On polyquadratic twists of X0(N)

JOURNAL OF NUMBER THEORY

卷:133

Article

Ozman, Ekin

关键词: Modular curves;    Twists;    Mordell-Weil sieve;   

DOI  :  10.1016/j.jnt.2013.03.015

来源: Elsevier

【 摘 要 】

Let K = Q(root d(1), ..., root 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 sigma(i) such that sigma(i)(root d(i)) = - root d(i). Let zeta : Gal(K/Q) -> Aut(X-0(N)) be the cocycle that sends sigma(i) to w(m1) where w(m1) 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(zeta)(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. (C) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

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