Ribet K A, Takahashi S
Mathematics Department, University of California, Berkeley, CA 94720-3840, USA.
Proc Natl Acad Sci U S A. 1997 Oct 14;94(21):11110-4. doi: 10.1073/pnas.94.21.11110.
Fix an isogeny class of semistable elliptic curves over Q. The elements A of have a common conductor N, which is a square-free positive integer. Let D be a divisor of N which is the product of an even number of primes--i.e., the discriminant of an indefinite quaternion algebra over Q. To D we associate a certain Shimura curve X(0)D(N/D), whose Jacobian is isogenous to an abelian subvariety of J0(N). There is a unique A [symbol; see text] A in for which one has a nonconstant map piD : X(0)D(N/D) --> A whose pullback A --> Pic0(X(0)D(N/D)) is injective. The degree of piD is an integer deltaD which depends only on D (and the fixed isogeny class A). We investigate the behavior of deltaD as D varies.
固定有理数域上半稳定椭圆曲线的一个同构类。该类中的元素(A)有一个公共导子(N)，(N)是一个无平方因子的正整数。设(D)是(N)的一个因子，它是偶数个素数的乘积——即有理数域上一个不定四元数代数的判别式。对于(D)，我们关联一条特定的志村曲线(X_0(D)(N/D))，其雅可比簇与(J_0(N))的一个阿贝尔子簇同构。存在唯一的(A\in\mathcal{A})，使得有一个非常值映射(\pi_D:X_0(D)(N/D)\to A)，其拉回(A\to\mathrm{Pic}^0(X_0(D)(N/D)))是单射。(\pi_D)的次数是一个仅依赖于(D)（以及固定的同构类(\mathcal{A})）的整数(\delta_D)。我们研究当(D)变化时(\delta_D)的行为。
