Refined conjectures of the “Birch and Swinnerton-Dyer type”. (English) Zbl 0636.14004

Let \(A\) be an elliptic curve over \(\mathbb{Q}\) which admits a modular parametrization. In an earlier joint work with J. Teitelbaum [Invent. Math. 84, 1–48 (1986; Zbl 0699.14028)] the authors presented \(p\)-adic analogues of the Birch and Swinnerton-Dyer conjectures for \(A\) when \(p\) is a prime of ordinary reduction for \(A\).
In this paper the authors offer remarkable refinements of these conjectures which eliminate any reference to the prime \(p\). To be slightly more precise let \(M\) be a fixed positive integer. Using modular symbols for \(A\) the authors construct an element \(\theta_{A,M}\) in the group ring \(\mathbb{Q}[(\mathbb{Z}/M\mathbb{Z})^*/(\pm 1)]\). If \(R\) is a subring of \(\mathbb{Q}\) containing the coefficients of \(\theta_{A,M}\) then the analogue of having the \(L\)-series \(L(A,s)\) vanish to order \(\geq r\) at \(s=1\) is having \(\theta_{A,M}\) lie in the \(r\)-th power of the augmentation ideal \(I\subseteq R[(\mathbb{Z}/M\mathbb{Z})^*/(\pm 1)]\). The authors conjecture that \(\theta_{A,M}\) lies in \(I^r\) where \(r\) is the rank of \(A(\mathbb{Q})\) plus the number of primes dividing \(M\) which are also primes of split multiplicative reduction for \(A\). In this case the image of \(\theta_{A,M}\) in \(I^r/I^{r+1}\) plays the role of the \(r\)-th coefficient of the Taylor expansion of the \(L\)-function at \(s=1\). Under certain conditions the authors conjecture a precise formula for this image. They also show that if \(r=0\) then their conjecture is implied by the classical Birch and Swinnerton-Dyer conjecture.


11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11G05 Elliptic curves over global fields
14G25 Global ground fields in algebraic geometry
14K15 Arithmetic ground fields for abelian varieties


Zbl 0699.14028
Full Text: DOI


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