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 $latex {\mathop{\mathbb P}^2}&fg=000000$.
Definition 1 (Weierstrass form) $latex {E \hookrightarrow \mathop{\mathbb P}^2 }&fg=000000$, In general the form is given by,
$latex \displaystyle E: y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6 \ \ \ \ \ (1)&fg=000000$
If $latex {char F \neq 2,3}&fg=000000$, then, we have a much more simper form,
$latex \displaystyle y^2=x^3+ax+b, \Delta:=4a^3+27b^2\neq 0. \ \ \ \ \ (2)&fg=000000$
Remark 1
$latex \displaystyle \Delta(E)=\prod_{1\leq i,\neq j\leq 3}(z_i-z_j)&fg=000000$
Where $latex {z_i^3+az_i+b=0, \forall 1\leq i\leq 3}&fg=000000$.
We have two way to classify the elliptic curve $latex {E}&fg=000000$ living in a fix field $latex {F}&fg=000000$. \paragraph{j-invariant} The first one is by the isomorphism in $latex {\bar F}&fg=000000$. i.e. we say two elliptic curves $latex {E_1,E_2}&fg=000000$ is equivalent iff
$latex \displaystyle \exists \rho:\bar F\rightarrow \bar F&fg=000000$
is a isomorphism such that $latex {\rho(E_1)=E_2}&fg=000000$.
Definition 2 (j-invariant) For a elliptic curve $latex {E}&fg=000000$, we have a j-invariant of $latex {E}&fg=000000$, given by,
$latex \displaystyle j(E)=1728\frac{4a^3}{4a^3+27b^2} \ \ \ \ \ (3)&fg=000000$
Why j-invariant is important, because j-invariant is the invariant depend the equivalent class of $latex {E}&fg=000000$ under the classify of isomorphism induce by $latex {\bar F}&fg=000000$. But in one equivalent class, there also exist a structure, called twist.
Definition 3 (Twist) For a elliptic curve $latex {E:y^2=x^3+ax+b}&fg=000000$, all elliptic curve twist with $latex {E}&fg=000000$ is given by,
$latex \displaystyle E^{(d)}:y^2=x^3+ad^2x+bd^3 \ \ \ \ \ (4)&fg=000000$
So the twist of a given elliptic curve $latex {E}&fg=000000$ is given by:
$latex \displaystyle H^1(Gal(\bar F/ F), Aut(E_{\bar F})) \ \ \ \ \ (5)&fg=000000$
Remark 2 Of course a elliptic curve $latex {E:y^2=x^3+ax+b}&fg=000000$ is the same as $latex {E:y^2=x^3+ad^2x+bd^4}&fg=000000$, induce by the map $latex {\mathop{\mathbb P}^1\rightarrow \mathop{\mathbb P}^1, (x,y,1)\rightarrow (x,dy,1)}&fg=000000$.
But this moduli space induce by the isomorphism of $latex {F}&fg=000000$ is not good, morally speaking is because of the abandon of universal property. see [1]. \paragraph{Level $latex {n}&fg=000000$ structure} We need a extension of the elliptic curve $latex {E}&fg=000000$, this is given by the integral model.
Definition 4 (Integral model) $latex {s:=Spec(\mathcal{O}_F)}&fg=000000$, $latex {E\rightarrow E_s}&fg=000000$. $latex {E_s}&fg=000000$ is regular and minimal, the construction of $latex {E_s}&fg=000000$ is by the following way, we first construct $latex {\widetilde{E_s} }&fg=000000$ and then blow up. $latex {\widetilde E_s}&fg=000000$ is given by the Weierstrass equation with coefficent in $latex {\mathcal{O}_F}&fg=000000$.
Remark 3 The existence of integral model need Zorn’s lemma.
Definition 5 (Semistable) the singularity of the minimal model of $latex {E}&fg=000000$ are ordinary double point.
Remark 4 Semistable is a crucial property, related to Szpiro’s conjecture.
