Arithmetic applications of Kudla-Millson theta lifts

This proposal is motivated by the Langlands programme, a series of conjectures made by the mathematician Robert Langlands in the 1960s and 70s. They predict precise links between three seemingly unrelated classes of objects. These come from representation theory (in the form of modular forms), number theory (Galois representations) and algebraic geometry (motives e.g. elliptic curves). For each of these disparate objects you can determine something akin to an underlying DNA, the so-called L-function. (The most famous example of an L-function is the "Riemann zeta function", which contains a host of arithmetic information about prime numbers). You can use this DNA to match the objects, e.g. for every modular form there should be a Galois representation with the same L-function. Establishing these links enables mathematicians to understand more deeply the properties of the objects involved and allows them to prove theorems, such as the famous example of the proof of Fermat's last theorem by Wiles and Taylor in 1994.

Automorphic forms (examples of which include modular forms) are special kinds of analytic functions and can be studied for groups of matrices with entries in different fields. Fields are typically sets of "numbers" in which the operations of addition, subtraction, multiplication and division are defined. An example is the field of rational numbers but there are many other fields besides this.

Much progress has been made in the theory of automorphic forms over the rational numbers (and other totally real fields) in the last two decades. This has led to successes such as the proof of the Sato-Tate conjectures for elliptic curves. This proposal wants to move to new ground in the hope that it will prove similarly fertile. New phenomena occur with Bianchi modular forms (automorphic forms for 2x2 invertible matrices over imaginary quadratic fields), a considerably different case in which previously developed tools from algebraic geometry are not applicable. This case is therefore an important testing ground for finding new techniques that could apply in the general context of the Langlands programme.

So what techniques will I try? Recent progress in the theory of Siegel modular forms (4x4 symplectic matrices over the rational numbers) has led me to consider Kudla and Millson's theta lift. This is a construction that can be used to transfer Bianchi modular forms to Siegel modular forms. I propose to study the finer properties of this theta lift with the goal of applying it to answer questions about Bianchi modular forms. In particular, I want to prove results about their associated L-functions and Galois representations.

Specific aims of the proposal include proving a relation between L-values of Bianchi modular forms and the squares of Fourier coefficients of the Kudla-Millson theta lifts (an analogue of a famous formula by Waldspurger); proving one direction of the Bloch-Kato conjecture for the Asai Galois representation (a result explaining the significance of the value of a particular L-function); and studying the theta lift in the context of a "p-adic Langlands functoriality" conjectured by Calegari and Mazur.

This proposal will lead to a much better understanding of Bianchi modular forms and will help other members of the large community of mathematicians working on the Langlands programme.

As with many projects in pure mathematics, the main benefits of this proposal will be academic in the short and medium term. Longer term, as yet unforeseen applications might be possible, as the example of elliptic curves over finite fields and their use in public-key cryptosystems on the internet or in mobile devices shows. The pathway to such economic impact relies on the widest and clearest possible dissemination of the research output.

There are three groups that will benefit from this project:

1. researchers in number theory;
2. researchers in topology;
3. PhD students in both of these fields.

Researchers in number theory will benefit directly from this project by the advances in the theory of Bianchi modular forms and by the development of the Kudla-Millson theta lifts as a tool in the Langlands programme. In addition, researchers in topology will benefit via the applications of Bianchi modular forms in 3-manifold theory.

PhD students in both number theory and topology will benefit from this work primarily by learning skills and knowledge related to the research subject and exposure to a wider research community. In addition they will also attain the sort of analytical, methodical skills necessary for future careers also outside academia, thus benefitting their employability and the UK economy.