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Unitary Friedberg--Jacquet periods and anticyclotomic p-adic L-functions

Andrew Graham

TL;DR

<3-5 sentence high-level summary> This work constructs a higher-dimensional p-adic L-function interpolating unitary Friedberg--Jacquet periods by integrating p-adic variation of Maass--Shimura differential operators into the framework of nearly overconvergent automorphic forms on unitary Shimura varieties. It develops a robust analytic and geometric toolkit—abstract branching laws, higher Coleman theory, and p-adic operator interpolation—to realize coherent-cohomology pairings as p-adically interpolable inputs. The main results provide an explicit interpolation formula with a p-adic Euler-type factor, extendable to Coleman families, and conjecturally linked to Euler systems and Bloch–Kato phenomena in anticyclotomic twists. This framework generalizes prior work for modular and Hilbert modular cases to a genuinely higher-rank unitary setting, with potential applications to explicit reciprocity laws and regulator formulas for special cycles.

Abstract

We extend the construction of the $p$-adic $L$-function interpolating unitary Friedberg--Jacquet periods in previous work of the author to include the $p$-adic variation of Maass--Shimura differential operators. In particular, we develop a theory of nearly overconvergent automorphic forms in higher degrees of coherent cohomology for unitary Shimura varieties generalising previous work for modular curves. The construction of this $p$-adic $L$-function can be viewed as a higher-dimensional generalisation of the work of Bertolini--Darmon--Prasanna and Castella--Hsieh, and the inclusion of this extra variable arising from the $p$-adic iteration of differential operators will play a key role in relating values of this $p$-adic $L$-function to $p$-adic regulators of special cycles on unitary Shimura varieties.

Unitary Friedberg--Jacquet periods and anticyclotomic p-adic L-functions

TL;DR

<3-5 sentence high-level summary> This work constructs a higher-dimensional p-adic L-function interpolating unitary Friedberg--Jacquet periods by integrating p-adic variation of Maass--Shimura differential operators into the framework of nearly overconvergent automorphic forms on unitary Shimura varieties. It develops a robust analytic and geometric toolkit—abstract branching laws, higher Coleman theory, and p-adic operator interpolation—to realize coherent-cohomology pairings as p-adically interpolable inputs. The main results provide an explicit interpolation formula with a p-adic Euler-type factor, extendable to Coleman families, and conjecturally linked to Euler systems and Bloch–Kato phenomena in anticyclotomic twists. This framework generalizes prior work for modular and Hilbert modular cases to a genuinely higher-rank unitary setting, with potential applications to explicit reciprocity laws and regulator formulas for special cycles.

Abstract

We extend the construction of the -adic -function interpolating unitary Friedberg--Jacquet periods in previous work of the author to include the -adic variation of Maass--Shimura differential operators. In particular, we develop a theory of nearly overconvergent automorphic forms in higher degrees of coherent cohomology for unitary Shimura varieties generalising previous work for modular curves. The construction of this -adic -function can be viewed as a higher-dimensional generalisation of the work of Bertolini--Darmon--Prasanna and Castella--Hsieh, and the inclusion of this extra variable arising from the -adic iteration of differential operators will play a key role in relating values of this -adic -function to -adic regulators of special cycles on unitary Shimura varieties.
Paper Structure (96 sections, 85 theorems, 553 equations, 1 figure)

This paper contains 96 sections, 85 theorems, 553 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Statement of the main results
  3. Results in families
  4. Expected relation with Euler systems
  5. Method of proof
  6. Structure of the article
  7. Acknowledgements
  8. Notation and conventions
  9. Abstract computations
  10. Recap on the groups and notation
  11. Algebraic and analytic groups
  12. Algebraic and locally analytic weights
  13. Branching laws
  14. Classical branching laws
  15. Differential operators
  16. ...and 81 more sections

Key Result

Theorem 1

There exists a locally analytic distribution $\mathscr{L}_{p, \phi}(\pi, -) \in \mathscr{D}^{\operatorname{la}}(\mathop{\mathrm{Gal}}\nolimits(F_{p^{\infty}}/F), L)$ such that for any $\chi \in \Sigma_{\pi}$ where:

Figures (1)

  • Figure 1: Regions of twists when $[F^+:\mathbb{Q}] = 2$.

Theorems & Definitions (267)

  • Remark 1.0.1
  • Theorem 1
  • Remark 1.1.2
  • Remark 1.1.3
  • Remark 1.1.4
  • Remark 1.1.5
  • Theorem 2
  • Remark 1.1.7
  • Definition 2.1.2
  • Lemma 2.1.3
  • ...and 257 more