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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Rationality of twists of the Siegel modular variety of genus $2$ and level $3$
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by Frank Calegari and Shiva Chidambaram PDF
Proc. Amer. Math. Soc. 150 (2022), 1975-1984 Request permission


Let $\overline {\rho }: G_{\mathbf {Q}} \rightarrow \operatorname {GSp}_4(\mathbf {F}_3)$ be a continuous Galois representation with cyclotomic similitude character. Equivalently, consider $\overline {\rho }$ to be the Galois representation associated to the $3$-torsion of a principally polarized abelian surface $A/\mathbf {Q}$. We prove that the moduli space $\mathcal {A}_2(\overline {\rho })$ of principally polarized abelian surfaces $B/\mathbf {Q}$ admitting a symplectic isomorphism $B[3] \simeq \overline {\rho }$ of Galois representations is never rational over $\mathbf {Q}$ when $\overline {\rho }$ is surjective, even though it is both rational over $\mathbf {C}$ and unirational over $\mathbf {Q}$ via a map of degree $6$.
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Additional Information
  • Frank Calegari
  • Affiliation: The University of Chicago, 5734 S University Ave, Chicago, Illinois 60637
  • MR Author ID: 678536
  • Email:
  • Shiva Chidambaram
  • Affiliation: Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, Massachusetts 02139
  • MR Author ID: 1341026
  • ORCID: 0000-0001-7325-6363
  • Email:
  • Received by editor(s): October 2, 2020
  • Received by editor(s) in revised form: April 11, 2021, and September 20, 2021
  • Published electronically: February 18, 2022
  • Additional Notes: Both authors were supported in part by NSF Grants DMS-1701703. The first author was supported in part by DMS-2001097, and the second author was supported in part by the Simons Foundation (grant 550033).
  • Communicated by: Rachel Pries
  • © Copyright 2022 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 150 (2022), 1975-1984
  • MSC (2020): Primary 11G18; Secondary 14K10, 11F80, 14E08
  • DOI:
  • MathSciNet review: 4392333