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

1 Answer

I do not know enough to answer questions regarding your quest to tackle the conjecture of Birch and Swinnerton-Dyer. I do however have a suggestion for you, after seeing the general pattern of your questions. It is to try to understand the Class number formula first, before attempting deep investigations on BSD. This is a result quite analogous to the latter; but is already(long ago) proven true and moreover the proof is much simpler than the attempts to crack BSD.

You will notice that each term of class number formula has a more sophisticated analogue in BSD. For example, the Tate-Shafarevich group is analogous to the class group of a number field. A particular part of my humble suggestion is therefore to understand the classgroup, and understand the proof of finiteness of the classgroup. This is not for mere analogy's sake. I will stick my neck out and claim that for someone to prove BSD it is necessary that at some point (s)he read about the finiteness of classgroup, given its importance in algebraic number theory.

If you are able to understand the proof of the class number formula, which is typically the culmination of a first course in algebraic number theory, then you will be able to return to tackle BSD from a more enlightened viewpoint, so to speak.. :) Seeing how involved even that proof is, would give you a better appreciation for the difficulties of number theory.

You could for instance try to read Hecke's lectures on algebraic number theory. It will be accessible for you although you are an engineer and perhaps never studied abstract algebra. Class number formula is approximately among propositions 120 -130 of that book; I do not recollect exactly. Learn one proposition and proof per day; it is very doable in your free time. The reward for you is that by the end of the year, or earlier if you are faster, you are going to have a solid foundation in algebraic number theory.

• +1 for Friendly tone :) , quite lacking on this SE I noticed – ldog Jan 27 '12 at 7:36