Stark-Heegner points on elliptic curves defined over imaginary quadratic fields. (English) Zbl 1111.14025

Let \(K\) be a number field and \(E\) be an elliptic curve defined over \(K\). One can define the Hasse-Weil \(L\)-series \(L(E/K,s)\) of \(E/K\) by an Euler product which converges when the real part of \(s\) is greater than \(3/2\). It is conjectured that the function \(L(E/K,s)\) has an analytic continuation to all the complex plane, which is known to be true in case \(K=\mathbb Q\). In the general case, one can consider conjecturally the leading term of the Taylor expansion of \(L(E/K,s)\) at \(s=1\). The Birch and Swinnerton-Dyer conjecture predicts a description of it in terms of arithmetic invariants of \(E/K\). A weak version of this conjecture is the following: for any natural integer \(r\), if \(\text{ord}_{s=1} L(E/K,s)=r\), then the rank of the group of the \(K\)-rational points of \(E\) is equal to \(r\). Moreover, the Shafarevich-Tate group of \(E/K\) is finite. This conjecture is proved if \(r\leq 1\) and \(K=\mathbb Q\).
As the author mentions, any approach to proving this conjecture should involve a method for constructing points on \(E\). If \(r\leq 1\) and \(K=\mathbb Q\), the proof uses variants of the Heegner point construction. In case \(K\) is an imaginary quadratic field, the author presents a \(p\)-adic analytic construction of points on \(E\), which he conjectures to be global, following ideas of H. Darmon [Ann. Math. (2) 154, No. 3, 589–639 (2001; Zbl 1035.11027)] to produce an analog of Heegner points. Furthermore, the author provides numerical evidence for his construction.


11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11G05 Elliptic curves over global fields
11R37 Class field theory
11G15 Complex multiplication and moduli of abelian varieties
14G05 Rational points
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)


Zbl 1035.11027
Full Text: DOI


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