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 $ {\mathop{\mathbb P}^2} $.
Definition 1 (Weierstrass form) $ {E \hookrightarrow \mathop{\mathbb P}^2 } $, In general the form is given by,
$ 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,
$ y^2=x^3+ax+b, \Delta:=4a^3+27b^2\neq 0. \ \ \ \ \ (2) $
Remark 1
$ \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
$ \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,
$ 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,
$ E^{(d)}:y^2=x^3+ad^2x+bd^3 \ \ \ \ \ (4) $
So the twist of a given elliptic curve $ {E} $ is given by:
$ 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 [1]. \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)
$ \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. $ {<P,Q>=\zeta_N\in \mu_{N}(s)} $
What happen if $ {k={\mathbb C}} $? In this case we have a analytic isomorphism:
$ E({\mathbb C})\simeq {\mathbb C}/\Lambda \ \ \ \ \ (7) $
Given by,
$ {\mathbb C}/\Lambda \longrightarrow \mathop{\mathbb P}^2 \ \ \ \ \ (8) $
$ 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.
$ P=\frac{1}{N}, Q=\frac{\tau}{N} \ \ \ \ \ (10) $
Where $ {\tau} $ is induce by
$ \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
$ \mu_N^*\times H/\Gamma(N) \ \ \ \ \ (12) $
Now we discuss the Mordell-Weil theorem.
Theorem 8 (Mordell-Weil theorem)
$ E(F)\simeq {\mathbb Z}^r\oplus E(F)_{tor} $
The proof of the theorem divide into two part:
- Weak Mordell-Weil theorem, i.e. $ {\forall m\in {\mathbb N}} $, $ {E(F)/mE(F)} $ is finite.
- There is a quadratic function,
$ \|\cdot\|: E(F)\longrightarrow {\mathbb R} \ \ \ \ \ (13) $
$ {\forall c\in {\mathbb R}} $, $ {E(F)_c=\{P\in E(F), \|P\|<c\}} $ 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,
$ 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})} $,
$ |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)
$ |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.
$ 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:
$ \# (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.
$ 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:
$ 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
$ Frob{\nu}x\equiv x^{q_{\nu}} (mod w),\forall x\in \mathcal{O}_{\bar Q} \ \ \ \ \ (20) $
So we can define
$ 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)} $.
$ 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,
- $ {ord_{s=1}L(s,E)=rank E(F)} $.
- $ {|Sha(E)|<\infty} $.
- $ {\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} $.