12 research outputs found
Moduli spaces and Modular forms (hybrid meeting)
The relation between moduli spaces and modular forms goes back
to the theory of elliptic curves. On the one hand both topics
experience their own growth and development, but from time to
time new unexpected links show up and usually these lead to progress on both
sides. One subject where there has been a lot of progress concerns
the moduli of abelian varieties and K3 surfaces and especially
on the Kodaira dimension of these spaces. The idea of the workshop
was to bring together the experts of the two areas in the hope that
discussion, interaction and lectures would spur the development
of new ideas. The lectures of the workshop gave ample evidence
of the interaction and provided opportunities for further interaction.
Besides the lectures participants interacted via zoom in smaller groups
Modular Forms
The theory of Modular Forms has been central in mathematics with a rich history and connections to many other areas of mathematics. The workshop explored recent developments and future directions with a particular focus on connections to the theory of periods
Abelian Surfaces over totally real fields are Potentially Modular
We show that abelian surfaces (and consequently curves of genus 2) over
totally real fields are potentially modular. As a consequence, we obtain the
expected meromorphic continuation and functional equations of their Hasse--Weil
zeta functions. We furthermore show the modularity of infinitely many abelian
surfaces A over Q with End_C(A)=Z. We also deduce modularity and potential
modularity results for genus one curves over (not necessarily CM) quadratic
extensions of totally real fields.Comment: 285 page
Eisenstein series and automorphic representations
We provide an introduction to the theory of Eisenstein series and automorphic
forms on real simple Lie groups G, emphasising the role of representation
theory. It is useful to take a slightly wider view and define all objects over
the (rational) adeles A, thereby also paving the way for connections to number
theory, representation theory and the Langlands program. Most of the results we
present are already scattered throughout the mathematics literature but our
exposition collects them together and is driven by examples. Many interesting
aspects of these functions are hidden in their Fourier coefficients with
respect to unipotent subgroups and a large part of our focus is to explain and
derive general theorems on these Fourier expansions. Specifically, we give
complete proofs of the Langlands constant term formula for Eisenstein series on
adelic groups G(A) as well as the Casselman--Shalika formula for the p-adic
spherical Whittaker function associated to unramified automorphic
representations of G(Q_p). In addition, we explain how the classical theory of
Hecke operators fits into the modern theory of automorphic representations of
adelic groups, thereby providing a connection with some key elements in the
Langlands program, such as the Langlands dual group LG and automorphic
L-functions. Somewhat surprisingly, all these results have natural
interpretations as encoding physical effects in string theory. We therefore
also introduce some basic concepts of string theory, aimed toward
mathematicians, emphasising the role of automorphic forms. In particular, we
provide a detailed treatment of supersymmetry constraints on string amplitudes
which enforce differential equations of the same type that are satisfied by
automorphic forms. Our treatise concludes with a detailed list of interesting
open questions and pointers to additional topics which go beyond the scope of
this book.Comment: 326 pages. Detailed and example-driven exposition of the subject with
highlighted applications to string theory. v2: 375 pages. Substantially
extended and small correction