Questions tagged [birch-swinnerton-dyer]

Questions related to the Birch and Swinnerton-Dyer conjecture about the vanishing order and first Taylor coefficient of the L-functions of elliptic curves at the point 1.

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57
votes
3answers
6k views

Is there a “Basic Number Theory” for elliptic curves?

Tate's thesis showed how to profitably analyze $\zeta$ functions of number fields in terms of adelic points on the multiplicative group. In particular, combining Fourier analysis and topology, Tate ...
10
votes
5answers
1k views

Clarification on the weak BSD conjecture

It is usually told that Birch and Swinnerton-Dyer developped their famous conjecture after studying the growth of the function $$ f_E(x) = \prod_{p \le x}\frac{|E(\mathbb{F}_p)|}{p} $$ as $x$ tends to ...
14
votes
3answers
3k views

Recent progress toward Birch and Swinnerton-Dyer conjecture

Has there been any progress toward the Birch and Swinnerton-Dyer conjecture after The current status of the Birch & Swinnerton-Dyer Conjecture
7
votes
2answers
633 views

Power series expansions of $L$-series

Let $\zeta_K(s)$ be the Dedekind zeta function for a number field $K$. We can understand the first non-vanishing coefficient of its Laurent series via the class number formula. Is anything known/...