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Tamely ramified Hida theory

Assaf Goldberger, Ehud de Shalit (2002)

Annales de l’institut Fourier

Let $J_{1}$ be the Jacobian of the modular curve associated with $Γ_{1} (N p), (p, N) = 1$ and $J_{0}$ the one associated with $Γ_{1} (N) \cap Γ_{0} (p)$ . We study $J_{1} [p - 1]$ as a Hecke and Galois-module. We relate a certain matrix of $p$ -adic periods to the infinitesimal deformation of the $U_{p}$ -operator.

Tetragonal modular curves

Daeyeol Jeon, Euisung Park (2005)

Acta Arithmetica

The conjecture of Langlands and Rapoport for certain Shimura varieties of non-rational weight.

Matthias Pfau (1996)

Journal für die reine und angewandte Mathematik

The dimension of some affine Deligne–Lusztig varieties

Eva Viehmann (2006)

Annales scientifiques de l'École Normale Supérieure

The field of moduli of quaternionic multiplication on abelian varieties.

Rotger, Victor (2004)

International Journal of Mathematics and Mathematical Sciences

The formal completion of the Néron model of J0(p).

Enric Nart (1991)

Publicacions Matemàtiques

For any prime number p > 3 we compute the formal completion of the Néron model of J0(p) in terms of the action of the Hecke algebra on the Z-module of all cusp forms (of weight 2 with respect to Γ0(p)) with integral Fourier development at infinity.

The modified diagonal cycle on the triple product of a pointed curve

Benedict H. Gross, Chad Schoen (1995)

Annales de l'institut Fourier

Let $X$ be a curve over a field $k$ with a rational point $e$ . We define a canonical cycle $Δ_{e} \in Z^{2} (X^{3})_{hom}$ . Suppose that $k$ is a number field and that $X$ has semi-stable reduction over the integers of $k$ with fiber components non-singular. We construct a regular model of $X^{3}$ and show that the height pairing $⟨ τ_{*} (Δ_{e}), τ_{*}^{'} (Δ_{e}) ⟩$ is well defined where $τ$ and $τ^{'}$ are correspondences. The paper ends with a brief discussion of heights and $L$ -functions in the case that $X$ is a modular curve.