Early Number Theory Researchers Workshop

Holomorphic modular forms are fruitful and therefore ubiquitous objects in analytic number theory. Real analytic modular forms are a natural generalization of holomorphic modular forms. Among the possible relaxations of holomorphicity and growth conditions, Maass forms have proven to be of interest as well, in particular after Zweger’s breakthrough on Ramanujan’s mock theta functions.

Theta functions play an essential role in many areas of number theory. One exciting application is their use for the functional equation of the Riemann Zeta function and Zeta function of a number field. Further, theta functions are deeply intertwined with the theory of quadratic forms and their arithmetic properties.

A theta lift is an explicit application of the theta correspondence and can be seen as integrating automorphic forms against a theta function with similar automorphic properties. The result is again an automorphic form whose Fourier expansion encodes arithmetic information related to the input automorphic form.

L-functions are generalisations of the Riemann Zeta-function and may be attached to many geometric and arithmetic objects. They play a key role in the Langlands programme, which is one of the most ambitious mathematical programmes of our time.

Elliptic curves are smooth projective curves of genus 1 with a fixed point. They are famously related to special values of L-functions by the Birch and Swinnerton-Dyer conjecture.