The rationality of Stark-Heegner points over genus fields of real quadratic fields
نویسندگان
چکیده
We study the algebraicity of Stark-Heegner points on a modular elliptic curve E. These objects are p-adic points on E given by the values of certain p-adic integrals, but they are conjecturally defined over ring class fields of a real quadratic field K. The present article gives some evidence for this algebraicity conjecture by showing that linear combinations of Stark-Heegner points weighted by certain genus characters of K are defined over the predicted quadratic extensions of K. The non-vanishing of these combinations is also related to the appropriate twisted Hasse-Weil L-series of E over K, in the spirit of the Gross-Zagier formula for classical Heegner points. Introduction 1. A review of Stark-Heegner points 1.1. Modular symbols 1.2. Double integrals 1.3. Indefinite integrals 1.4. Stark-Heegner points 2. Hida theory 2.1. Hida families 2.2. Periods attached to Hida families 2.3. Indefinite integrals revisited 3. p-adic L-functions 3.1. The Mazur-Kitagawa p-adic L-function 3.2. p-adic L-functions attached to real quadratic fields 3.3. A factorisation formula 4. Proof of the main result References
منابع مشابه
Heegner points, Stark-Heegner points, and values of L-series
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