Remarks on the Birch-Swinnerton-Dyer conjecture

TL;DR

This paper discusses the Birch-Swinnerton-Dyer conjecture, explores related conjectures, and examines the connection between nilpotent orbits of SL(2,R) and CM points, providing insights into these deep number theory topics.

Contribution

It offers a brief overview of the conjecture and introduces new perspectives on the relation between nilpotent orbits and CM points.

Findings

Relation between nilpotent orbits of SL(2,R) and CM points elucidated

Related conjectures described and analyzed

Provides a concise overview of the Birch-Swinnerton-Dyer conjecture

Abstract

We give a brief description of the Birch-Swinnerton-Dyer conjecture and present related conjectures. We describe the relation between the nilpotent orbits of SL(2,R) and CM points.

Equations

$$y^2=f(x),$$

$$E(\mathbb{Q})=\left\{(x,y)\in {\mathbb{Q}}^2|y^2=f(x)\right\}\cup \{\infty \}$$

$$E(\mathbb{Q})\cong {\mathbb{Z}}^r\oplus E_{\text{tor}}(\mathbb{Q}),$$

$$\mathbb{Z}/n\mathbb{Z}(1\le n\le 10,n=12),$$

$$\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2n\mathbb{Z}(1\le n\le 4).$$

$$E:y^2=x^2+ax+b,a,b\in \mathbb{Z},4a^3+27b^2\ne 0.$$

$$E_e:y^2+xy+y=x^3-x^2-\alpha x+\beta$$

$$\alpha =20067762415575526585033208209338542750930230312178956502$$

$$\beta =34481611795030556467032985690390720374855944359319180361266008296291939448732243429.$$

$$\Gamma_{0}(N):=\left\{\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in SL(2,\mathbb{Z})\,\big{|}\ N|c\ \right\}$$

$$Y_0(N)={\Gamma }_0(N)\backslash \mathbb{H}$$

$$X_0(N)={\Gamma }_0(N)\backslash \mathbb{H}\cup \mathbb{Q}\cup \{\infty \}$$

$$f((az+b)/(cz+d))=(cz+d{)}^{k}f(z)$$

$$f(z)=\sum _{n=1}^{\infty }{c}_n(f)q^n,q:=e^{2\pi iz}$$

$$L(f,s)=\sum _{n=1}^{\infty }{c}_n(f)n^{-s}.$$

$$(f|T_p)(z)=p^{-1}\sum _{i=0}^{p-1}f((z+i)/p)+p^{k-1}(cpz+d{)}^k\cdot f((apz+d)/(cpz+d))$$

$${\rho }_{f,\lambda }:\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})⟶G{L}_2({\bar{\mathbb{Q}}}_{\lambda })$$

$${\bar{\rho }}_{f,\lambda }:\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})⟶G{L}_2({\bar{\mathbb{F}}}_{\ell }),$$

$$[n]P:=nP=P+\cdots +P(n\text{times}),P\in E(\mathbb{C}).$$

$$\text{End}(E)=\left\{\phi :E(\mathbb{C})⟶E(\mathbb{C}),\text{an isogeny}\right\}$$

$$\text{End}(E)⊈\mathbb{Z}\cong \left\{[n]|n\in \mathbb{Z}\right\},$$

$$L(E,s)=\prod _{p|{\Delta }_E}(1-a_p{p}^{-s}{)}^{-1}\cdot \prod _{p\nmid {\Delta }_E}(1-a_p{p}^{-s}+p^{1-2s}{)}^{-1},$$

$$a_p:=p+1-|\bar{E}({\mathbb{F}}_p)|,$$

$$\Lambda (E,s):=N(E{)}^{s/2}(2\pi {)}^{-s}\Gamma (s)L(E,s),\text{Re}s>\frac{3}{2}.$$

