EPSRC Reference: 
EP/T001615/1 
Title: 
Constructions and properties of padic Lfunctions for GL(n) 
Principal Investigator: 
Williams, Dr CD 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics 
Organisation: 
University of Warwick 
Scheme: 
EPSRC Fellowship 
Starts: 
01 October 2019 
Ends: 
30 September 2022 
Value (£): 
310,002

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
Lfunctions 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 SwinnertonDyer (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 Lfunction. This prediction, providing a mysterious bridge between the fields of arithmetic geometry and complex analysis, has since been hugely generalised in the BlochKato 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 BlochKato.
The distance in question is the 'padic' 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 2adically, since their difference is divisible by 2 nine times). For each prime p, there should be a padic version of the BlochKato 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 padic versions of Lfunctions.
In addition to their utility in solving important conjectures, padic Lfunctions are beautiful objects in their own right. It is expected that for every Lfunction there is a padic 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 padic picture by constructing new padic Lfunctions, 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 higherdimensional automorphic forms.

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Organisation Website: 
http://www.warwick.ac.uk 