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On maps between modular Jacobians and Jacobians of Shimura curves

机译:在模块化雅可比人和志村曲线的雅可比人之间的地图上

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Fix a squarefree integer N, divisible by an even number of primes, and let Gamma' be a congruence subgroup of level M, where M is prime to N. For each D dividing N and divisible by an even number of primes, the Shimura curve X-D (Gamma(0)(N/D) boolean AND Gamma') associated to the indefinite quaternion algebra of discriminant D and Gamma(0)(N/D) boolean AND Gamma'-level structure is well defined, and we can consider its Jacobian J(D) (Gamma(0)(N/D) boolean AND Gamma'). Let J(D) denote the N/D-new subvariety of this Jacobian. By the Jacquet-Langlands correspondence [J-L] and Faltings' isogeny theorem [Fa], there are Hecke-equivariant isogenies among the various varieties J(D) defined above. However, since the isomorphism of Jacquet-Langlands is noncanonical, this perspective gives no information about the isogenies so obtained beyond their existence. In this paper, we study maps between the varieties J(D) in terms of the maps they induce on the character groups of the tori corresponding to the mod p reductions of these varieties for p dividing N. Our characterization of such maps in these terms allows us to classify the possible kernels of maps from J(D) to J(D'), for D dividing D', up to support on a small finite set of maximal ideals of the Hecke algebra. This allows us to compute the Tate modules T-m J(D) of J(D) at all non-Eisenstein m of residue characteristic l > 3. These computations have implications for the multiplicities of irreducible Galois representations in the torsion of Jacobians of Shimura curves; one such consequence is a "multiplicity one" result for Jacobians of Shimura curves.
机译:固定无平方整数N(可被偶数个素数整除),并使Gamma'是级M的全等子组,其中M是N的素数。对于每个D,除以N并且可被偶数个素数整除的Shimura曲线与判别式D的不定四元数代数相关的XD(Gamma(0)(N / D)布尔AND Gamma')和Gamma(0)(N / D)布尔AND Gamma'级结构定义明确,我们可以考虑其Jacobian J(D)(Gamma(0)(N / D)布尔AND Gamma')。令J(D)表示该雅可比行列的N / D-新子变量。根据雅克-兰格兰兹对应关系[J-L]和法尔廷斯等距定理[Fa],在上面定义的各种品种J(D)中存在Hecke等价同构。但是,由于Jacquet-Langlands的同构是非规范的,因此该观点未提供有关如此存在的同构异构体的信息。在本文中,我们研究了品种J(D)之间的图谱,即它们在花托字符组上诱导的图谱上,这些图谱对应于这些品种对p除N的mod p归约化。允许我们对从J(D)到J(D')的可能映射核进行分类,以D划分D',以支持一小部分有限的Hecke代数的最大理想。这使我们能够在残差特征l> 3的所有非爱森斯坦m处计算J(D)的泰特模Tm J(D)。这些计算对Shimura曲线的Jacobian扭转中的不可约Galois表示的多重性具有影响;这样的结果之一就是Shimura曲线的Jacobian曲线的“多重性”结果。

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