A crash introduction to BSD conjecture

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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. {<P,Q>=\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\|<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,

\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}.




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