Open AccessParametrizations of elliptic curves by Shimura curves and by classical modular curves
Author(s) -
Kenneth A. Ribet,
Shuzo Takahashi
Publication year - 1997
Publication title -
proceedings of the national academy of sciences of the united states of america
Language(s) - English
Resource type - Journals
eISSN - 1091-6490
pISSN - 0027-8424
DOI - 10.1073/pnas.94.21.11110
Subject(s) - mathematics , elliptic curve , isogeny , twists of curves , supersingular elliptic curve , modular curve , discriminant , pure mathematics , divisor (algebraic geometry) , hessian form of an elliptic curve , integer (computer science) , quaternion algebra , jacobian curve , abelian group , prime (order theory) , schoof's algorithm , combinatorics , algebra over a field , division algebra , artificial intelligence , computer science , filtered algebra , quarter period , programming language
Fix an isogeny class of semistable elliptic curves over Q. The elements of have a common conductorN , which is a square-free positive integer. LetD be a divisor ofN which is the product of an even number of primes—i.e., the discriminant of an indefinite quaternion algebra over Q. ToD we associate a certain Shimura curveX 0 D (N /D ), whose Jacobian is isogenous to an abelian subvariety ofJ 0 (N ). There is a uniqueA ∈ for which one has a nonconstant map πD :X 0 D (N /D ) →A whose pullbackA → Pic0 (X 0 D (N /D )) is injective. The degree of πD is an integer δD which depends only onD (and the fixed isogeny class ). We investigate the behavior of δD asD varies.