The tag has no usage guidance.

learn more… | top users | synonyms

2
votes
1answer
222 views

Is Ш a good parameter for the failure of Global-Local principle for abelian varieties?

(Comparing to class group cases : we have an isomorphism $Cl(K)\rightarrow \prod \left(K^\times \backslash K_p^\times /O_p^\times \right)$ for a number field $K$. Similarly, for an elliptic curve ...
8
votes
1answer
306 views

Tate-Shafarevich groups over finitely generated fields

Let $G$ be an algebraic group over a number field $k$. One defines the Tate-Shafarevich set of $G$ to be $$Ш(k,G) = \ker\left(H^1(k,G) \to \prod_{v} H^1(k_v,G)\right),$$ where the product is over all ...
11
votes
1answer
348 views

$S$-Tate-Shafarevich groups of elliptic curves

Let $S$ be a finite set of places of a number field $k$ and let $E$ be an elliptic curve over $k$. Define the ''$S$-Tate-Shafarevich group" of $E$ to be $$Ш(E,S) = \ker\left(H^1(k,E) \to \prod_{v ...
35
votes
0answers
782 views

The exponent of Ш of y^2 = x^3 + px, where p is a Fermat prime

For $d$ a non-zero integer, let $E_d$ be the elliptic curve $$ E_d : y^2 = x^3+dx. $$ When we let $d$ be $p = 2^{2^k}+1$, for $k \in \{1,2,3,4\}$, sage tells us that, conditionally on BSD, $$ \# ...
-2
votes
1answer
931 views

Why should I believe in the Siegel's and Hasse's rationale ?

Hello everyone, I was deeply attracted by the Hasse and Siegel's theorems while studying $p$-adic analysis. While reading a paper B.J. Birch and H.P.F. Swinnerton-Dyer - Notes on elliptic curves. I, ...
14
votes
1answer
943 views

What's the Hilbert class field of an elliptic curve?

My question points in a direction similar to Qiaochu's, but it's not the same (or so I think). Let me provide you with a little bit of background first. Let E be an elliptic curve defined over some ...