Further contributions to the study of Shimura curves

AbstractThe aim of this paper is to prove inequalities towards instances of the Bloch–Kato conjecture for Hilbert modular forms of parallel weight two, when the order of vanishing of the L-function at the central point is zero or one. We achieve this implementing an inductive Euler system argument which relies on explicit reciprocity laws for cohomology classes constructed using congruences of automorphic forms and special points on several Shimura curves.

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Automorphic forms and cohomology theories on Shimura curves of small discriminant

Advances in Mathematics ◽

10.1016/j.aim.2010.03.009 ◽

2010 ◽

Vol 225 (2) ◽

pp. 1013-1045 ◽

Cited By ~ 5

Author(s):

Michael Hill ◽

Tyler Lawson

Keyword(s):

Automorphic Forms ◽

Shimura Curves

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Intersection Numbers of Heegner Divisors on Shimura Curves

Pure and Applied Mathematics Quarterly ◽

10.4310/pamq.2008.v4.n4.a8 ◽

2008 ◽

Vol 4 (4) ◽

pp. 1165-1204

Author(s):

Kevin Keating ◽

David P. Roberts

Keyword(s):

Shimura Curves ◽

Intersection Numbers

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Reduction modulo 𝔓 of Shimura curves

Hokkaido Mathematical Journal ◽

10.14492/hokmj/1381758092 ◽

1981 ◽

Vol 10 (2) ◽

pp. 209-238 ◽

Cited By ~ 6

Author(s):

Yasuo MORITA

Keyword(s):

Shimura Curves ◽

Reduction Modulo

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Diagrams in the mod p cohomology of Shimura curves

Compositio Mathematica ◽

10.1112/s0010437x21007375 ◽

2021 ◽

Vol 157 (8) ◽

pp. 1653-1723

Author(s):

Andrea Dotto ◽

Daniel Le

Keyword(s):

Shimura Curves ◽

Shimura Curve ◽

Langlands Program ◽

Mod P

Abstract We prove a local–global compatibility result in the mod $p$ Langlands program for $\mathrm {GL}_2(\mathbf {Q}_{p^f})$ . Namely, given a global residual representation $\bar {r}$ appearing in the mod $p$ cohomology of a Shimura curve that is sufficiently generic at $p$ and satisfies a Taylor–Wiles hypothesis, we prove that the diagram occurring in the corresponding Hecke eigenspace of mod $p$ completed cohomology is determined by the restrictions of $\bar {r}$ to decomposition groups at $p$ . If these restrictions are moreover semisimple, we show that the $(\varphi ,\Gamma )$ -modules attached to this diagram by Breuil give, under Fontaine's equivalence, the tensor inductions of the duals of the restrictions of $\bar {r}$ to decomposition groups at $p$ .

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