Christopher Williams (Warwick)

will speak on

p-adic L-functions in higher dimensions

Time: 2:00PM
Date: Thu 19th November 2020
Location: Online [map]

Further information

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 ${\rm GL}_{2n}$.

(This talk is part of the Algebra and Number Theory series.)

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