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Details of Grant 

EPSRC Reference: EP/T001615/2
Title: Constructions and properties of p-adic L-functions for GL(n)
Principal Investigator: Williams, Dr CD
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Researcher Co-Investigators:
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Department: Sch of Mathematical Sciences
Organisation: University of Nottingham
Scheme: EPSRC Fellowship
Starts: 15 December 2022 Ends: 30 December 2023 Value (£): 26,513
EPSRC Research Topic Classifications:
Algebra & Geometry
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
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Summary on Grant Application Form
L-functions are fundamental mathematical objects that encode deep arithmetic information. Their study goes back centuries, and they are the subject of the two biggest unsolved problems in modern number theory, namely the Riemann hypothesis and the Birch and Swinnerton-Dyer (BSD) conjecture.

The BSD conjecture predicts that the number of rational solutions of a cubic equation (defining an 'elliptic curve') is controlled by a value of an analytic L-function. This prediction, providing a mysterious bridge between the fields of arithmetic geometry and complex analysis, has since been hugely generalised in the Bloch-Kato conjectures. There has been much recent success in attacking such problems by changing the way we look at this bridge. In particular, by considering different notions of 'distance' between two numbers, we are able to build a whole array of different algebraic connections between arithmetic and analysis, and these have allowed us to build parts of the bridge required for BSD and Bloch-Kato.

The distance in question is the 'p-adic' distance, where two numbers are very close if their difference is very divisible by a prime p (for example, the numbers 1 and 1,000,000,001 are very close 2-adically, since their difference is divisible by 2 nine times). For each prime p, there should be a p-adic version of the Bloch-Kato conjectures - known as 'Iwasawa main conjectures' - and each of these gives another crucial connection between arithmetic and analysis. Such connections depend absolutely on the existence of p-adic versions of L-functions.

In addition to their utility in solving important conjectures, p-adic L-functions are beautiful objects in their own right. It is expected that for every L-function there is a p-adic version, but as they can be extremely difficult to construct, we are very far from reaching this goal.

The aim of this proposal is to extensively push forward our understanding of this p-adic picture by constructing new p-adic L-functions, drawing together novel techniques from algebraic topology, geometry and representation theory to attack fundamental but historically intractable cases. In particular, I will use powerful new methods developed in my recent research to give some of the first constructions for higher-dimensional automorphic forms.
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Organisation Website: http://www.nottingham.ac.uk