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

Trifković, Mak

Duke Math. J. 135, No. 3, 415-453 (2006).

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.

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

References:

[1]	M. Bertolini and H. Darmon, The rationality of Stark-Heegner points over genus fields of real quadratic fields , to appear in Ann. of Math. (2). · Zbl 1203.11045
[2]	C. Breuil, B. Conrad, F. Diamond, and R. Taylor, On the modularity of elliptic curves over Q : Wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), 843–939. JSTOR: · Zbl 0982.11033
[3]	H. Cohen, A Course in Computational Algebraic Number Theory , Grad. Texts in Math. 138 , Springer, Berlin, 1993. · Zbl 0786.11071
[4]	J. Cremona, Hyperbolic tessellations, modular symbols, and elliptic curves over complex quadratic fields, Compositio Math. 51 (1984), 275–324. · Zbl 0546.14027
[5]	J. Cremona and E. Whitley, Periods of cusp forms and elliptic curves over imaginary quadratic fields, Math. Comp. 62 (1994), 407–429. JSTOR: · Zbl 0798.11015
[6]	H. Darmon, Integration on \(\mathcal H_p\,\times\,\mathcal H\) and arithmetic applications, Ann. of Math. (2) 154 (2001), 589–639. JSTOR: · Zbl 1035.11027
[7]	H. Darmon and R. Pollack, The efficient calculation of Stark-Heegner points via overconvergent modular symbols, to appear in Israel J. Math. · Zbl 1157.11028
[8]	M. Greenberg, Lifting modular symbols of non-critical slope, preprint, 2005. · Zbl 1165.11049
[9]	R. Greenberg and G. Stevens, “On the conjecture of Mazur, Tate, and Teitelbaum” in \(p\)-adic Monodromy and the Birch and Swinnerton-Dyer Conjecture, (Boston, 1991) , Contemp. Math. 165 , Amer. Math. Soc., Providence, 1994, 183, –211. · Zbl 0846.11030
[10]	B. H. Gross and D. B. Zagier, Heegner points and derivatives of \(L\)-series , Invent. Math. 84 (1986), 225, –320. · Zbl 0608.14019
[11]	V. A. Kolyvagin, Finiteness of \(E(Q)\) and \sha \((E,Q)\) for a subclass of Weil curves (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 52 , no. 3 (1998), 522–540.; English translation in Math. USSR-Izv. 32 (1989), 523–541. · Zbl 0662.14017
[12]	P. KurčAnov, The cohomology of discrete groups and Dirichlet series that are related to Jacquet-Langlands cusp forms (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 42 , no. 3 (1978), 588–601.; English translation in Math. USSR-Izv. 12 (1978), 543–555.
[13]	D. Lemelin, Mazur-Tate type conjectures for elliptic curves defined over quadratic imaginary fields, Ph.D. dissertation, McGill University, Montréal, 2001.
[14]	D. E. Rohrlich, Galois theory, elliptic curves, and root numbers, Compositio Math. 100 (1996), 311–349. · Zbl 0860.11033
[15]	D. Simon, PARI, www.math.unicaen.fr/\(\sim\)simon/
[16]	G. Stevens, Overconvergent modular symbols, unpublished notes. · Zbl 1268.11075
[17]	R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), 553–572. JSTOR: · Zbl 0823.11030
[18]	J. T. Teitelbaum, Values of \(p\)-adic \(L\)-functions and a \(p\)-adic Poisson kernel, Invent. Math. 101 (1990), 395–410. · Zbl 0731.11065
[19]	A. Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), 443–551. JSTOR: · Zbl 0823.11029
[20]	S. Zhang, Heights of Heegner points on Shimura curves, Ann. of Math. (2) 153 (2001), 27–147. JSTOR: · Zbl 1036.11029