Two dimensional adelic analysis

Lead Research Organisation: University of Nottingham
Department Name: Sch of Mathematical Sciences


This research proposal is in number theory and its interaction with algebraic geometry, analysis, algebraic K-theory, topology, model theory. The subject of the research is the study of fundamental properties of zeta functions of elliptic curves over global fields and their regular minimal models using new analytical and arithmetical theory, methods and tools in geometrical setting, which form the two dimensional adelic analysis programme. Zeta functions is the most important object in number theory.The Langlands correspondence for zeta and L-functions (interelations of automorphicand Galois theoretical aspects of zeta functions), the Riemann hypothesis for zeta functions (location of their zeros or poles), the Birch and Swinnerton-Dyer conjecture (special behaviour of the zeta function at integer points) are major open and very difficult problems in number theory. The research of this proposal aims to study the fundamental issues of the zeta functionsof elliptic curves and regular models using new complex analytic methods, ideas and tools. This is a two dimensional extension of the fundamental work of Tate and Iwasawa. The research of this proposal will have applications to the meromorphic continuation and functional equation of the zeta and L-function of elliptic curves over global fields, the Riemann hypothesis for the zeta function, the Birch and Swinnerton-Dyer conjecture. It also aims to initiate and develop the theory of automorphic functions on arithmetic surfaces, in the central case of elliptic surfaces. The International Review of UK Mathematics indicates one direction that has played a major role outside the UK, and which seems never to have developed a critical mass in the UK is the modern theory of automorphic forms, for example, the Langlands programme. The subject and methods of the two dimensional adelic analysis are interrelated with various aspects of the Langlands correspondence in many different ways. The work on the project includes support of visits to the UK of several world leaders in the Langlands correspondence.


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Fesenko I (2010) Analysis on arithmetic schemes. II in Journal of K-theory

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Morrow Matthew (2010) Integration on Valuation Fields over Local Fields in TOKYO JOURNAL OF MATHEMATICS

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Morrow Matthew (2008) Integration on product spaces and GL(n) of a valuation field over a local field in COMMUNICATIONS IN NUMBER THEORY AND PHYSICS

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Morrow Matthew (2010) An explicit approach to residues on and dualizing sheaves of arithmetic surfaces in NEW YORK JOURNAL OF MATHEMATICS

Description The zeta functions of arithmetic schemes is the most fundamental object in number theory.

There are two main methods to study it: either work with its L-factors and use methods which come from the Langlands programme and local methods (in particular, Iwasawa theory) or work directly with the zeta function and study it using the new powerful tool of the zeta integral. This proposal is a pioneering work in the second direction.

With exception of the classical Riemann hypothesis, which is still out of reach, main fundamental problems in number theory start with geometric dimension two. Despite recent achievements in number theory, we still know very little in higher dimensions in characteristic zero. The proposal has significantly extended the area of known in two-dimensional number theory.

A theory which studies the zeta functions of arithmetic schemes via higher zeta integrals had been a dream of several great mathematicians for 60 years. Such a theory for arithmetic surfaces, a commutative theory of two-dimensional zeta integrals has been developed in the last ten years by the PI. This is the first theory which directly works with the zeta functions of arithmetic objects in dimension two using complex analytic, functional analysis, analytic adelic and geometric adelic methods and structures.

In dimension two, unlike dimension one, there are two different adelic structures, a geometric one and an analytic one. The latter was discovered by the PI and used to define a new zeta integral. The main theorem on a two dimensional unramified Tate-Iwasawa formula for regular models of elliptic curves over global fields was established. Its applications in the three main directions were studied and extended.

The first direction is the study of the meromorphic continuation and functional equation of the zeta function. A new correspondence between the zeta functions and mean-periodic functions in the space of smooth functions on the real line of not more than exponential growth has been proposed. This new correspondence complements the existing Langlands programme and there are many interactions between the two which are useful for each of them. The main theorem was proved: if the boundary function of the zeta integral is mean-periodic then the zeta and L function of the elliptic curve over a global fields has meromorphic continuation and functional equation. Moreover, a new adelic interpretation of the conductor of the model was obtained and a better understanding of the functional equation of the zeta function was achieved.

The second direction was the study of the location of the poles of the zeta function, in other words the Generalized Riemann Hypothesis. New phenomena in dimension two were discovered which allow to understand the GRH using methods not available in the classical case. A new hypothesis on positivity of the fourth derivative of the boundary function was stated and studied. This hypothesis plus the real part of the GRH implies the full GRH. Computer verification of the hypothesis were conducted and an explicit analytic representation of the fourth derivative of the boundary function in the language of Bessel's series have been obtained. M. Suzuki proved that the GRH and some technical and widely expected condition imply the hypothesis. Thus, the hypothesis which naturally originated in work of the PI is entirely compatible with the existing theories and the GRH.

Towards the BSD conjecture the PI proved a theorem which uses the Tate-Shioda unscrewing method and reduces the rank part of the conjecture to an auxiliary property of adelic integrals which is quite natural from the adelic and higher class field theory point of view. Thus, for the first time in 50 years long study of aspects of the BSD, there is a conceptual explanation of its rank part. The new method is not local but adelic and compex analytic. Work on the proof of the auxiliary property of adelic integrals is being conducted.

In addition, towards automorphic theory in geometric dimension two, for every algebraic group G a new adelic object was proposed; functions on it should be G-automorphic functions on the surface.

18 papers and volumes have been published by the PI, his students and collaborators on various aspects of the proposal.
The PI has delivered more around 100 hours of invited lectures, seminar, talks at various international venues.
The PI has served an a coorganizer of 6 international conferences including a large symposium on modern analytic number theory in Edinburgh in 2008.

A number of world leading mathematicians visited to discuss the PI work and its relations to several other branches of modern number theory and representation theory. The list of senior visitors includes A. Beilinson (Univ. of Chicago), and D. Gaitsgory (Harvard) and D. Goldfeld (Columbia Univ.) each visited twice, and the PI visited A. Beilinson and D. Goldfeld.
Exploitation Route As with all great achievements in pure mathematics, sooner or later this research will find various applications in non-academic contexts. In particular, the new higher Haar measure invented by the PI is closely related to the Feynman path integral which is a key tool in modern quantum physics and its applications. This research deals with the most fundamental structures in modern number theory and mathematics. It greatly extends the area of known in two-dimensional number theory. Its methods are entirely novel and results and objects are pioneering.
Sectors Education

Description impact for mathematics, mathematical community, as well as some physical areas
Sector Education
Impact Types Cultural