# 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 ${\mathop{\mathbb P}^2}$.

Definition 1 (Weierstrass form) ${E \hookrightarrow \mathop{\mathbb P}^2 }$, In general the form is given by,

$\displaystyle E: y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6 \ \ \ \ \ (1)$

If ${char F \neq 2,3}$, then, we have a much more simper form,

$\displaystyle y^2=x^3+ax+b, \Delta:=4a^3+27b^2\neq 0. \ \ \ \ \ (2)$

Remark 1

$\displaystyle \Delta(E)=\prod_{1\leq i,\neq j\leq 3}(z_i-z_j)$

Where ${z_i^3+az_i+b=0, \forall 1\leq i\leq 3}$.

We have two way to classify the elliptic curve ${E}$ living in a fix field ${F}$. \paragraph{j-invariant} The first one is by the isomorphism in ${\bar F}$. i.e. we say two elliptic curves ${E_1,E_2}$ is equivalent iff

$\displaystyle \exists \rho:\bar F\rightarrow \bar F$

is a isomorphism such that ${\rho(E_1)=E_2}$.

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

$\displaystyle j(E)=1728\frac{4a^3}{4a^3+27b^2} \ \ \ \ \ (3)$

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

Definition 3 (Twist) For a elliptic curve ${E:y^2=x^3+ax+b}$, all elliptic curve twist with ${E}$ is given by,

$\displaystyle E^{(d)}:y^2=x^3+ad^2x+bd^3 \ \ \ \ \ (4)$

So the twist of a given elliptic curve ${E}$ is given by:

$\displaystyle H^1(Gal(\bar F/ F), Aut(E_{\bar F})) \ \ \ \ \ (5)$

Remark 2 Of course a elliptic curve ${E:y^2=x^3+ax+b}$ is the same as ${E:y^2=x^3+ad^2x+bd^4}$, induce by the map ${\mathop{\mathbb P}^1\rightarrow \mathop{\mathbb P}^1, (x,y,1)\rightarrow (x,dy,1)}$.

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

Definition 4 (Integral model) ${s:=Spec(\mathcal{O}_F)}$, ${E\rightarrow E_s}$. ${E_s}$ is regular and minimal, the construction of ${E_s}$ is by the following way, we first construct ${\widetilde{E_s} }$ and then blow up. ${\widetilde E_s}$ is given by the Weierstrass equation with coefficent in ${\mathcal{O}_F}$.

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

Definition 5 (Semistable) the singularity of the minimal model of ${E}$ are ordinary double point.

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

Definition 6 (Level ${n}$ structure)

$\displaystyle \phi: ({\mathbb Z}/n{\mathbb Z})_s^2\longrightarrow E[N] \ \ \ \ \ (6)$

${P=\phi(1,0), Q=\phi(o,1)}$ The weil pairing of ${P,Q}$ is given by a unit in cycomotic fields, i.e. ${=\zeta_N\in \mu_{N}(s)}$

What happen if ${k={\mathbb C}}$? In this case we have a analytic isomorphism:

$\displaystyle E({\mathbb C})\simeq {\mathbb C}/\Lambda \ \ \ \ \ (7)$

Given by,

$\displaystyle {\mathbb C}/\Lambda \longrightarrow \mathop{\mathbb P}^2 \ \ \ \ \ (8)$

$\displaystyle z\longrightarrow (\mathfrak{P}(z), \mathfrak{P}'(z), 1 ) \ \ \ \ \ (9)$

Where ${\mathfrak{P(z)}=\frac{1}{z^2}+\sum_{\lambda\in \Lambda,\lambda\neq 0}(\frac{1}{(z-\lambda)^2}-\frac{1}{\lambda^2})}$, and the Weierstrass equation ${E}$ is given by ${y^2=4x^3-60G_4(\Lambda)x-140G_6(\Lambda)}$. The full n tructure of it is given by ${{\mathbb Z}+{\mathbb Z}\lambda}$ and the value of ${P,Q}$, i.e.

$\displaystyle P=\frac{1}{N}, Q=\frac{\tau}{N} \ \ \ \ \ (10)$

Where ${\tau}$ is induce by

$\displaystyle \Gamma(N):=ker(SL_2({\mathbb Z})\rightarrow SL_2({\mathbb Z}/n{\mathbb Z})) \ \ \ \ \ (11)$

The key point is following:

Theorem 7 ${k={\mathbb C}}$, the moduli of elliptic curves with full level n-structure is identified with

$\displaystyle \mu_N^*\times H/\Gamma(N) \ \ \ \ \ (12)$

Now we discuss the Mordell-Weil theorem.