$$\Lambda (E,s)=ϵ\Lambda (E,2-s),ϵ=\pm 1.$$

$$\prod _{p<x}\frac{N(p)+1}{p}\sim C(\text{log}p{)}^r\text{as}x⟶\infty$$

$$L(E,s)=c_0(s-1{)}^m+c_1(s-1{)}^{m+1}+\cdots$$

$$c_0=\underset{s\to 1}{\lim }\frac{L(E,s)}{(s-1{)}^m}=\alpha \cdot R\cdot |E_{\text{tor}}(\mathbb{Q}){|}^{-1}\cdot \Omega \cdot S,$$

$${\int }_{\gamma }^{\infty }\frac{dx}{\sqrt{f(x)}}(\gamma =\text{the largest root of}f(x)=0)$$

$$\tilde{E}:-139y^2=x^3+10x^2-20x+8$$

Full text

1. Introduction

On May 24, 2000, the Clay Mathematics Institute (CMI for short) announced that it would award prizes of 1 million dollars each for solutions to seven mathematics problems. These seven problems are

1. Problem 1.

The “P versus NP" Problem 2. Problem 2.

The Riemann Hypothesis 3. Problem 3.

The Poincaré Conjecture 4. Problem 4.

The Hodge Conjecture 5. Problem 5.

The Birch-Swinnerton-Dyer Conjecture (briefly, the BSD conjecture) 6. Problem 6.

The Navier-Stokes Equations : Prove or disprove the existence and

smoothness of solutions to the three dimensional Navier-Stokes equations. 7. Problem 7.

Yang-Mills Theory : Prove that quantum Yang-Mills fields exist

and have a mass gap.

Problem 1 is arisen from theoretical computer science, Problem 2 and Problem 5 from number theory, Problem 3 from topology, Problem 4 from algebraic geometry and topology, and finally problem 6 and 7 are related to physics. For more details on some stories about these problems, we refer to Notices of AMS, vol. 47, no. 8, pp. 877-879 (September 2000) and the homepage of CMI. In 2003, Problem 3 was solved by Grisha Perelman [35, 36, 37]. We refer to [12, 13, 21, 32, 33] for more details on Perelman’s work. Recently Bhargava’s school computed the Selmer groups of an elliptic curve and so solved Problem 5 partially.

The purpose of this paper is to describe the relation between the nilpotent orbits of $SL(2,\mathbb{R})$ and CM points and to present several conjectures relating to the BSD Conjecture.

The paper is organized as follows. From Section 2 to Section 5, we will explain Problem 5, that is, the Birch-Swinnerton-Dyer conjecture which was proposed by the English mathematicians, B. Birch and H. P. F. Swinnerton-Dyer [7, 8] around 1960s in some detail. This conjecture says that if $E$ is an elliptic curve defined over $\mathbb{Q}$, then the algebraic rank of $E$ equals the analytic rank of $E.$ In 2001, the Shimura-Taniyama conjecture stating that any elliptic curve defined over $\mathbb{Q}$ is modular was shown to be true by Breuil, Conrad, Diamond and Taylor [10]. This fact shed some lights on the solution of the BSD conjecture. In Section 6, we describe the connection between the heights of Heegner points on modular curves $X_{0}(N)$ and Fourier coefficients of modular forms of half integral weight or of the Jacobi forms corresponding to them by the Skoruppa-Zagier correspondence. Most of the materials in Section 2–6 were already printed in [46]. In Section 7, we briefly review the works done recently by the school of Manjul Bhargava [1, 2, 3, 4, 5, 6]. In Section 8, we describe the adjoint orbits of $SL(2,\mathbb{R})$ in its Lie algebra $\mathfrak{s}\mathfrak{l}(2,\mathbb{R})$ explicitly. In the final section, we describe the relation between the nilpotent orbits of $SL(2,\mathbb{R})$ and CM points. We propose several conjectures relating to the Birch-Swinnerton-Dyer conjecture.

Notations : We denote by $\mathbb{Q},\,\mathbb{R}$ and $\mathbb{C}$ the fields of rational numbers, real numbers and complex numbers respectively. $\mathbb{Z}$ and $\mathbb{Z}^{+}$ denotes the ring of integers and the set of positive integers respectively. $\mathbb{H}$ denotes the Poincaré upper half plane.

