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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].

##### MSC:
 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