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**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.

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.

Reviewer: Alain Kraus (Paris)

### MSC:

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) |

### Citations:

Zbl 1035.11027
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\textit{M. Trifković}, Duke Math. J. 135, No. 3, 415--453 (2006; Zbl 1111.14025)

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### References:

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