Thirteenth Algorithmic Number Theory Symposium ANTS-XIII University of Wisconsin, Madison July 16 – 20, 2018 Thirteenth Algorithmic Number Theory Symposium (ANTS-XIII) July 16 – 20, 2018

Principally polarized squares of elliptic curves with field of moduli equal to Q

Alexandre Gélin, Everett W. Howe and Christophe Ritzenthaler

Abstract: We give equations for 13 genus-2 curves over \bar(Q), with models over \bar(Q), whose unpolarized Jacobians are isomorphic to the square of an elliptic curve with complex multiplication by a maximal order. If the Generalized Riemann Hypothesis is true, there are no further examples of such curves. More generally, we prove under the Generalized Riemann Hypothesis that there exist exactly 46 genus-2 curves over \bar(Q) with field of moduli Q whose Jacobians are isomorphic to the square of an elliptic curve with complex multiplication by a maximal order.

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