2. The Mordell-Weil Group

A curve $E$ is said to be an elliptic curve over $\mathbb{Q}$ if it is a nonsingular projective curve of genus 1 with its affine model

[TABLE]

where $f(x)$ is a polynomial of degree 3 with integer coefficients and with 3 distinct roots over $\mathbb{C}$. An elliptic curve over $\mathbb{Q}$ has an abelian group structure with distinguished element $\infty$ as an identity element. The set $E(\mathbb{Q})$ of rational points given by

[TABLE]

also has an abelian group structure.

L. J. Mordell (1888-1972) [31] proved the following theorem in 1922.

Theorem 2.1** (Mordell,1922).**

$E(\mathbb{Q})$ is finitely generated, that is,

[TABLE]

where $r$ is a nonnegative integer and $E_{\text{tor}}(\mathbb{Q})$ is the torsion subgroup of $E(\mathbb{Q})$.

Definition 2.2**.**

Around 1930, A. Weil (1906-1998) proved that the set $A(\mathbb{Q})$ of rational points on an abelian variety $A$ defined over $\mathbb{Q}$ is finitely generated. An elliptic curve is an abelian variety of dimension one. Therefore $E(\mathbb{Q})$ is called the Mordell-Weil group and the integer $r$ is said to be the algebraic rank of $E$.

In 1977, B. Mazur (1937- )[29] discovered the structure of the torsion subgroup $E_{\text{tor}}(\mathbb{Q})$ completely using a deep theory of modular curves.

Theorem 2.3** (Mazur, 1977).**

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. Then the torsion subgroup $E_{\text{tor}}(\mathbb{Q})$ is isomorphic to the following 15 groups

[TABLE]

[TABLE]

E. Lutz (1914-?) and T. Nagell (1895-?) obtained the following result independently.

Theorem 2.4** (Lutz, 1937; Nagell, 1935).**

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ given by

[TABLE]

Suppose that $P=(x_{0},y_{0})$ is an element of the torsion subgroup $E_{\text{tor}}(\mathbb{Q})$. Then

(a) $\quad x_{0},y_{0}\in\mathbb{Z}$, and

(b) $2P=0$  or  $y_{0}^{2}|(4a^{3}+27b^{2}).$

We observe that the above theorem gives an effective method for bounding $E_{\text{tor}}(\mathbb{Q})$. According to Theorem B and C, we know the torsion part of $E(\mathbb{Q})$ satisfactorily. But we have no idea of the free part of $E(\mathbb{Q})$ so far. As for the algebraic rank $r$ of an elliptic curve $E$ over $\mathbb{Q}$, Noam Elkies found an example of an elliptic curve of rank 28 in 2006. Indeed, that elliptic curve is given by

[TABLE]

has its algebraic rank 28. Here

[TABLE]

and

[TABLE]

Elkies also computed 28 generators of $E_{e}(\mathbb{Q})$ (cf. https://web.math.pmf.unizg.hr/ duje/tors/rk28.html).

Conjecture A**.**

Given a nonnegative integer $n$, there is an elliptic curve $E$ over $\mathbb{Q}$ with its algebraic rank $n$.

The algebraic rank of an elliptic curve is an invariant under an isogeny. Here an isogeny of an elliptic curve $E$ means a holomorphic map $\varphi:E(\mathbb{C})\longrightarrow E(\mathbb{C})$ satisfying the condition $\varphi(0)=0.$

3. Modular Elliptic Curves

For a positive integer $N\in\mathbb{Z}^{+},$ we let

[TABLE]

be the Hecke subgroup of $SL(2,\mathbb{Z})$ of level $N$. Let $\mathbb{H}$ be the Poincaré upper half plane. Then

