Duke Mathematical Journal

Stark-Heegner points and the cohomology of quaternionic Shimura varieties

Matthew Greenberg

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Abstract

Let $F$ be a totally real field of narrow class number one, and let $E/F$ be a modular, semistable elliptic curve of conductor $N\neq(1)$. Let $K/F$ be a non-CM quadratic extension with $({\rm Disc} K, N)=1$ such that the sign in the functional equation of $L(E/K,s)$ is $-1$. Suppose further that there is a prime $\mathfrak{p}|N$ that is inert in $K$. We describe a $\mathfrak{p}$-adic construction of points on $E$ which we conjecture to be rational over ring class fields of $K/F$ and satisfy a Shimura reciprocity law. These points are expected to behave like classical Heegner points and can be viewed as new instances of the Stark-Heegner point construction of [5]. The key idea in our construction is a reinterpretation of Darmon's theory of modular symbols and mixed period integrals in terms of group cohomology

Article information

Source
Duke Math. J. Volume 147, Number 3 (2009), 541-575.

Dates
First available in Project Euclid: 1 April 2009

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1238592865

Digital Object Identifier
doi:10.1215/00127094-2009-017

Mathematical Reviews number (MathSciNet)
MR2510743

Zentralblatt MATH identifier
05550767

Subjects
Primary: 14G05: Rational points
Secondary: 14G35: Modular and Shimura varieties [See also 11F41, 11F46, 11G18]

Citation

Greenberg, Matthew. Stark-Heegner points and the cohomology of quaternionic Shimura varieties. Duke Math. J. 147 (2009), no. 3, 541--575. doi:10.1215/00127094-2009-017. http://projecteuclid.org/euclid.dmj/1238592865.


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