## Duke Mathematical Journal

### Stark-Heegner points and the cohomology of quaternionic Shimura varieties

Matthew Greenberg

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

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

Mathematical Reviews number (MathSciNet)
MR2510743

Zentralblatt MATH identifier
1183.14030

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

#### References

• M. Bertolini and H. Darmon, The rationality of Stark-Heegner points over genus fields of real quadratic fields, to appear in Ann. of Math.
• M. Bertolini, H. Darmon, and P. Green, Periods and points attached to quadratic algebras'' in Heegner Points and Rankin $L$-series (Berkeley, Calif., 2001), Math. Sci. Res. Inst. Publ. 49, Cambridge Univ. Press, Cambridge, 2004, 323--367.
• H. Carayol, Sur les représentations $l$-adiques associées aux formes modulaires de Hilbert, Ann. Sci. École Norm. Sup. (4) 19 (1986), 409--468.
• P. Charollois and H. Darmon, Arguments des unités de Stark et périodes de séries d'Eisenstein, Algebra Number Theory 2 (2008), 655--688.
• H. Darmon, Integration on $\cH_p\cp\cH$ and arithmetic applications, Ann. of Math. (2) 154 (2001), 589--639.
• —, Rational Points on Modular Elliptic Curves, CBMS Regional Conf. Ser. Math. 101, Amer. Math. Soc., Providence, 2004.
• H. Darmon and R. Pollack, The efficient calculation of Stark-Heegner points via overconvergent modular symbols, Israel J. Math. 153 (2006), 319--354.
• S. Dasgupta, Stark-Heegner points on modular Jacobians, Ann. Sci. École Norm. Sup. (4) 38 (2005), 427--469.
• N. D. Elkies, Shimura curves for level-$3$ subgroups of the ($2,3,7$) triangle group, and some other examples'' in Algorithmic Number Theory (Portland, Ore., 1998), Lecture Notes in Comput. Sci. 1423, Springer, Berlin, 1998, 1--47.
• E. Freitag, Hilbert Modular Forms, Springer, Berlin, 1990.
• R. Greenberg and G. Stevens, $p$-adic $L$-functions and $p$-adic periods of modular forms, Invent. Math. 111 (1993), 407--447.
• B. H. Gross, Heights and the special values of $L$-series'' in Number Theory (Montreal, 1985), CMS Conf. Proc. 7, American Math. Soc., Providence, 1987, 115--187.
• B. W. Jordan and R. Livné, The Ramanujan property for regular cubical complexes, Duke Math. J. 105 (2000), 85--103.
• Y. Matsushima and G. Shimura, On the cohomology groups attached to certain vector valued differential forms on the product of upper half planes, Ann. of Math. (2) 78 (1963), 417--449.
• B. Mazur, J. Tate, and J. Teitelbaum, On $p$-adic analogues of the conjecture of Birch and Swinnerton-Dyer, Invent. Math. 84 (1986), 1--48.
• G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, reprint, Publ. Math. Soc. Japan 11, Kanô Memorial Lectures 1, Princeton Univ. Press, Princeton, 1994.
• J. T. Teitelbaum, Values of $p$-adic $L$-functions and a $p$-adic Poisson kernel, Invent. Math. 101 (1990), 395--410.
• M. Trifković, Stark-Heegner points on elliptic curves defined over imaginary quadratic fields, Duke Math. J. 135 (2006), 415--453.
• M.-F. Vigneras, Arithmétique des algèbres de quaternions, Lecture Notes in Math. 800, Springer, Berlin, 1980.
• S. Zhang, Heights of Heegner points on Shimura curves, Ann. of Math. (2) 153 (2001), 27--147.