[TABLE]

is a noncompact surface, and

[TABLE]

is a compactification of $Y_{0}(N).$ We recall that a cusp form of weight $k\geq 1$ and level $N\geq 1$ is a holomorphic function $f$ on $\mathbb{H}$ such that for all $\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\Gamma_{0}(N)$ and for all $z\in\mathbb{H}$, we have

[TABLE]

and $|f(z)|^{2}(\text{Im}\,z)^{k}$ is bounded on $\mathbb{H}$. We denote the space of all cusp forms of weight $k$ and level $N$ by $S_{k}(N)$. If $f\in S_{k}(N)$, then it has a Fourier expansion

[TABLE]

convergent for all $z\in\mathbb{H}.$ We note that there is no constant term due to the boundedness condition on $f$. Now we define the $L$-series $L(f,s)$ of $f$ to be

[TABLE]

For each prime $p\nmid N$, there is a linear operator $T_{p}$ on $S_{k}(N)$, called the Hecke operator, defined by

[TABLE]

for any $\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in SL(2,\mathbb{Z})$ with $c\equiv 0\,(N)$ and $d\equiv p\,(N).$ The Hecke operators $T_{p}$ for $pN$ can be diagonalized on the space $S_{k}(N)$ and a simultaneous eigenvector is called an eigenform. If $f\in S_{k}(N)$ is an eigenform, then the corresponding eigenvalues, $a_{p}(f)$, are algebraic integers and we have $c_{p}(f)=a_{p}(f)\,c_{1}(f).$

Let $\lambda$ be a place of the algebraic closure ${\bar{\mathbb{Q}}}$ in $\mathbb{C}$ above a rational prime $\ell$ and ${\bar{\mathbb{Q}}}_{\lambda}$ denote the algebraic closure of $\mathbb{Q}_{\ell}$ considered as a ${\bar{\mathbb{Q}}}$-algebra via $\lambda$. It is known that if $f\in S_{k}(N),$ there is a unique continuous irreducible representation

[TABLE]

such that for any prime ${p\nmid N}{\ell}$, $\rho_{f,\lambda}$ is unramified at $p$ and $\text{tr}\,\rho_{f,\lambda}(\text{Frob}_{p})=a_{p}(f).$ The existence of $\rho_{f,\lambda}$ is due to G. Shimura (1930- ) if $k=2$ [39], to P. Deligne (1944- ) if $k>2$ [15] and to P. Deligne and J.-P. Serre (1926- ) if $k=1$ [16]. Its irreducibility is due to Ribet if $k>1$ [38], and to Deligne and Serre if $k=1$ [16]. Moreover $\rho_{f,\lambda}$ is odd and potentially semi-stable at $\ell$ in the sense of Fontaine. We may choose a conjugate of $\rho_{f,\lambda}$ which is valued in $GL_{2}(\mathcal{O}_{{\bar{\mathbb{Q}}}_{\lambda}})$, and reducing modulo the maximal ideal and semi-simplifying yields a continuous representation

[TABLE]

which, up to isomorphism, does not depend on the choice of conjugate of $\rho_{f,\lambda}$.

Definition 3.1**.**

Let $\rho:\text{Gal}({\bar{\mathbb{Q}}}/\mathbb{Q})\longrightarrow GL_{2}({\bar{\mathbb{Q}}}_{\ell})$ be a continuous representation which is unramified outside finitely many primes and for which the restriction of $\rho$ to a decomposition group at $\ell$ is potentially semi-stable in the sense of Fontaine. We call $\rho$ modular if $\rho$ is isomorphic to $\rho_{f,\lambda}$ for some eigenform $f$ and some $\lambda|\ell.$

Definition 3.2**.**

An elliptic curve $E$ defined over $\mathbb{Q}$ is said to be modular if there exists a surjective holomorphic map $\varphi:X_{0}(N)\longrightarrow E(\mathbb{C})$ for some positive integer $N$.

In 2001 C. Breuil, B. Conrad, F. Diamond and R. Taylor [10] proved that the Taniyama-Shimura conjecture is true.

