The pdf version is A crash introduction to BSD conjecture .

We begin with the Weierstrass form of elliptic equation, i.e. look it as an embedding cubic curve in .

Definition 1 (Weierstrass form), In general the form is given by,

If , then, we have a much more simper form,

Remark 1

Where .

We have two way to classify the elliptic curve living in a fix field . \paragraph{j-invariant} The first one is by the isomorphism in . i.e. we say two elliptic curves is equivalent iff

is a isomorphism such that .

Definition 2 (j-invariant)For a elliptic curve , we have a j-invariant of , given by,

Why j-invariant is important, because j-invariant is the invariant depend the equivalent class of under the classify of isomorphism induce by . But in one equivalent class, there also exist a structure, called twist.

Definition 3 (Twist)For a elliptic curve , all elliptic curve twist with is given by,

So the twist of a given elliptic curve is given by:

Remark 2Of course a elliptic curve is the same as , induce by the map .

But this moduli space induce by the isomorphism of is not good, morally speaking is because of the abandon of universal property. see \cite{zhang}. \paragraph{Level structure} We need a extension of the elliptic curve , this is given by the integral model.

Definition 4 (Integral model), . is regular and minimal, the construction of is by the following way, we first construct and then blow up. is given by the Weierstrass equation with coefficent in .

Remark 3The existence of integral model need Zorn’s lemma.

Definition 5 (Semistable)the singularity of the minimal model of are ordinary double point.

Remark 4Semistable is a crucial property, related to Szpiro’s conjecture.

Definition 6 (Level structure)

The weil pairing of is given by a unit in cycomotic fields, i.e.

What happen if ? In this case we have a analytic isomorphism:

Given by,

Where , and the Weierstrass equation is given by . The full n tructure of it is given by and the value of , i.e.

Where is induce by

The key point is following:

Theorem 7, the moduli of elliptic curves with full level n-structure is identified with

Now we discuss the Mordell-Weil theorem.

Theorem 8 (Mordell-Weil theorem)

The proof of the theorem divide into two part:

Remark 5The proof is following the ideal of infinity descent first found by Fermat. The height is called Faltings height, introduce by Falting. On the other hand, I point out, for elliptic curve , there is a naive height come from the coefficient of Weierstrass representation, i.e. .

While the torsion part have a very clear understanding, thanks to the work of Mazur. The rank part of is still very unclear, we have the BSD conjecture, which is far from a fully understanding until now.

But to understanding the meaning of the conjecture, we need first constructing the zeta function of elliptic curve, .

\paragraph{Local points} We consider a local field , and a locally value map , then we have the short exact sequences,

Topologically, we know are union of disc indexed by ,

. Define , then we have Hasse principle:

Theorem 9 (Hasse principle)

Remark 6I need to point out, the Hasse principle, in my opinion, is just a uncertain principle type of result, there should be a partial differential equation underlying mystery.

So count the points in reduce to count points in , reduce to count the Selmer group . We have a short exact sequences to explain the issue.

I mention the Goldfold-Szipiro conjecture here. , there such that:

\paragraph{L-series} Now I focus on the construction of , there are two different way to construct the L-series, one approach is the Euler product.

Where or when has bad reduction on .

The second approach is the Galois presentation, one of the advantage is avoid the integral model. Given is a fixed prime, we can consider the Tate module:

Then by the transform of different embedding of , we know , decompose it into a lots of orbits, so we can define , the decomposition group of (extension of to ). We define is the inertia group of .

Then is generated by some Frobenius elements

So we can define

And then .

Faltings have proved is the invariant depending the isogenous class in the follwing meaning:

Theorem 10 (Faltings)is an isogenous ivariant, i.e. isogenous to iff , .

Where come from an automorphic representation for . Now we give the statement of BSD onjecture. is the regulator of , i.e. the volume of fine part of with respect to the Neron-Tate height pairing. be the volume of Then we have,

- .
- .

Here is an explictly positive integer depending only on for dividing .