Definition 6 (Level $latex {n}&fg=000000$ structure)
$latex \displaystyle \phi: ({\mathbb Z}/n{\mathbb Z})_s^2\longrightarrow E[N] \ \ \ \ \ (6)&fg=000000$
$latex {P=\phi(1,0), Q=\phi(o,1)}&fg=000000$ The weil pairing of $latex {P,Q}&fg=000000$ is given by a unit in cycomotic fields, i.e. $latex {<P,Q>=\zeta_N\in \mu_{N}(s)}&fg=000000$
What happen if $latex {k={\mathbb C}}&fg=000000$? In this case we have a analytic isomorphism:
$latex \displaystyle E({\mathbb C})\simeq {\mathbb C}/\Lambda \ \ \ \ \ (7)&fg=000000$
Given by,
$latex \displaystyle {\mathbb C}/\Lambda \longrightarrow \mathop{\mathbb P}^2 \ \ \ \ \ (8)&fg=000000$
$latex \displaystyle z\longrightarrow (\mathfrak{P}(z), \mathfrak{P}'(z), 1 ) \ \ \ \ \ (9)&fg=000000$
Where $latex {\mathfrak{P(z)}=\frac{1}{z^2}+\sum_{\lambda\in \Lambda,\lambda\neq 0}(\frac{1}{(z-\lambda)^2}-\frac{1}{\lambda^2})}&fg=000000$, and the Weierstrass equation $latex {E}&fg=000000$ is given by $latex {y^2=4x^3-60G_4(\Lambda)x-140G_6(\Lambda)}&fg=000000$. The full n tructure of it is given by $latex {{\mathbb Z}+{\mathbb Z}\lambda}&fg=000000$ and the value of $latex {P,Q}&fg=000000$, i.e.
$latex \displaystyle P=\frac{1}{N}, Q=\frac{\tau}{N} \ \ \ \ \ (10)&fg=000000$
Where $latex {\tau}&fg=000000$ is induce by
$latex \displaystyle \Gamma(N):=ker(SL_2({\mathbb Z})\rightarrow SL_2({\mathbb Z}/n{\mathbb Z})) \ \ \ \ \ (11)&fg=000000$
The key point is following:
Theorem 7 $latex {k={\mathbb C}}&fg=000000$, the moduli of elliptic curves with full level n-structure is identified with
$latex \displaystyle \mu_N^*\times H/\Gamma(N) \ \ \ \ \ (12)&fg=000000$
Now we discuss the Mordell-Weil theorem.
Theorem 8 (Mordell-Weil theorem)
$latex \displaystyle E(F)\simeq {\mathbb Z}^r\oplus E(F)_{tor}&fg=000000$
The proof of the theorem divide into two part:
- Weak Mordell-Weil theorem, i.e. $latex {\forall m\in {\mathbb N}}&fg=000000$, $latex {E(F)/mE(F)}&fg=000000$ is finite.
- There is a quadratic function,
$latex \displaystyle \|\cdot\|: E(F)\longrightarrow {\mathbb R} \ \ \ \ \ (13)&fg=000000$
$latex {\forall c\in {\mathbb R}}&fg=000000$, $latex {E(F)_c=\{P\in E(F), \|P\|<c\}}&fg=000000$ 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 $latex {E}&fg=000000$, there is a naive height come from the coefficient of Weierstrass representation, i.e. $latex {\max\{|4a^3|,|27b^2|\}}&fg=000000$.
While the torsion part have a very clear understanding, thanks to the work of Mazur. The rank part of $latex {E({\mathbb Q})}&fg=000000$ 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, $latex {L(s,E)}&fg=000000$.
\paragraph{Local points} We consider a local field $latex {F_v}&fg=000000$, and a locally value map $latex {F\rightarrow F_{\nu}}&fg=000000$, then we have the short exact sequences,
$latex \displaystyle 0\longrightarrow E^0(F_{\nu})\longrightarrow E(F_{\nu})=E_s(\mathcal{O}_F)\longrightarrow E_s(K_0)\longrightarrow 0 \ \ \ \ \ (14)&fg=000000$
Topologically, we know $latex {E(F_{\nu})}&fg=000000$ are union of disc indexed by $latex {E_s(k_{\nu})}&fg=000000$,
$latex \displaystyle |E_s(k_{\nu})| \sim q_{\nu}+1=\# \mathop{\mathbb P}^1(k_{\nu})&fg=000000$
. Define $latex {a_{\nu}=\# \mathop{\mathbb P}^1(k_{\nu})-|E_s(k_{\nu})|}&fg=000000$, then we have Hasse principle:
Theorem 9 (Hasse principle)
$latex \displaystyle |a_{\nu}|\leq 2\sqrt{q_{\nu}} \ \ \ \ \ (15)&fg=000000$
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 $latex {E(F)}&fg=000000$ reduce to count points in $latex {H^1(F_{\nu},E(m))}&fg=000000$, reduce to count the Selmer group $latex {S(E)[m]}&fg=000000$. We have a short exact sequences to explain the issue.
