2025 Theses Doctoral

𝑝-adic L-functions for 𝑃-ordinary Hida families on unitary groups

Marcil, David

We construct a 𝑝-adic L-function for 𝑃-ordinary Hida families of cuspidal automorphic representations on a unitary group G. The main new idea of our work is to incorporate the theory of Schneider-Zink types for the Levi quotient of 𝑃, to allow for the possibility of higher ramification at primes dividing p, into the study of (𝑝-adic) modular forms and automorphic representations on G. For instance, we describe the local structure of such a 𝑃-ordinary automorphic representation 𝜋 at 𝑝 using these types, allowing us to analyze the geometry of 𝑃-ordinary Hida families.

Furthermore, these types play a crucial role in the construction of certain Siegel Eisenstein series designed to be compatible with such Hida families in two specific ways : Their Fourier coefficients can be p-adically interpolated into a 𝑝-adic Eisenstein measure on 𝑑+1 variables and, via the doubling method of Garrett and Piatetski--Shapiro-Rallis, the corresponding zeta integrals yield special values of standard 𝐿-functions. Here, 𝑑 is the rank of the Levi quotient of 𝑃. Lastly, the doubling method is reinterpreted algebraically as a pairing between modular forms on 𝐺, whose nebentype are types, and viewed as the evaluation of our 𝑝-adic 𝐿-function at classical points of a 𝑃-ordinary Hida family.