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Details of Grant 

EPSRC Reference: EP/W009838/1
Title: Strong subconvexity and an optimal large sieve inequality for PGL(2)
Principal Investigator: Petrow, Dr I
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Texas A and M University Tsinghua University
Department: Mathematics
Organisation: UCL
Scheme: New Investigator Award
Starts: 01 July 2022 Ends: 30 June 2025 Value (£): 360,853
EPSRC Research Topic Classifications:
Algebra & Geometry
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
31 Aug 2021 EPSRC Mathematical Sciences Prioritisation Panel September 2021 Announced
Summary on Grant Application Form
This project seeks to develop applications of automorphic forms (certain highly symmetric wavesforms) to number theory (the study of the properties of whole numbers). Automorphic forms play the role of fundamental building blocks in a wide-ranging web of ideas and conjectures known as the Langlands program, which ties together such diverse areas of mathematics as number theory, algebraic geometry, dynamics, analysis, and mathematical physics.

One of the most important ways that automorphic forms can apply to number theory is via the theory of L-functions, which act as avatars of automorphic forms. The prototypical example of an L-function is the Riemann zeta function. We care about the Riemann zeta function because it was found in 1850s by Berhnard Riemann to control the distribution of prime numbers on the number line. Riemann mentioned in his 1859 memoir that it is very probable that all nontrivial roots of zeta(s) lie on the line Re(s)=1/2, but he was unable to prove this. This conjecture is now known as the "Riemann Hypothesis". It is today one of the greatest unsolved conjectures in mathematics. The research in this proposal aims to establish several of the consequences of the Riemann Hypothesis without using any unproven conjectures.

One application that we aim to develop is to give estimates for L-functions at the center point of their symmetry. This is a highly active and exciting area of reserach known as the subconvexity problem. There are many number theoretic applications of the subconvexity problem, for instance approximate formulas for the number of representations of a large integer n by a quadratic equation in several variables taking only integer values. Another application of subconvex estimates for L-functions is to a question in mathematical physics known as quantum unique ergodicity. One project in this proposal seeks to prove very strong subconvex bounds (of 'Weyl' strength) for L-functions attached to any automorphic form arising from the group of 2x2 matrices up to scaling.

A second application of the generalised Riemann hypothesis is to the number of primes in an arithmetic progression (i.e. a sequence of whole numbers having a fixed common difference). Using L-functions, one can give an approximate formula for the number of primes less than a given bound which are of remainder a after dividing by some coprime number q. The generalised Riemann hypothesis gives a very strong control on the size of the error made in this approximation. Even though we cannot (without proving the GRH) say that the error is always so controlled, we can say that the number of q's for which there is an exeptionally large error is exceedingly small. This is a by now classical result from the 1960s known as the Bombieri-Vinogradov theorem.

The main technical input to proving the Bombieri-Vinogradov theorem is an inequality known as the large sieve inequality. What this inequality says is that all multiplicative harmonics are approximately orthogonal to each other, in a precise quantitative sense. The Bombieri-Vinogradov theorem is only one of a large number of applications of the large sieve inequality - it is extremely flexible and ubiquitous tool in number theory. The second major goal of this reserach grant is to prove a large sieve inequality for automorphic forms on the group of 2x2 matrices up to scaling, a well-known outstanding problem in analytic number theory.

The two goals of Weyl-strength subconvexity and the large sieve inequality have only very recently come within striking range of our current tools due to recent work of the PI and Young in 2019 and a new trace formula developed by Hu in 2020, who will be a project partner. The projects are highly timely and at the cutting edge of research in number theory.
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