2025/01/08 by Ambrosi, Emiliano, Coppola, Nirvana, Fité, Francesc
#11G10 #11G35 #14G20 #14G25 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2501.04803
Two abelian varieties A and B over a number field K are said to be strongly locally quadratic twists if they are quadratic twists at every completion of K. While it was known that this does not imply that A and B are quadratic twists over K, the only known counterexamples (necessarily of dimension ≥ 4) are not geometrically simple. We show that, for every prime p≡ 13 \pmod24, there exists a pair of geometrically simple abelian varieties of dimension p-1 over ℚ that are strongly locally quadratic twists but not quadratic twists. The proof is based on Galois cohomology computations and class field theory.