$latex \displaystyle 0\longrightarrow E(F)/mE(F) \longrightarrow Sha(E)[m] \longrightarrow E(F)/mE(F)\longrightarrow 0 \ \ \ \ \ (16)&fg=000000$
I mention the Goldfold-Szipiro conjecture here. $latex {\forall \epsilon>0}&fg=000000$, there $latex {\exists C_{\epsilon}(E)}&fg=000000$ such that:
$latex \displaystyle \# (E)\leq c_{\epsilon}(E)N_{E/{\mathbb Q}}(N)^{\frac{1}{2}+\epsilon} \ \ \ \ \ (17)&fg=000000$
\paragraph{L-series} Now I focus on the construction of $latex {L(s,E)}&fg=000000$, there are two different way to construct the L-series, one approach is the Euler product.
$latex \displaystyle L(s,E)=\prod_{\nu: bad}(1-a_{\nu}q_{\nu}^{-s})^{-1}\cdot \prod_{\nu:good}(1-a_{\nu}q_{\nu}^{-s}+q_{\nu}^{1-2s})^{-1} \ \ \ \ \ (18)&fg=000000$
Where $latex {a_{\nu}=0,1}&fg=000000$ or $latex {-1}&fg=000000$ when $latex {E_s}&fg=000000$ has bad reduction on $latex {\nu}&fg=000000$.
The second approach is the Galois presentation, one of the advantage is avoid the integral model. Given $latex {l}&fg=000000$ is a fixed prime, we can consider the Tate module:
$latex \displaystyle T_l(E):=\varprojlim_{l^n} E[l^n] \ \ \ \ \ (19)&fg=000000$
Then by the transform of different embedding of $latex {F\hookrightarrow \bar F}&fg=000000$, we know $latex { T_{l}(E)/Gal(\bar F/F)}&fg=000000$, decompose it into a lots of orbits, so we can define $latex {D_{\nu}}&fg=000000$, the decomposition group of $latex {w}&fg=000000$(extension of $latex {\nu}&fg=000000$ to $latex {\bar F}&fg=000000$). We define $latex {I_{\nu}}&fg=000000$ is the inertia group of $latex {D_{\nu}}&fg=000000$.
Then $latex {D_{\nu}/I_{\nu}}&fg=000000$ is generated by some Frobenius elements
$latex \displaystyle Frob{\nu}x\equiv x^{q_{\nu}} (mod w),\forall x\in \mathcal{O}_{\bar Q} \ \ \ \ \ (20)&fg=000000$
So we can define
$latex \displaystyle L_{\nu}(s,E)=(1-q_{\nu}^{-s}Frob_{\nu}|T_{l}(E)^{I_{\nu}})^{-1} \ \ \ \ \ (21)&fg=000000$
And then $latex {L(s,E)=\prod_{\nu}L_{\nu}(s,E)}&fg=000000$.
Faltings have proved $latex {L_{\nu}(s,E)}&fg=000000$ is the invariant depending the isogenous class in the follwing meaning:
Theorem 10 (Faltings) $latex {L_{\nu}(s,E)}&fg=000000$ is an isogenous ivariant, i.e. $latex {E_1}&fg=000000$ isogenous to $latex {E_2}&fg=000000$ iff $latex {\forall a.e. \nu}&fg=000000$, $latex {L_{\nu}(s,E_1)=L_{\nu}(s,E_2)}&fg=000000$.
$latex \displaystyle L(s,E)=L(s-\frac{1}{2},\pi ) \ \ \ \ \ (22)&fg=000000$
Where $latex {\pi}&fg=000000$ come from an automorphic representation for $latex {GL_2(A_F)}&fg=000000$. Now we give the statement of BSD onjecture. $latex {R}&fg=000000$ is the regulator of $latex {E}&fg=000000$, i.e. the volume of fine part of $latex {E(F)}&fg=000000$ with respect to the Neron-Tate height pairing. $latex {\Omega}&fg=000000$ be the volume of $latex {\prod_{v|\infty}F(F_v)}&fg=000000$ Then we have,
- $latex {ord_{s=1}L(s,E)=rank E(F)}&fg=000000$.
- $latex {|Sha(E)|<\infty}&fg=000000$.
- $latex {\lim_{s\rightarrow 0}L(s,E)(s-1)^{-rank(E)}=c\cdot \Omega(E)\cdot R(E)\cdot |Sha(E)|\cdot |E(F)_{tor}|^{-2}}&fg=000000$
Here $latex {c}&fg=000000$ is an explictly positive integer depending only on $latex {E_{\nu}}&fg=000000$ for $latex {\nu}&fg=000000$ dividing $latex {N}&fg=000000$.