# 分类： Algebraic number theory

# A crash introduction to BSD conjecture

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 .

# SL_2(Z) and its congruence subgroups

The pdf version is SL_2(Z) and its congruence subgroups.

We know we can always do the following thing:

Remark 1Why it is but not 1? if it is 1, then the action distribute is not trasitive on , i.e. every element in unite group present a connected component

Now we consider the subgroup .

We are most interested in the case . So how to investigate ? We can look at the action of it on something, for particular, we look at the action of it on Riemann sphere, i.e. given by fraction linear map:

Remark 2What is fraction linear map? This action carry much more information than the action on vector, thanks for the exist of multiplication in and the algebraic primitive theorem. Due to I always looks the fraction linear map as something induce by the permutation of the roots of polynomial of degree 2, this is true at least for fix points, and could natural extension. So how about the higher dimension generate? consider the transform of tuples induce by polynomial with degree ?

Remark 3

- , then action faithful on , i.e. except identity, every action is nontrivial. This is easy to be proved, observed,
- Up half plane is invariant under the action of , i.e. , . The proof is following,

Now we focus on or the same,. All the argument for make sense for

Fix , define,

Then is the kernel of map , i.e. we have short exact sequences,

Remark 4The relationship of is just like .

Definition 1 (Congruence group)A subgroup of is called a congruence group iff , .

Example 1We give two examples of congruence subgroups here.

Definition 2 (Fundamental domain)

Now here is a theorem charistization the fundamental domain.

Theorem 3This domain is a fundamental domain of

*Proof:* Ths key point is have two generators,

- .
- .

Thanks to this two generator exactly divide the action of on into a lots of scales, then is a fundamental domain is a easy corollary.

Remark 5This is not rigorous, need be replace by , but this is very natural to get a modification to a right one.

Remark 6are equivalent iff and or if on the unit circle and

Remark 7If , then expect in the following three case:

- if .
- if .
- if .
Where .

Remark 8The group is generated by the two elements , . In other word, any fraction linear transform is a “word” induce by . But not free group, we have relationship .

The natural function space on is the memorphic function, under the map: , it has a -expension,

And there are only finite many negative such that .