1998/09/03 by Imin Chen, Chen, Imin
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.math/9809209
arxiv created 1998/09/03 · openalex publication_date 1998/09/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The topic of this paper concerns a certain relation between the jacobians of various quotients of the modular curve X(p), which relates the jacobian of the quotient of X(p) by the normaliser of a non-split Cartan subgroup of GL2(Fp) to the jacobians of more standard modular curves. In this paper, we confirm a conjecture of Merel found in a paper of Darmon-Merel, "Winding quotients and some variants of Fermat's Last Theorem", Crelle, v. 490, p. 81-100, 1997, which describes this relation in terms of explicit correspondences. The method used is to reduce the conjecture to showing a certain Z[GL2(Fp)]-module homomorphism is an isomorphism. This is accomplished by using some peculiar relations between double coset operators to find a expression for the eigenvalues of this homomorphism in terms of Legendre character sums and Soto-Andrade sums. A ramification argument then shows that these eigenvalues are non-zero.