STNB2023(36th edition)

Local-global principles for quadratic and polyquadratic twists of abelian varieties


Francesc Fité Naya


Let A and A' be abelian varieties defined over a number field k. In the talk I will consider the following question: Is it true that A and A' are quadratic twists of one another if and only if they are quadratic twists modulo p for almost every prime p of k? Serre and Ramakrishnan have given a positive answer in the case of elliptic curves and a result of Rajan implies the validity of the principle when the endomorphism ring of A (and A') over an algebraic closure of Q is just Z. More in general, I will show that the answer is affirmative up to dimension 3, but that it becomes negative in dimension 4. The proof builds on Rajan's result and uses a Tate module tensor decomposition of geometrically isotypic abelian varieties obtained jointly with X. Guitart. I will also discuss a similar result concerning polyquadratic twists of abelian varieties obtained in collaboration with Antonella Perucca.


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