ncts

Directors

International Advisory Committee

Executive Committee

Academic Committee & Related

Sponsored by

Events

News

[ Events ]

Activity Search

Sort out

Field

Year

．Seminars

	On the Conjecture of Birch and Swinnerton-Dyer for Quadratic Twists of X₀(49) I/II

10:45-15:30
R202, Astronomy-Mathematics Building, NTU

Speaker(s):
John H. Coates (University of Cambridge)

Organizer(s):
Ming-Lun Hsieh (National Taiwan University)

On the Conjecture of Birch and Swinnerton-Dyer for Quadratic Twists of X<sub>0</sub>(49) I/II

Abstract:

Let $X_0(49)$ be the modular elliptic curve with equation

$y^2+xy=x^3-x^2-2x-1,$

and let $E$ be any elliptic curve defined over $\mathbb{Q}$ which is a quadratic twist of $X_0(49)$ . Let $E(\mathbb{Q})$ be the group of rational points of $E$ , $\mathrm{III}$ $(E/\mathbb{Q})$ its Tate-Shafarevich group, and $L(E/\mathbb{Q},s)$ its complex $L$ -series. The aim of my four lectures will be to discuss in some detail the proof of the following theorem.

Theorem 0.1. We have $L(E/\mathbb{Q},1) \neq 0$ if and only if both $E(\mathbb{Q})$ and the 2-primary subgroup of $\mathrm{III}$ $(E/\mathbb{Q})$ are finite. When these equivalent conditions hold, the full Birch-Swinnerton-Dyer conjecture is valid for E.

To my knowledge, a result like this is not konown for the family of all quadratic twists of any other elliptic curve defined over $\mathbb{Q}$ . The proof involves earlier work of K. Rubin and C. Gonzalez-Aviles, as well as some joint recent work by Y. Kezuka, Y. Li, Y. Tian and myself. If time allows, I will also explain a partial generalization to a large family of quadratic twists of the Gross curves $A(q)$ , with complex multiplication by the ring of integers of an imaginary quadratic field $K = \mathbb{Q} ( \sqrt{-q} )$ , where $q$ is any prime which is congruent to 7 modulo 8.

Poster: events_3_551606084034142805.pdf