## 2017-02-25

### Theoretical and Computational Aspects of the Birch and Swinnerton-Dyer Conjecture (2016, ICTS Bangalore)

# click the upper-left icon to select videos from the playlist

source: International Centre for Theoretical Sciences   2016年12月12日
12 December 2016 to 22 December 2016
VENUE
The Birch and Swinnerton-Dyer conjecture is a striking example of conjectures in number theory, specifically in arithmetic geometry, that has abundant numerical evidence but not a complete general solution. An elliptic curve, say E, can be represented by points on a cubic equation as below with certain A, B ∈ Q:
y2 = x3 + Ax +B
A Theorem of Mordell says that that E(Q), the set of rational points of E, is a finitely generated abelian group, and thus,
E(Q) = Zr ⊕ T,
for some non-negative integer r and a finite group T. Here, r is called the algebraic rank of E.

The Birch and Swinnerton-Dyer conjecture relates the algebraic rank of E to the value of the L-function, L(E, s), attached to E at s = 1.
Further theoretical understanding, corroborated by computations lead to a stronger version of the BSD conjecture. This refined version of the BSD conjecture provides a very precise formula for the leading term of L(E, s) at s = 1, the coefficient of (s − 1)r, in terms of various arithmetical data attached to E. Thus, the computational side of the BSD conjecture goes hand in hand with the advanced concepts in the theory of Elliptic curves.
In this program, the computational aspects of the BSD conjecture with various illustrative examples, as well as p-adic L-functions, which are the p-adic analogues of the L-functions and other theoretical aspects which are important for the BSD conjecture will be discussed.

Introduction to elliptic curves and BSD Conjecture by Sujatha Ramadorai 1:16:20
[private video]
Coates-Wiles Theorem by Anupam Saikia 1:10:56
Introduction to elliptic curves and BSD Conjecture by Sujatha Ramadorai 1:14:53
[private video]
Coates-Wiles Theorem by Anupam Saikia 1:07:24
Solving Diophantine equations using elliptic curves + Introduction to SAGE by Chandrakant Aribam 59:15
Solving Diophantine equations using elliptic curves + Introduction to SAGE by Chandrakant Aribam 1:07:26
Simultaneous non-vanishing of L-values by Soumya Das 1:04:09
On the 2-part of the Birch–Swinnerton-Dyer conjecture for elliptic curves by Zhibin Liang 1:05:29
[private video]
On Class Number of Number Fields by Debopam Chakraborty 23:26
K-groups and Global Fields by Haiyan Zhou 46:18
Stark-Heegner points and generalised Kato classes by Henri Darmon 53:35
two variables p-adic L function by Shanwen Wang 1:07:06
Torsion points of the Jacobian of modular curves X0(p2 ) and non- by Debargha Banerjee 57:59
Rigidity of p-adic local systems and Abapplications to Shimura varieties by Ruochuan Liu 1:10:27
Comparing the corank of fine Selmer group and Selmer group of elliptic curves by Sudhanshu Shekhar 52:59
[private video]