A p-adic Eisenstein measure for vector-weight automorphic forms

We construct a p-adic Eisenstein measure with values in the space of vector-weight p-adic automorphic forms on certain unitary groups. This measure allows us to p-adically interpolate special values of certain vector-weight C-infinity automorphic forms, including Eisenstein series, as their weights vary. We also explain how to extend our methods to the case of Siegel modular forms and how to recover Nicholas Katz's p-adic families of Eisenstein series for Hilbert modular forms.Comment: Accepted for publication in Algebra & Number Theor

Differential operators, pullbacks, and families of automorphic forms

This paper has two main parts. First, we construct certain differential operators, which generalize operators studied by G. Shimura. Then, as an application of some of these differential operators, we construct certain p-adic families of automorphic forms. Building on the author's earlier work, these differential operators map automorphic forms on a unitary group of signature (n,n) to (vector-valued) automorphic forms on the product $U^\varphi\times U^{-\varphi}$ of two unitary groups, where $U^\varphi$ denotes the unitary group associated to a Hermitian form $\varphi$ of arbitrary signature on an n-dimensional vector space. These differential operators have both a p-adic and a C-infinity incarnation. In the scalar-weight, C-infinity case, these operators agree with ones studied by Shimura. In the final section of the paper, we also discuss some generalizations to other groups and settings. The results from this paper apply to the author's paper-in-preparation with J. Fintzen, E. Mantovan, and I. Varma and to her ongoing joint project with M. Harris, J. -S. Li, and C. Skinner; they also relate to her recent paper with X. Wan.Comment: Accepted for publication in special issue of Annales Mathematiques du Quebec in honor of Glenn Stevens's sixtieth birthda

A p-adic Eisenstein measure for unitary groups

We construct a p-adic Eisenstein measure with values in the space of p-adic automorphic forms on certain unitary groups. Using this measure, we p-adically interpolate certain special values of both holomorphic and non-holomorphic Eisenstein series, as both the archimedean and the p-adic weights of the Eisenstein series vary.Comment: Accepted for publication in the Journal f\"ur die reine und angewandte Mathematik (Crelle's Journal

Automorphic Forms on Unitary Groups

This manuscript provides a more detailed treatment of the material from my lecture series at the 2022 Arizona Winter School on Automorphic Forms Beyond $GL_2$. The main focus of this manuscript is automorphic forms on unitary groups, with a view toward algebraicity results. I also discuss related aspects of automorphic $L$-functions in the setting of unitary groups.Comment: 60-page manuscript that gives a more detailed treatment of the material from my lecture series at the 2022 Arizona Winter Schoo

p-adic Differential Operators on Automorphic Forms on Unitary Groups

The goal of this paper is to study certain p-adic differential operators on automorphic forms on U(n,n). These operators are a generalization to the higher-dimensional, vector-valued situation of the p-adic differential operators constructed for Hilbert modular forms by N. Katz. They are a generalization to the p-adic case of the C^{\infty}-differential operators first studied by H. Maass and later studied extensively by M. Harris and G. Shimura. The operators should be useful in the construction of certain p-adic L-functions attached to p-adic families of automorphic forms on the unitary groups U(n) x U(n)

A BRIEF REPORT ON pp-ADIC SPIN LL-FUNCTIONS FOR GSp6_{6} (Automorphic forms, Automorphic representations, Galois representations, and its related topics)

This document summarizes the content of the author's presentation at the remote RIMS conference "Automorphic forms, Automorphic representations, Galois representations, and its related topics." In particular, we report on a paper-inpreparation (joint with S. Shah and G. Rosso) on p-adic Spin L-functions for GSp₆

-ADIC EISENSTEIN SERIES and -FUNCTIONS of CERTAIN CUSP FORMS on DEFINITE UNITARY GROUPS

We construct $p$ -adic families of Klingen–Eisenstein series and $L$ -functions for cusp forms (not necessarily ordinary) unramified at an odd prime $p$ on definite unitary groups of signature $(r,0)$ (for any positive integer $r$ ) for a quadratic imaginary field ${\mathcal{K}}$ split at $p$ . When $r=2$ , we show that the constant term of the Klingen–Eisenstein family is divisible by a certain $p$ -adic $L$ -function