On polyquadratic twists ofX0(N)

Abstract

Let K = Q ( d 1 , … , d k ) be a polyquadratic number field and N be a squarefree positive integer with at least k distinct factors. The Galois group, Gal ( K / Q ) is an elementary abelian two-group generated by σ i such that σ i ( d i ) = − d i . Let ζ : Gal ( K / Q ) → Aut ( X 0 ( N ) ) be the cocycle that sends σ i to w m i where w m i are the Atkin–Lehner involutions of X 0 ( N ) . In this paper, we study the Q p -rational points of the twisted modular curve X 0 ζ ( N ) and give an algorithm to produce such curves which has Q p -rational points for all primes p . Then we investigate violations of the Hasse principle for these curves and give an asymptotic for the number of such violations. Finally, we study reasons of such violations.