Theorem 3.3** ([10], 2001).**

An elliptic curve defined over $\mathbb{Q}$ is modular.

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. For a positive integer $n\in\mathbb{Z}^{+}$, we define the isogeny $[n]:E(\mathbb{C})\longrightarrow E(\mathbb{C})$ by

[TABLE]

For a negative integer $n$, we define the isogeny $[n]:E(\mathbb{C})\longrightarrow E(\mathbb{C})$ by $[n]P:=-[-n]P,\ P\in E(\mathbb{C})$, where $-[-n]P$ denotes the inverse of the element $[-n]P$. And $[0]:E(\mathbb{C})\longrightarrow E(\mathbb{C})$ denotes the zero map. For an integer $n\in\mathbb{Z},\ [n]$ is called the multiplication-by-$n$ homomorphism. The kernel $E[n]$ of the isogeny $[n]$ is isomorphic to $\mathbb{Z}/n\mathbb{Z}\oplus\mathbb{Z}/n\mathbb{Z}.$ Let

[TABLE]

be the endomorphism group of $E$. An elliptic curve $E$ over $\mathbb{Q}$ is said to have complex multiplication (or CM for short) if

[TABLE]

that is, there is a nontrivial isogeny $\varphi:E(\mathbb{C})\longrightarrow E(\mathbb{C})$ such that $\varphi\not=[n]$ for all integers $n\in\mathbb{Z}.$ Such an elliptic curve is called a CM curve. For most of elliptic curves $E$ over $\mathbb{Q}$, we have $\text{End}(E)\cong\mathbb{Z}.$

4. The $L$-Series of an Elliptic Curve

Let $E$ be an elliptic curve over $\mathbb{Q}$. The $L$-series $L(E,s)$ of $E$ is defined as the product of the local $L$-factors :

[TABLE]

where $\Delta_{E}$ is the discriminant of $E$, $p$ is a prime, and if $p\nmid\Delta_{E}$,

[TABLE]

and if $p|\Delta_{E},$ we set $a_{p}:=0,\,1,\,-1$ if the reduced curve ${\bar{E}}/\mathbb{F}_{p}$ has a cusp at $p$, a split node at $p$, and a nonsplit node at $p$ respectively. Then $L(E,s)$ converges absolutely for $\text{Re}\,s>{\frac{3}{2}}$ from the classical result that $|a_{p}|<2\sqrt{p}$ for each prime $p$ due to H. Hasse (1898-1971) and is given by an absolutely convergent Dirichlet series. We remark that $x^{2}-a_{p}x+p$ is the characteristic polynomial of the Frobenius map acting on ${\bar{E}}(\mathbb{F}_{p})$ by $(x,y)\mapsto(x^{p},y^{p}).$

Conjecture B**.**

Let $N(E)$ be the conductor of an elliptic curve $E$ over $\mathbb{Q}$ ([S], p. 361). We set

[TABLE]

Then $\Lambda(E,s)$ has an analytic continuation to the whole complex plane and satisfies the functional equation

[TABLE]

The above conjecture is now true because the Shimura-Taniyama conjecture is true (cf. Theorem E). We have some knowledge about analytic properties of $E$ by investigating the $L$-series $L(E,s)$. The order of $L(E,s)$ at $s=1$ is called the analytic rank of $E$.

Now we explain the connection between the modularity of an elliptic curve $E$, the modularity of the Galois representation and the $L$-series of $E$. For a prime $\ell$, we let $\rho_{E,\ell}$ (resp. ${\bar{\rho}}_{E,\ell})$ denote the representation of $\text{Gal}({\bar{\mathbb{Q}}}/\mathbb{Q})$ on the $\ell$-adic Tate module (resp. the $\ell$-torsion) of $E({\bar{\mathbb{Q}}}).$ Let $N(E)$ be the conductor of $E$. Then it is known that the following conditions are equivalent :

1. (1)

The $L$-function $L(E,s)$ of $E$ equals the $L$-function $L(f,s)$ for some eigenform $f$. 2. (2)

