. p-adic interpolation of modular and de Rham sheaves
  16. ...and 21 more sections

Key Result

Theorem A

For suitable choice of $r$, there are interpolation sheaves $\mathfrak{w}^0_k$ and $\mathbb{W}^0_k$ on $\mathfrak{X}_r$, that interpolates modular forms and symmetric powers of de Rham classes for weight $k^0$ respectively. The sheaf $\mathbb{W}_{k}^0$ on $\mathfrak{X}_{r}$ is equipped with an incre

Theorems & Definitions (241)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem
  • Theorem
  • Theorem D
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3
  • ...and 231 more