Duke Mathematical Journal

Stark-Heegner points and the cohomology of quaternionic Shimura varieties

Matthew Greenberg

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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(1). Let K/F be a non-CM quadratic extension with (DiscK,N)=1 such that the sign in the functional equation of L(E/K,s) is 1. Suppose further that there is a prime p|N that is inert in K. We describe a 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

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

First available in Project Euclid: 1 April 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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. https://projecteuclid.org/euclid.dmj/1238592865

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