The $L$-function $L(E,s)$ of $E$ equals the $L$-function $L(f,s)$ for some eigenform $f$

of weight 2 and level $N(E)$. 3. (3)

For some prime $\ell$, the representation $\rho_{E,\ell}$ is modular. 4. (4)

For all primes $\ell$, the representation $\rho_{E,\ell}$ is modular. 5. (5)

There is a non-constant holomorphic map $X_{0}(N)\longrightarrow E(\mathbb{C})$ for some

positive integer $N$. 6. (6)

There is a non-constant morphism $X_{0}(N(E))\longrightarrow E$ which is defined over $\mathbb{Q}.$ 7. (7)

$E$ admits a hyperbolic uniformization of arithmetic type (cf. [30] and [42]).

5. The Birch-Swinnerton-Dyer conjecture

Now we state the BSD conjecture.

The BSD Conjecture. Let $E$ be an elliptic curve over $\mathbb{Q}$. Then the algebraic rank of $E$ equals the analytic rank of $E.$

I will describe some historical backgrounds about the BSD conjecture. Around 1960, Birch (1931- ) and Swinnerton-Dyer (1927- ) formulated a conjecture which determines the algebraic rank $r$ of an elliptic curve $E$ over $\mathbb{Q}$. The idea is that an elliptic curve with a large value of $r$ has a large number of rational points and should therefore have a relatively large number of solutions modulo a prime $p$ on the average as $p$ varies. For a prime $p$, we let $N(p)$ be the number of pairs of integers $x,y\,(\text{mod}\,p)$ satisfying (2.1) as a congruence (mod $p$). Then the BSD conjecture in its crudest form says that we should have an asymptotic formula

[TABLE]

for some constant $C>0.$ If the $L$-series $L(E,s)$ has an analytic continuation to the whole complex plane (this fact is conjectured; cf.  Conjecture F), then $L(E,s)$ has a Taylor expansion

[TABLE]

at $s=1$ for some non-negative integer $m\geq 0$ and constant $c_{0}\not=0.$ The BSD conjecture says that the integer $m$, in other words, the analytic rank of $E$, should equal the algebraic rank $r$ of $E$ and furthermore the constant $c_{0}$ should be given by

[TABLE]

where $\alpha>0$ is a certain constant, $R$ is the elliptic regulator of $E,\ |E_{\text{tor}}(\mathbb{Q})|$ denotes the order of the torsion subgroup $E_{\text{tor}}(\mathbb{Q})$ of $E(\mathbb{Q})$, $\Omega$ is a simple rational multiple (depending on the bad primes) of the elliptic integral

[TABLE]

and $S$ is an integer square which is supposed to be the order of the Tate-Shafarevich group $\text{III}(E)$ of $E$.

The Tate-Shafarevich group $\text{III}(E)$ of $E$ is a very interesting subject to be investigated in the future. Unfortunately $\text{III}(E)$ is still not known to be finite. So far an elliptic curve whose Tate-Shafarevich group is infinite has not been discovered. So many mathematicians propose the following.

Conjecture C**.**

The Tate-Shafarevich group $\text{III}(E)$ of $E$ is finite.

There are some evidences supporting the BSD conjecture. I will list these evidences chronologically.

Result 1** (Coates-Wiles [14], 1977).**

Let $E$ be a CM curve over $\mathbb{Q}$. Suppose that the analytic rank of $E$ is zero. Then the algebraic rank of $E$ is zero.

Result 2** (Rubin [38], 1981).**

Let $E$ be a CM curve over $\mathbb{Q}$. Assume that the analytic rank of $E$ is zero. Then the Tate-Shafarevich group $\text{III}(E)$ of $E$ is finite.

Result 3** (Gross-Zagier [19], 1986 ; [10], 2001).**

Let $E$ be an elliptic curve over $\mathbb{Q}$. Assume that the analytic rank of $E$ is equal to one and $\epsilon=-1$ (cf. Conjecture F). Then the algebraic rank of $E$ is equal to or bigger than one.

