On polyquadratic twists of X0(N)X0(N)

Let K=Q(d1,…,dk) be a polyquadratic number field and N be a squarefree positive integer with at least k   distinct factors. The Galois group, Gal(K/Q)Gal(K/Q) is an elementary abelian two-group generated by σiσi such that σi(di)=−di. Let ζ:Gal(K/Q)→Aut(X0(N))ζ:Gal(K/Q)→Aut(X0(N)) be the cocycle that sends σiσi to wmiwmi where wmiwmi are the Atkin–Lehner involutions of X0(N)X0(N). In this paper, we study the QpQp-rational points of the twisted modular curve X0ζ(N) and give an algorithm to produce such curves which has QpQp-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.