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Search: All articles in the CJM digital archive with keyword modular curves

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1. CJM 2016 (vol 68 pp. 1362)

Papikian, Mihran; Rabinoff, Joseph
Optimal Quotients of Jacobians with Toric Reduction and Component Groups
Let $J$ be a Jacobian variety with toric reduction over a local field $K$. Let $J \to E$ be an optimal quotient defined over $K$, where $E$ is an elliptic curve. We give examples in which the functorially induced map $\Phi_J \to \Phi_E$ on component groups of the Néron models is not surjective. This answers a question of Ribet and Takahashi. We also give various criteria under which $\Phi_J \to \Phi_E$ is surjective, and discuss when these criteria hold for the Jacobians of modular curves.

Keywords:Jacobians with toric reduction, component groups, modular curves
Categories:11G18, 14G22, 14G20

2. CJM 2013 (vol 66 pp. 924)

Stankewicz, James
Twists of Shimura Curves
Consider a Shimura curve $X^D_0(N)$ over the rational numbers. We determine criteria for the twist by an Atkin-Lehner involution to have points over a local field. As a corollary we give a new proof of the theorem of Jordan-Livné on $\mathbf{Q}_p$ points when $p\mid D$ and for the first time give criteria for $\mathbf{Q}_p$ points when $p\mid N$. We also give congruence conditions for roots modulo $p$ of Hilbert class polynomials.

Keywords:Shimura curves, complex multiplication, modular curves, elliptic curves
Categories:11G18, 14G35, 11G15, 11G10

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