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Elliptic curves and \(p\)-adic deformations. (English) Zbl 0821.14021
Kisilevsky, Hershy (ed.) et al., Elliptic curves and related topics. Providence, RI: American Mathematical Society. CRM Proc. Lect. Notes. 4, 101-110 (1994).
Let \(E\) be an elliptic curve over \(\mathbb{Q}\). Let \(E(\mathbb{Q})\) be the Mordell- Weil group, \(S_ E (\mathbb{Q})\) the Selmer group, and \(L(E,s)\) the \(L\)- series of \(E\). If one assumes that \(E\) is modular (which is not a major restriction thanks to Wiles’ result), the \(L\)-series has an analytic continuation and satisfies the functional equation \(\Lambda (E,2 - s) = w(E) \Lambda (E,s)\) where \(\Lambda (E,s) : = (2 \pi/ \sqrt N)^{-s} \Gamma (s) L(E,s)\), \(N =\) the conductor of \(E\), and \(w(E) = \pm 1\). The Birch and Swinnerton-Dyer conjecture predicts that if \(w(E) = - 1\), then \(\text{rank}_ \mathbb{Z} E (\mathbb{Q})\) is odd. In this paper, the following result is proposed.
Proposed theorem. Assume that \(w(E) = - 1\). Then the \(p\)-primary subgroup of \(S_ E (\mathbb{Q})\) is infinite for all primes \(p \geq 5\) where \(E\) has good, ordinary reduction.
Note that the conclusion in the above statement is equivalent to the assertion that either \(E(\mathbb{Q})\) is infinite or the \(p\)-primary subgroup of the Tate-Shafarevich group of \(E\) is infinite for all primes \(p\) in the statement. The proposed theorem was previously known to be true for elliptic curves with complex multiplication by the author [Invent. Math. 72, 241-265 (1983; Zbl 0546.14015)]. – This article contains two new results about the proposed theorem:
(1) a new proof of the proposed theorem in the case when elliptic curves have CM: the proof makes use of Rohrlich’s result [D. F. Rohrlich, Invent. Math. 75, 383-408 (1984; Zbl 0565.14008)] and
(2) the conjecture stated below implies the proposed theorem in the case when elliptic curves have no CM; a proof makes use of Hida’s theory on \(p\)-adic deformation of the Tate module \(T_ p (E)\) [H. Hida, Ann. Sci. Éc. Norm. Supér., IV. Sér. 19, 231-273 (1986; Zbl 0607.10022) and Invent. Math. 85, 545-613 (1986; Zbl 0612.10021)]. – The conjecture is formulated as follows:
Conjecture. Let \(\Sigma\) be a finite set of primes. Let \(f\) vary over all normalized newforms of weight 2 for \(\Gamma_ 0 (M)\), where \(M\) is divisible only by primes in \(\Sigma\). Then the order of vanishing of the \(L\)-function \(L(f,s)\) at \(s = 1\) is 0 or 1, except of at most finitely many such \(f\)’s.
A brief discussion is presented for Selmer groups for higher dimensional newforms of even weight \(2k\) that the same approach gives a proof of the analogue of the proposed theorem assuming the above conjecture for newforms of weight \(2k\).
For the entire collection see [Zbl 0788.00052].

14H52 Elliptic curves
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14G05 Rational points
11G05 Elliptic curves over global fields