p-adic L-functions in higher dimensions

Abstract: There are lots of theorems and conjectures relating special values of complex analytic L-functions to arithmetic data; for example, celebrated examples include the class number formula and the BSD conjecture. These conjectures predict a surprising (complex) bridge between the fields of analysis and arithmetic. However, these conjectures are extremely difficult to prove. Most recent progress has come from instead trying to build analogous $p$-adic bridges, constructing a $p$-adic version of the $L$-function and relating it to $p$-adic arithmetic data via "Iwasawa main conjectures". From the $p$-adic bridge, one can deduce special cases of the complex bridge; this strategy has, for example, led to the current state-of-the-art results towards the BSD conjecture.

Essential in this strategy is the construction of a $p$-adic L-function. In this talk I will give an introduction to $p$-adic L-functions, focusing first on the $p$-adic analogue of the Riemann zeta function (the case of ${\rm GL}_1$), then describing what one expects in a more general setting. At the end of the talk I will state some recent results from joint work with Daniel Barrera and Mladen Dimitrov on the construction of $p$-adic L-functions for certain automorphic representations of ${\rm GL}_{2n}$.

number theory

Audience: researchers in the topic

Dublin Algebra and Number Theory Seminar

Series comments: Passcode: The 3-digit prime numerator of Riemann zeta at -11