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arXiv · 1510.06193

Equations of hyperelliptic Shimura curves

Jia-Wei Guo·Yifan Yang
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Abstract

By constructing suitable Borcherds forms on Shimura curves and using Schofer's formula for norms of values of Borcherds forms at CM-points, we determine all the equations of hyperelliptic Shimura curves $X_0^D(N)$. As a byproduct, we also address the problem of whether a modular form on Shimura curves $X_0^D(N)/W_{D,N}$ with a divisor supported on CM-divisors can be realized as a Borcherds form, where $X_0^D(N)/W_{D,N}$ denotes the quotient of $X_0^D(N)$ by all the Atkin-Lehner involutions. The construction of Borcherds forms is done by solving certain integer programming problems.

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Jia-Wei Guo, Yifan Yang. 2015-10-21. Equations of hyperelliptic Shimura curves. https://doi.org/10.1112/s0010437x16007739

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