Theorem 8 (Mordell-Weil theorem)

$\displaystyle E(F)\simeq {\mathbb Z}^r\oplus E(F)_{tor}$

The proof of the theorem divide into two part:

1. Weak Mordell-Weil theorem, i.e. ${\forall m\in {\mathbb N}}$, ${E(F)/mE(F)}$ is finite.
2. There is a quadratic function,

$\displaystyle \|\cdot\|: E(F)\longrightarrow {\mathbb R} \ \ \ \ \ (13)$

${\forall c\in {\mathbb R}}$, ${E(F)_c=\{P\in E(F), \|P\| is finite.

Remark 5 The 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 ${E}$, there is a naive height come from the coefficient of Weierstrass representation, i.e. ${\max\{|4a^3|,|27b^2|\}}$.

While the torsion part have a very clear understanding, thanks to the work of Mazur. The rank part of ${E({\mathbb Q})}$ 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, ${L(s,E)}$.

\paragraph{Local points} We consider a local field ${F_v}$, and a locally value map ${F\rightarrow F_{\nu}}$, then we have the short exact sequences,

$\displaystyle 0\longrightarrow E^0(F_{\nu})\longrightarrow E(F_{\nu})=E_s(\mathcal{O}_F)\longrightarrow E_s(K_0)\longrightarrow 0 \ \ \ \ \ (14)$

Topologically, we know ${E(F_{\nu})}$ are union of disc indexed by ${E_s(k_{\nu})}$,

$\displaystyle |E_s(k_{\nu})| \sim q_{\nu}+1=\# \mathop{\mathbb P}^1(k_{\nu})$

. Define ${a_{\nu}=\# \mathop{\mathbb P}^1(k_{\nu})-|E_s(k_{\nu})|}$, then we have Hasse principle:

Theorem 9 (Hasse principle)

$\displaystyle |a_{\nu}|\leq 2\sqrt{q_{\nu}} \ \ \ \ \ (15)$

Remark 6 I 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 ${E(F)}$ reduce to count points in ${H^1(F_{\nu},E(m))}$, reduce to count the Selmer group ${S(E)[m]}$. We have a short exact sequences to explain the issue.

$\displaystyle 0\longrightarrow E(F)/mE(F) \longrightarrow Sha(E)[m] \longrightarrow E(F)/mE(F)\longrightarrow 0 \ \ \ \ \ (16)$

I mention the Goldfold-Szipiro conjecture here. ${\forall \epsilon>0}$, there ${\exists C_{\epsilon}(E)}$ such that:

$\displaystyle \# (E)\leq c_{\epsilon}(E)N_{E/{\mathbb Q}}(N)^{\frac{1}{2}+\epsilon} \ \ \ \ \ (17)$

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

$\displaystyle L(s,E)=\prod_{\nu: bad}(1-a_{\nu}q_{\nu}^{-s})^{-1}\cdot \prod_{\nu:good}(1-a_{\nu}q_{\nu}^{-s}+q_{\nu}^{1-2s})^{-1} \ \ \ \ \ (18)$

Where ${a_{\nu}=0,1}$ or ${-1}$ when ${E_s}$ has bad reduction on ${\nu}$.

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

$\displaystyle T_l(E):=\varprojlim_{l^n} E[l^n] \ \ \ \ \ (19)$

Then by the transform of different embedding of ${F\hookrightarrow \bar F}$, we know ${ T_{l}(E)/Gal(\bar F/F)}$, decompose it into a lots of orbits, so we can define ${D_{\nu}}$, the decomposition group of ${w}$(extension of ${\nu}$ to ${\bar F}$). We define ${I_{\nu}}$ is the inertia group of ${D_{\nu}}$.

Then ${D_{\nu}/I_{\nu}}$ is generated by some Frobenius elements

$\displaystyle Frob{\nu}x\equiv x^{q_{\nu}} (mod w),\forall x\in \mathcal{O}_{\bar Q} \ \ \ \ \ (20)$

So we can define

$\displaystyle L_{\nu}(s,E)=(1-q_{\nu}^{-s}Frob_{\nu}|T_{l}(E)^{I_{\nu}})^{-1} \ \ \ \ \ (21)$

And then ${L(s,E)=\prod_{\nu}L_{\nu}(s,E)}$.

