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
![](https://chart.googleapis.com/chart?cht=tx&chl=X_0(49)&chf=bg,s,333333&chco=ffffff)
be the modular elliptic curve with equation
and let
![](https://chart.googleapis.com/chart?cht=tx&chl=E&chf=bg,s,333333&chco=ffffff)
be any elliptic curve defined over
![](https://chart.googleapis.com/chart?cht=tx&chl=%5Cmathbb%7BQ%7D&chf=bg,s,333333&chco=ffffff)
which is a quadratic twist of
![](https://chart.googleapis.com/chart?cht=tx&chl=X_0(49)&chf=bg,s,333333&chco=ffffff)
. Let
![](https://chart.googleapis.com/chart?cht=tx&chl=E(%5Cmathbb%7BQ%7D)&chf=bg,s,333333&chco=ffffff)
be the group of rational points of
![](https://chart.googleapis.com/chart?cht=tx&chl=E&chf=bg,s,333333&chco=ffffff)
,
![](https://chart.googleapis.com/chart?cht=tx&chl=%5Cmathrm%7BIII%7D&chf=bg,s,333333&chco=ffffff)
![](https://chart.googleapis.com/chart?cht=tx&chl=(E%2F%5Cmathbb%7BQ%7D)&chf=bg,s,333333&chco=ffffff)
its Tate-Shafarevich group, and
![](https://chart.googleapis.com/chart?cht=tx&chl=L(E%2F%5Cmathbb%7BQ%7D%2Cs)&chf=bg,s,333333&chco=ffffff)
its complex
![](https://chart.googleapis.com/chart?cht=tx&chl=L&chf=bg,s,333333&chco=ffffff)
-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
![](https://chart.googleapis.com/chart?cht=tx&chl=L(E%2F%5Cmathbb%7BQ%7D%2C1)%20%5Cneq%200&chf=bg,s,333333&chco=ffffff)
if and only if both
![](https://chart.googleapis.com/chart?cht=tx&chl=E(%5Cmathbb%7BQ%7D)&chf=bg,s,333333&chco=ffffff)
and the 2-primary subgroup of
![](https://chart.googleapis.com/chart?cht=tx&chl=%5Cmathrm%7BIII%7D&chf=bg,s,333333&chco=ffffff)
![](https://chart.googleapis.com/chart?cht=tx&chl=(E%2F%5Cmathbb%7BQ%7D)&chf=bg,s,333333&chco=ffffff)
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
![](https://chart.googleapis.com/chart?cht=tx&chl=%5Cmathbb%7BQ%7D&chf=bg,s,333333&chco=ffffff)
. 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
![](https://chart.googleapis.com/chart?cht=tx&chl=A(q)&chf=bg,s,333333&chco=ffffff)
, with complex multiplication by the ring of integers of an imaginary quadratic field
![](https://chart.googleapis.com/chart?cht=tx&chl=K%20%3D%20%5Cmathbb%7BQ%7D%20(%20%5Csqrt%7B-q%7D%20)&chf=bg,s,333333&chco=ffffff)
, where
![](https://chart.googleapis.com/chart?cht=tx&chl=q&chf=bg,s,333333&chco=ffffff)
is any prime which is congruent to 7 modulo 8.
Poster: events_3_551606084034142805.pdf