215 research outputs found
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
Can a Drinfeld module be modular?
Let be a global function field with field of constants \Fr and let
be a fixed place of . In his habilitation thesis \cite{boc2},
Gebhard B\"ockle attaches abelian Galois representations to characteristic
valued cusp eigenforms and double cusp eigenforms \cite{go1} such that Hecke
eigenvalues correspond to the image of Frobenius elements. In the case where
k=\Fr(T) and corresponds to the pole of , it then becomes
reasonable to ask whether rank 1 Drinfeld modules over are themselves
``modular'' in that their Galois representations arise from a cusp or double
cusp form. This paper gives an introduction to \cite{boc2} with an emphasis on
modularity and closes with some specific questions raised by B\"ockle's work.Comment: Final corrected versio
Iwasawa Theory and Motivic L-functions
We illustrate the use of Iwasawa theory in proving cases of the (equivariant) Tamagawa number conjecture
Modular symbols in Iwasawa theory
This survey paper is focused on a connection between the geometry of
and the arithmetic of over global fields,
for integers . For over , there is an explicit
conjecture of the third author relating the geometry of modular curves and the
arithmetic of cyclotomic fields, and it is proven in many instances by the work
of the first two authors. The paper is divided into three parts: in the first,
we explain the conjecture of the third author and the main result of the first
two authors on it. In the second, we explain an analogous conjecture and result
for over . In the third, we pose questions for general
over the rationals, imaginary quadratic fields, and global function fields.Comment: 43 page
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