Faltings have proved ${L_{\nu}(s,E)}$ is the invariant depending the isogenous class in the follwing meaning:

Theorem 10 (Faltings) ${L_{\nu}(s,E)}$ is an isogenous ivariant, i.e. ${E_1}$ isogenous to ${E_2}$ iff ${\forall a.e. \nu}$, ${L_{\nu}(s,E_1)=L_{\nu}(s,E_2)}$.

$\displaystyle L(s,E)=L(s-\frac{1}{2},\pi ) \ \ \ \ \ (22)$

Where ${\pi}$ come from an automorphic representation for ${GL_2(A_F)}$. Now we give the statement of BSD onjecture. ${R}$ is the regulator of ${E}$, i.e. the volume of fine part of ${E(F)}$ with respect to the Neron-Tate height pairing. ${\Omega}$ be the volume of ${\prod_{v|\infty}F(F_v)}$ Then we have,

1. ${ord_{s=1}L(s,E)=rank E(F)}$.
2. ${|Sha(E)|<\infty}$.
3. ${\lim_{s\rightarrow 0}L(s,E)(s-1)^{-rank(E)}=c\cdot \Omega(E)\cdot R(E)\cdot |Sha(E)|\cdot |E(F)_{tor}|^{-2}}$

Here ${c}$ is an explictly positive integer depending only on ${E_{\nu}}$ for ${\nu}$ dividing ${N}$.

# SL_2(Z) and its congruence subgroups

We know we can always do the following thing:

$\displaystyle R\ commutative\ ring \longrightarrow \ "general\ linear\ group" \ GL_2(R) \ \ \ \ \ (1)$

Where

$\displaystyle GL_2(R):=\{\begin{pmatrix} a & b \\ c & d \end{pmatrix}: det \begin{pmatrix} a & b \\ c & d \end{pmatrix}=R^*, a,b,c,d\in R\} \ \ \ \ \ (2)$

Remark 1 Why it is ${R^*}$ but not 1? if it is 1, then the action ${R/GL_2(R)}$ distribute is not trasitive on ${R}$, i.e. every element in unite group present a connected component

Now we consider the subgroup ${SL_2(R)\subset GL_2(R)}$.

$\displaystyle SL_2(R)=\{\begin{pmatrix} a & b\\ c & d \end{pmatrix}: det \begin{pmatrix} a & b\\ c & d \end{pmatrix}=1, a,b,c,d\in R \} \ \ \ \ \ (3)$

We are most interested in the case ${R={\mathbb Z}, {\mathbb Z}/n{\mathbb Z}}$. So how to investigate ${SL_2({\mathbb R})}$? We can look at the action of it on something, for particular, we look at the action of it on Riemann sphere, i.e. ${ \hat {\mathbb C}/({\mathbb R})}$ given by fraction linear map:

$\displaystyle g(z):=\frac{az+b}{cz+d}, g(\infty)=\frac{a}{c} \ \ \ \ \ (4)$

Remark 2 What is fraction linear map? This action carry much more information than the action on vector, thanks for the exist of multiplication in ${{\mathbb C}}$ 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 ${k-1}$ tuples induce by polynomial with degree ${k}$?

Remark 3

1. ${SL_2({\mathbb R})/\pm I:=PSL_2({\mathbb R})}$, then ${PSL_2({\mathbb R}) }$ action faithful on ${\hat C}$, i.e. except identity, every action is nontrivial. This is easy to be proved, observed,

$\displaystyle \frac{az+b}{cz+d}=z,\forall z\in \hat{\mathbb C}\Longrightarrow \begin{pmatrix} a & b\\ c & d \end{pmatrix}=\begin{pmatrix} 1 & 0\\ 0&1 \end{pmatrix} or \begin{pmatrix} -1&0\\ 0&-1 \end{pmatrix} \ \ \ \ \ (5)$

2. Up half plane ${H}$ is invariant under the action of ${PSL_2({\mathbb R})}$, i.e. ${\forall g\in PSL_2({\mathbb R})}$, ${gH=H}$. The proof is following,
3. $\displaystyle \begin{array}{rcl} Im(\frac{az+b}{cz+d}) & = & Im(\frac{(az+b)(c\bar z+d)}{|cz+d|^2})\\ & = & Im(\frac{ac|z|^2+bc\bar z+adz+bd}{|cz+d|^2})\\ & > & 0,\ due\ to\ ad=bc+1. \end{array}$

