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arXiv · 2404.01449

On converse theorems for Hilbert modular forms assuming unramified twists

Pengcheng Zhang
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Abstract

We prove two results on converse theorems for Hilbert modular forms over totally real fields of degree $r>1$. The first result recovers a Hilbert modular form (of some level) from an $L$-series satisfying functional equations twisted by all the unramified Hecke characters. The second result assumes both the 'unramified' functional equations and an Euler product, and recovers a Hilbert modular form of the expected level predicted by the shape of the functional equations. Our result generalizes the current converse theorems for $\mathrm{GL}_2$ in the case of Hilbert modular forms in that we completely remove the assumptions on ramified twists.

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Pengcheng Zhang. 2024-04-01. On converse theorems for Hilbert modular forms assuming unramified twists. https://doi.org/10.1093/qmath/haaf040

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