Seminars this semester

The celebrated connection between elliptic curves and weight 2 newforms over the rationals has a conjectural extension to general number fields. For example, over odd degree totally real fields, one knows how to associate an elliptic curve to a weight 2 newform with integer Hecke eigenvalues. Beyond totally real fields, we are at a loss at associating elliptic curves to weight 2 newforms. The best one can do is to "search" for the elliptic curve.

In joint work with X.Guitart (Essen) and M.Masdeu (Warwick), we generalize Darmon's conjectural construction of algebraic points on elliptic curves to general number fields and then use this conjectural construction to construct the elliptic curve starting from a weight 2 newform over a general number field from its p-adic periods, under some hypothesis. In the talk, I will start with a discussion of the first paragraph and then will sketch our method.

We establish an explicit relation between the $K$-homology of boundary crossed product algebras associated to groups of hyperbolic isometries, and the cohomology of such groups. We show that the notion of Hecke operator for arithmetic groups has a natural definition in terms of Kasparov's $KK$-theory.

In the case of Bianchi groups, we establish an explicit Hecke equivariant isomorphism between the first cohomology of $\Gamma$ and the first K-homology group of the boundary cross-product algebra associated to $\Gamma$. A similar result holds for cocompact arithmetic Kleinian groups as well. These results are achieved in the context of unbounded Fredholm modules, shedding light on noncommutative geometric aspects of the boundary crossed product.

In 1983, Cohen and Lenstra introduced a measure on the set of finite abelian p-groups (p odd) and conjectured that this gives the distribution of p-class groups of imaginary quadratic fields. Bush, Hajir, and I introduced a measure on the set of finitely generated pro-p groups (p odd) and conjecture (BBH) that this gives the distribution of p-class tower groups of imaginary quadratic fields. These heuristics can be succinctly rephrased in terms of moments as recently shown by Wood and me. We also prove some function field cases of the BBH conjecture.

Cohomology of arithmetic groups plays a prominent role in modern number theory. The standard way to study the cohomology of an arithmetic group G goes through studying its action on the associated global symmetric space X. In this talk, we instead consider the action of G on the "boundary" of X. As this action is topologically not good, we employ the approach of Noncommutative Geometry in the style of Connes. In joint work with Haluk Sengun (Sheffield), we show that the cohomology of G, as a Hecke module, can be captured in the K-groups of a certain noncommutative C*-algebra which encodes the action of G on the boundary of X.

In their influential 1965 article, Artin and Mazur introduced a dynamical zeta function that can be used to encode the periodic point data of a discrete time dynamical system. This is now an intensely studied function and diverse generalisations and analogues have been developed. In this talk, I will discuss such a generalisation for dynamical systems arising from group actions. The case for this generalised function is made more compelling by its relationship with the zeta function of the acting group which is of independent interest. I will also consider some examples and natural questions in the setting of actions by compact abelian group automorphisms. For example, in this context, when is the dynamical zeta function of a group action rational?

Class fields of imaginary quadratic fields are generated by the special values of elliptic modular forms (Kronecker’s Jugendtraum). In the 20th century, G. Shimura gave an extension of Kronecker’s Jugendtraum. His theory is called the theory of Shimura varieties.

Today, the theory of Shimura varieties is an important theme in number theory and theoretically well-studied. However, to obtain explicit models of Shimura varieties is still a non-trivial problem.

The main theme of this talk is to obtain an explicit model of a Shimura variety using the moduli theory of K3 surfaces. First, we will see basic properties of Shimura varieties. Afterwards, the speaker will present the canonical model of a Shimura variety via explicit periods of K3 surfaces.

Assuming two deep but standard reciprocity conjectures from the Langlands Programme, we prove that the asymptotic Fermat's Last Theorem holds for imaginary quadratic fields Q(\sqrt{-d}) with -d=2, 3 mod 4. For a general number field K, again assuming standard conjectures, we give a criterion based on the solutions to a certain S-unit equation, which if satisfied implies the asymptotic Fermat's Last Theorem.

Up to isomorphism, elliptic curves over $\mathbb{C}$ are classified by their j-invariant; their coarse moduli space is an affine line with the j-invariant as coordinate. Conversely, it is not difficult to construct an elliptic curve with a specified j-invariant.

In higher genus the situation is quite a bit more complicated. The moduli space of smooth genus 2 curves, as determined by Igusa, is already no longer a quasi-affine space, although it is still birational. In this genus Clebsch and Mestre have developed methods to reconstruct curves from their invariants, which also apply to hyperelliptic curves of higher genus. These methods are however very specific to the hyperelliptic case and do not at all generalize.

• Calculating endomorphism rings of Jacobians;
• Decomposing Jacobians into simple factors; and
• Reconstructing curves from period matrices.

My research in the recent years have been guided by the simple question: "Why not consider K-theory instead of ordinary cohomology in the study of arithmetic groups and automorphic forms?". Here I mean not only the topological K-theory or arithmetic manifolds but also the operator K-theory of the various C*-algebras associated to arithmetic groups; such as group C*-algebras, boundary crossed product algebras...

In this talk, I will sketch basics around cohomology of arithmetic groups and automorphic forms, and then will give about some samples from my K-theoretic works, but I will mainly be raising questions some of which I hope will lead to conversations between number theorists and algebraic topologists in the department.