Now we focus on ${SL_2({\mathbb Z})}$ or the same,${PSL({\mathbb Z})}$. All the argument for ${SL_2(R)}$ make sense for

$\displaystyle \Gamma:= SL_2({\mathbb Z}), \bar \Gamma:=SL_2({\mathbb Z})/\pm I \ \ \ \ \ (6)$

Fix ${N\in {\mathbb N}}$, define,

$\displaystyle \Gamma(N):=\{\begin{pmatrix} a &b\\ c&d \end{pmatrix}, a,d \equiv 1(mod N), b,c\equiv 0(mod N)\} \ \ \ \ \ (7)$

Then ${\Gamma(N)}$ is the kernel of map ${SL_2({\mathbb Z})\rightarrow SL_2({\mathbb Z}/n{\mathbb Z})}$, i.e. we have short exact sequences,

$\displaystyle 0\longrightarrow \Gamma(N)\longrightarrow \Gamma\longrightarrow SL_2({\mathbb Z}/n{\mathbb Z})\longrightarrow 0 \ \ \ \ \ (8)$

Remark 4 The relationship of ${\Gamma(N)\subset \Gamma}$ is just like ${N{\mathbb Z}+1\subset {\mathbb Z}}$.

Definition 1 (Congruence group) A subgroup of ${\Gamma}$ is called a congruence group iff ${\exists n\in {\mathbb N}}$, ${\Gamma(N)\subset G}$.

Example 1 We give two examples of congruence subgroups here.

1. $\displaystyle \Gamma_1(N)=\{\begin{pmatrix} 1& *\\ 0 &1 \end{pmatrix} mod N\} \ \ \ \ \ (9)$

2. $\displaystyle \Gamma_0(N)=\{\begin{pmatrix} * & *\\ 0 & * \end{pmatrix}mod N\} \ \ \ \ \ (10)$

Definition 2 (Fundamental domain)

$\displaystyle F=\{z\in H:-\frac{1}{2}\leq Re(z)\leq \frac{1}{2}\ and |z|\geq 1\} \ \ \ \ \ (11)$

Now here is a theorem charistization the fundamental domain.

Theorem 3 This domain ${F}$ is a fundamental domain of ${\hat {\mathbb H}/({\mathbb R})}$

Proof: Ths key point is ${SL_2({\mathbb Z})}$ have two generators,

1. ${\tau_a: z\rightarrow z+a, \forall a\in {\mathbb Z}}$.
2. ${s: z\rightarrow \frac{1}{z}}$.

Thanks to this two generator exactly divide the action of ${\Gamma}$ on ${H}$ into a lots of scales, then ${\Omega}$ is a fundamental domain is a easy corollary. $\Box$

Remark 5 This is not rigorous, ${H}$ need be replace by ${\hat H}$, but this is very natural to get a modification to a right one.

Remark 6 ${z_1,z_2\in \partial F}$ are ${\Gamma}$ equivalent iff ${Re(z)=\pm \frac{1}{2}}$ and ${z_2=z_1\pm 1}$ or if ${z_1}$ on the unit circle and ${z_2=-\frac{1}{z_1}}$

Remark 7 If ${z\in F}$, then ${\Gamma_z=\pm I}$ expect in the following three case:

1. ${\Gamma=\pm \{\tau,s\}}$ if ${z=i}$.
2. ${\Gamma=\pm\{ I,s\tau, (s\tau)^2\}}$ if ${z=w=-\frac{1}{2}+\frac{\sqrt{-3}}{2}}$.
3. ${\Gamma=\pm\{I,\tau s, (\tau s)^2\}}$ if ${z=-\bar w=\frac{1}{2}+\frac{\sqrt{-3}}{2}}$.

Where ${\tau=\tau_1}$.

Remark 8 The group ${\bar \Gamma=SL_2({\mathbb Z})/\pm I}$ is generated by the two elements ${s}$, ${\tau}$. In other word, any fraction linear transform is a “word” induce by ${s,\tau,s^{-1}.\tau^{-1}}$. But not free group, we have relationship ${s^2=-I,(s\tau)^3=-I}$.

The natural function space on ${F}$ is the memorphic function, under the map: ${H\rightarrow D-\{0\}}$, it has a ${q}$-expension,

$\displaystyle f(q)=\sum_{k\in {\mathbb Z}}a_kq^k \ \ \ \ \ (12)$

And there are only finite many negative ${k}$ such that ${a_k\neq 0}$.