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We introduce a method for producing vector-valued automorphic forms on unitary groups from scalar-valued ones. As an application, we construct an explicit example. Our strategy employs certain differential operators. It is inspired by work of Cl\'ery and van der Geer in the setting of Siegel modular forms, but it also requires overcoming challenges that do not arise in the Siegel setting., Comment: 32 pages. Comments welcome

We prove algebraicity of critical values of certain Spin $L$-functions. More precisely, our results concern $L(s, \pi \otimes \chi, \mathrm{Spin})$ for cuspidal automorphic representations $\pi$ associated to a holomorphic Siegel eigenform on $\mathrm{GSp}_6$, Dirichlet characters $\chi$, and critical points $s$ to the right of the center of symmetry. We use the strategy of relating the $L$-values to properties of Eisenstein series, and a significant portion of the paper concerns the Fourier coefficients of these Eisenstein series. Unlike in prior algebraicity results following this strategy, our Eisenstein series are on a group $G$ that has no known moduli problem, and the $L$-functions are related to the Eisenstein series through a non-unique model., Comment: 60 pages. Comments welcome

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: A more detailed treatment of the material from my lecture series at the 2022 Arizona Winter School

Gaussian periods are certain sums of roots of unity whose study dates back to Gauss's seminal work in algebra and number theory. Recently, large scale plots of Gaussian periods have been revealed to exhibit striking visual patterns, some of which have been explored in the second named author's prior work. In 2020, the first named author produced a new app, \texttt{Gaussian periods}, which allows anyone to create these plots much more efficiently and at a larger scale than before. In this paper, we introduce Gaussian periods, present illustrations created with the new app, and summarize how mathematics controls some visual features, including colorings left unexplained in earlier work.

We derive precise formulas for the archimedean Euler factors occurring in certain standard Langlands $L$-functions for unitary groups. In the 1980s, Paul Garrett, as well as Ilya Piatetski-Shapiro and Stephen Rallis (independently of Garrett), discovered integral representations of automorphic $L$-functions that are Eulerian but, in contrast to the Rankin--Selberg and Langlands--Shahidi methods, do not require that the automorphic representations to which the $L$-functions are associated are globally generic. Their approach, the doubling method, opened the door to a variety of applications that could not be handled by prior methods. For over three decades, though, the integrals occurring in the Euler factors at archimedean places for unitary groups eluded precise computation, except under particular simplifications (such as requiring certain representations to be one-dimensional, as Garrett did in the first major progress on this computation and only prior progress for general signatures). We compute these integrals for holomorphic discrete series of general vector weights for unitary groups of any signature. This has consequences not only for special values of $L$-functions in the archimedean setting, but also for $p$-adic $L$-functions, where the corresponding term had remained open., Comment: Accepted for publication in the Journal f\"ur die reine und angewandte Mathematik (Crelle's Journal)

This paper concerns certain $\mod p$ differential operators that act on automorphic forms over Shimura varieties of type A or C. We show that, over the ordinary locus, these operators agree with the $\mod p$ reduction of the $p$-adic theta operators previously studied by some of the authors. In the characteristic $0$, $p$-adic case, there is an obstruction that makes it impossible to extend the theta operators to the whole Shimura variety. On the other hand, our $\mod p$ operators extend ("analytically continue", in the language of de Shalit and Goren) to the whole Shimura variety. As a consequence, motivated by their use by Edixhoven and Jochnowitz in the case of modular forms for proving the weight part of Serre's conjecture, we discuss some effects of these operators on Galois representations. Our focus and techniques differ from those in the literature. Our intrinsic, coordinate-free approach removes difficulties that arise from working with $q$-expansions and works in settings where earlier techniques, which rely on explicit calculations, are not applicable. In contrast with previous constructions and analytic continuation results, these techniques work for any totally real base field, any weight, and all signatures and ranks of groups at once, recovering prior results on analytic continuation as special cases., Comment: Accepted for publication in Algebra & Number Theory

This paper completes the construction of $p$-adic $L$-functions for unitary groups. More precisely, in 2006, the last three named authors proposed an approach to constructing such $p$-adic $L$-functions (Part I). Building on more recent results, including the first named author's construction of Eisenstein measures and $p$-adic differential operators, Part II of the present paper provides the calculations of local $\zeta$-integrals occurring in the Euler product (including at $p$). Part III of the present paper develops the formalism needed to pair Eisenstein measures with Hida families in the setting of the doubling method., Comment: 152 pages. Accepted for publication in Forum of Mathematics, Pi

In the 1970's, Serre exploited congruences between $q$-expansion coefficients of Eisenstein series to produce $p$-adic families of Eisenstein series and, in turn, $p$-adic zeta functions. Partly through integration with more recent machinery, including Katz's approach to $p$-adic differential operators, his strategy has influenced four decades of developments. Prior papers employing Katz's and Serre's ideas exploiting differential operators and congruences to produce families of automorphic forms rely crucially on $q$-expansions of automorphic forms. The overarching goal of the present paper is to adapt the strategy to automorphic forms on unitary groups, which lack $q$-expansions when the signature is of the form $(a, b)$, $a\neq b$. In particular, this paper completely removes the restrictions on the signature present in prior work. As intermediate steps, we achieve two key objectives. First, partly by carefully analyzing the action of the Young symmetrizer on Serre-Tate expansions, we explicitly describe the action of differential operators on the Serre-Tate expansions of automorphic forms on unitary groups of arbitrary signature. As a direct consequence, for each unitary group, we obtain congruences and families analogous to those studied by Katz and Serre. Second, via a novel lifting argument, we construct a $p$-adic measure taking values in the space of $p$-adic automorphic forms on unitary groups of any prescribed signature. We relate the values of this measure to an explicit $p$-adic family of Eisenstein series. One application of our results is to the recently completed construction of $p$-adic $L$-functions for unitary groups by the first named author, Harris, Li, and Skinner., Comment: 39 pages

We formulate and prove certain vanishing theorems for p-adic automorphic forms on unitary groups of arbitrary signature. The p-adic q-expansion principle for p-adic modular forms on the Igusa tower says that if the coefficients of (sufficiently many of) the q-expansions of a p-adic modular form f are zero, then f vanishes everywhere on the Igusa tower. There is no p-adic q-expansion principle for unitary groups of arbitrary signature in the literature. By replacing q-expansions with Serre-Tate expansions (expansions in terms of Serre-Tate deformation coordinates) and replacing modular forms with automorphic forms on unitary groups of arbitrary signature, we prove an analogue of the p-adic q-expansion principle. More precisely, we show that if the coefficients of (sufficiently many of) the Serre-Tate expansions of a p-adic automorphic form f on the Igusa tower (over a unitary Shimura variety) are zero, then f vanishes identically on the Igusa tower. This paper also contains a substantial expository component. In particular, the expository component serves as a complement to Hida's extensive work on p-adic automorphic forms., Comment: 36 pages, accepted for publication in Directions in Number Theory: Proceedings for the 2014 WIN3 workshop

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 birthday

We construct p-adic families of Klingen Eisenstein series and L-functions for cuspforms (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 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., Comment: Accepted for publication in the Journal of the Institute of Mathematics of Jussieu

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 Theory

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)

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).