Ellen Eischen
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Constructing vector-valued automorphic forms on unitary groups Open
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…
Automorphic Forms on Unitary Groups Open
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 …
Differential operators mod p : analytic continuation and consequences Open
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…
A p -adic Eisenstein measure for vector-weight automorphic forms Open
We construct a
A p-adic Eisenstein measure for unitary groups Open
We construct a
Entire Theta Operators at Unramified Primes Open
Starting with the work of Serre, Katz, and Swinnerton-Dyer, theta operators have played a key role in the study of $p$-adic and $\textrm{mod}\; p$ modular forms and Galois representations. This paper achieves two main results for theta ope…
A Gallery of Gaussian Periods Open
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 o…
Archimedean Zeta Integrals for Unitary Groups Open
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 Garre…
-ADIC EISENSTEIN SERIES and -FUNCTIONS of CERTAIN CUSP FORMS on DEFINITE UNITARY GROUPS Open
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 …
Differential Operators and Families of Automorphic Forms on Unitary Groups of Arbitrary Signature Open
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,…
View article: p-adic L-functions for unitary groups, part II: zeta-integral calculations
p-adic L-functions for unitary groups, part II: zeta-integral calculations Open
This paper completes key steps toward a construction of $p$-adic $L$-functions for unitary groups. More precisely, in 2006, the last three authors proposed an approach to constructing $p$-adic $L$-functions for unitary groups. Building on …