# Intuition and Stumbling blocks in proving the finiteness of WC group

After reading many articles about the Tate-Shafarevich Group ,i understood that "in naive perspective the group is nothing but the measure of the failure of Hasse principle,

and coming to its definition we can say it as for an Elliptic curve $E$ over a number field $K$ $Ш(E/K)=\mathrm{Ker}(H^1(K,E)\mapsto \prod_{v}H^1(K_v,E))$,where the Galois cohomology comes into play,

and I got that it is the set of "the non-trivial elements of the Tate-Shafarevich group can be thought of as the homogeneous spaces of $A$(where $A$ is an Abelian Variety defined over $K$) that have $K_v$-rational points for every place $v$ of $K$, but no $K$-rational point."

I want the explanations to the following questions:

### Intuitive Questions about the Formulation:

• The Tate-Shafarevich Group has been conjectured to be finite,but it is not proven completely,I was not completely referring to the Tate-shafarevich group of the Elliptic curve ,but for any Abelian Variety $A$(Assume) over a number field, my question that came into my mind primarily was "If the Tate Shafarevich Group was infinite ,then there are infinitely many elements that have Local points ,but do not correspond to any Global point ,(local in the sense it has $K_v$ rational points for all places $v$ , And global means K-rational point)" which in turn leads to complete failure of Hasse principle,which says the existence of correspondence between local part and global part,then my Question is

"Was the Quest for Finiteness of Tate-Shafarevich Group account to support the Hasse-principle,and does the Conjecture about its finiteness imply that Hasse-principle is not False totally??(here totally refers that may be there is a small failure in Hasse principle ,i.e there may be finite amount elements which fail to account for the criteria established by Hasse -principle,but not all elements ,which inturn leads to complete failure of the principle ,so i referred it as totally)

• As we already Know that the Hasse-Minkowski theorem fails for Cubic forms,As elliptic curve is a cubic,then Hasse-principle may not hold good, then

"What is the use of knowing the extent of its failure??,(I mean what was the goal Behind introducing the Tate-Shafarevich Group),i always doubt that there is something to be done with that measure,i mean that measure certainly accounts for something,but i want to know what does it account for,and what can we do by knowing the Extent of Failure)

### Stumbling Blocks:

• Now the Above Question Concerns about the Formulation and background,while this one concerns about the stumbling blocks

"What are the Problems that one Encounter while Proving the Finiteness,to put the Question Differently,What are the Ingredients that one need to prove in order to prove the finiteness of the TS-Group??"(to understand my intention i give an analogue "suppose in order to prove the Fermat's Last Theorem the block was proving the Taniyama-Shimura conjecture ,and to prove the BSD conjecture the finiteness of Tate-Shafarevich Group is a block,so i was asking what are the blocks that occur while proving the finiteness ,I mean are there any such blocks,which if proved may imply the finiteness of the Group)

I end here, please do the following things if you are going to answer

1. Answer it with some tags/notation in which you specify which Question you were answering ,so that i can correspond the answer to that Question
2. If you are down-voting please tell your reason and comments so that i can rectify myself,
3. If you feel that this Question is a bit good question ,suggest me whether i can move it to MO so that i can get better answers

Thanks a lot for taking patience reading my Question, i am always in debt with all those who helped me

• The definition of an elliptic curve over a field $K$ includes a specified $K$-rational point on the curve. So it is not quite correct to say that Hasse principle may not hold for elliptic curves. The WC group looks rather at the torsors of the elliptic curve. For a gentle introduction to WC-group, Hasse principle, etc, you could try having a look at Cassels's "Lectures on elliptic curves". – George Sep 9 '11 at 12:45
• I'm only just starting to learn about the Tate-Shafarevich Group and these are similar questions that have crossed my mind too! – Haikal Yeo May 14 '14 at 22:26