Vol. 4, No. 1, 2020

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Abelian surfaces with fixed $3$-torsion

Frank Calegari, Shiva Chidambaram and David P. Roberts

Vol. 4 (2020), No. 1, 91–108

Given a genus two curve X : y2 = x5 + ax3 + bx2 + cx + d, we give an explicit parametrization of all other such curves Y with a specified symplectic isomorphism on three-torsion of Jacobians Jac(X)[3]Jac(Y )[3]. It is known that under certain conditions modularity of X implies modularity of infinitely many of the Y , and we explain how our formulas render this transfer of modularity explicit. Our method centers on the invariant theory of the complex reflection group C3 × Sp4(𝔽3). We discuss other examples where complex reflection groups are related to moduli spaces of curves, and in particular motivate our main computation with an exposition of the simpler case of the group Sp2(𝔽3) = SL2(𝔽3) and 3-torsion on elliptic curves.

abelian surfaces, three torsion, Galois representations
Mathematical Subject Classification
Primary: 11F80
Secondary: 11G10, 20F55
Received: 23 February 2020
Revised: 1 September 2020
Accepted: 2 September 2020
Published: 29 December 2020
Frank Calegari
Department of Mathematics
The University of Chicago
Chicago, IL
United States
Shiva Chidambaram
Department of Mathematics
The University of Chicago
Chicago, IL
United States
David P. Roberts
Division of Science and Mathematics
University of Minnesota
Morris, MN
United States