## Courses (3)

### Kronecker limit formalism and moonshine type groups

**Topic and importance:**

Kronecker’s limit formula refers to the evaluation of the first two terms in the Laurent expansion of the non-holomorphic Eisenstein series E(s,z) associated to the discrete group PSL(2,Z) at s=1. In this case, the first term is a pole with residue equal to 1/V, where V is the co-volume of PSL(2,Z), and the second term involves the weight twelve cusp form, suitably normalized, associated to PSL(2,Z).

More generally, one can consider generalizations of Kronecker’s limit formula to be any result which studies the Laurent expansion of any zeta or L function about its first pole. There are various examples where such considerations exist, including the study of spectral zeta functions (which yields determinants of the Laplacian and analytic torsion) and various types of Eisenstein series (which yields constructions of certain modular forms). Going further, the variation of the Kronecker limit function may be of additional interest, one example of which is the manifestation of Dedekind sums.

**Syllabus:**

In these talks the speakers will outline their point-of-view of Kronecker’s limit functions and Dedekind sums associated to various Eisenstein series for co-finite quotients of the hyperbolic upper half plane. Specific attention will be paid to the so-called “groups of moonshine type”, which we define as normalizer subgroups associated to Gamma_{0}(N). In lieu of a problem solving session, the speakers will offer a number of open problems.

### Explicit L-functions, automorphic forms and Galois representations

**Topic and importance:**

Automorphic forms and L-functions have played prominent roles invarious branches of mathematics and physics since the 20th century,mostly at a theoretical level. However, recent advances in explicitand computational methods have made a more hands-on approach possible,and with it a host of applications in explicit and algorithmic numbertheory have arisen.

**Syllabus:**

This course will survey some of these methods and applications, in twoparts. The first part will focus on the computational theory ofL-functions and applications of an analytic nature, such asHelfgott's proof of the ternary Goldbach conjecture. The second partwill focus on explicit equations for modular varieties, the arithmeticof Jacobians of curves, and applications of an algebraic nature, suchas Couveignes and Edixhoven's polynomial-time algorithm for computingthe Ramanujan \tau-function.

### Galois Representations and Diophantine Problems

**Topic and importance:**

Wiles' proof of Fermat's Last Theorem was one of the crowningachievements of 20th century mathematics, and has led to twomajor research programmes. The first aims to establish moregeneral and powerful modularity theorems for Galois representations,and has yielded the proof of modularity of elliptic curves over therationals by Wiles, Breuil, Conrad, Diamond and Taylor, the proof ofSerre's modularity conjecture by Khare and Wintenberger, and powerfulmodularity lifting theorems by Kisin, Barnet-Lamb, Gee, Geraghty andothers. The second programme aims to apply the proof strategy ofFermat's Last Theorem to solve Diophantine problems, especially casesof the notorious Beal conjecture (also known as the generalized Fermat conjecture).The original strategy of Hellegouarch, Frey, Serre andRibet has been refined and extended by many, most notably Bennett,Dahmen, Darmon and Kraus. The two programmes are still intricatelyconnected, as evident in the recent work of Freitas and Siksek on theFermat equation over totally real fields.

**Syllabus: **

The minicourse will use Diophantine equations as a vehicle tointroduce and motivate the study of Galois representations of ellipticcurves and of modular forms and their relationship (modularity). Thestudents will learn the basics of 2-dimensional Galoisrepresentations, understand the statements of modularity theorems(including the more recent modularity lifting theorems), and how toapply them to solve basic ternary Diophantine problems.