Hilbert modular forms with prescribed ramification

Details Hilbert modular forms with prescribed ramification Hilbert modular forms with prescribed ramification

2008-01-29

arxiv.org/abs/0801.4416v4

Int. Math. Res. Not., (2009) 1388-1420

Mathematics

Number Theory

30 pages, published version

Scientific paper

Let $K$ be a totally real field. In this article we present an asymptotic formula for the number of Hilbert modular cusp forms $f$ with given ramification at every place $v$ of $K$. When $v$ is an infinite place, this means specifying the weight of $f$ at $k$, and when $v$ is finite, this means specifying the restriction to inertia of the local Weil-Deligne representation attached to $f$ at $v$. Our formula shows that with essentially finitely many exceptions, the cusp forms of $K$ exhibit every possible sort of ramification behavior, thus generalizing a theorem of Khare and Prasad. From this fact we compute the minimal field over which a modular Jacobian becomes semi-stable.

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