Can a Drinfeld module be modular?

Details Can a Drinfeld module be modular? Can a Drinfeld module be modular?

2002-10-24

arxiv.org/abs/math/0210388v2

Journal of the Ramanujan Math. Soc. {\bf 17} No. 4 (2002) 221-260

Mathematics

Number Theory

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Scientific paper

Let $k$ be a global function field with field of constants $\Fr$ and let $\infty$ be a fixed place of $k$. In his habilitation thesis \cite{boc2}, Gebhard B\"ockle attaches abelian Galois representations to characteristic $p$ 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 $\infty$ corresponds to the pole of $T$, it then becomes reasonable to ask whether rank 1 Drinfeld modules over $k$ 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.

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