Result 4** (Gross-Zagier [19], 1986).**

There exists an elliptic curve $E$ over $\mathbb{Q}$ such that $\text{rank}\,E(\mathbb{Q})=\text{ord}_{s=1}L(E,s)=3$. For instance, the elliptic curve ${\tilde{E}}$ given by

[TABLE]

satisfies the above property.

Result 5** **(Kolyvagin [24], 1990 : Gross-Zagier [19], 1986 : Bump-Friedberg-Hoffstein [11], 

1990 : Murty-Murty [34], 1990 : [10], 2001).

Let $E$ be an elliptic curve over $\mathbb{Q}$. Assume that the analytic rank of $E$ is 1 and $\epsilon=-1$. Then algebraic rank of $E$ is equal to 1.

Result 6** **(Kolyvagin [24], 1990 : Gross-Zagier [19], 1986 : Bump-Friedberg-Hoffstein [11],

 1990 : Murty-Murty [34], 1990 : [10], 2001).

Let $E$ be an elliptic curve over $\mathbb{Q}$. Assume that the analytic rank of $E$ is zero and $\epsilon=1$. Then algebraic rank of $E$ is equal to zero.

Cassels proved the fact that if an elliptic curve over $\mathbb{Q}$ is isogeneous to another elliptic curve $E^{\prime}$ over $\mathbb{Q}$, then the BSD conjecture holds for $E$ if and only if th e BSD conjecture holds for $E^{\prime}$.

6. Jacobi Forms and Heegner Points

In this section, I shall describe the result of Gross-Kohnen-Zagier [20] roughly.

First we begin with giving the definition of Jacobi forms. By definition a Jacobi form of weight $k$ and index $m$ is a holomorphic complex valued function $\phi(z,w)\,(z\in\mathbb{H},\,z\in\mathbb{C})$ satisfying the transformation formula

[TABLE]

for all $\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in SL(2,\mathbb{Z})$ and $(\lambda,\mu)\in\mathbb{Z}^{2}$, and having a Fourier expansion of the form

[TABLE]

We remark that the Fourier coefficients $c(n,r)$ depend only on the discrimnant $D=r^{2}-4mn$ and the residue $r\,(\text{mod}\ 2m).$ From now on, we put $\Gamma_{1}:=SL(2,\mathbb{Z}).$ We denote by $J_{k,m}(\Gamma_{1})$ the space of all Jacobi forms of weight $k$ and index $m$. It is known that one has the following isomorphisms

[TABLE]

where $\Gamma_{2}$ denotes the Siegel modular group of degree 2, $[\Gamma_{2},k]^{M}$ denotes the Maass space introduced by H. Maass (1911-1993) (cf. [26, 27, 28]), $M_{k-{\frac{1}{2}}}^{+}(\Gamma_{0}(4))$ denotes the Kohnen space introduced by W. Kohnen [22] and $[\Gamma_{1},2k-2]$ denotes the space of modular forms of weight $2k-2$ with respect to $\Gamma_{1}$. We refer to [42] and [44], pp. 65-70 for a brief detail. The above isomorphisms are compatible with the action of the Hecke operators. Moreover, according to the work of Skoruppa and Zagier [41], there is a Hecke-equivariant correspondence between Jacobi forms of weight $k$ and index $m$, and certain usual modular forms of weight $2k-2$ on $\Gamma_{0}(N).$

Now we give the definition of Heegner points of an elliptic curve $E$ over $\mathbb{Q}$. By [10], $E$ is modular and hence one has a surjective holomorphic map $\phi_{E}:X_{0}(N)\longrightarrow E(\mathbb{C}).$ Let $K$ be an imaginary quadratic field of discriminant $D$ such that every prime divisor $p$ of $N$ is split in $K$. Then it is easy to see that $(D,N)=1$ and $D$ is congruent to a square $r^{2}$ modulo $4N$. Let $\Theta$ be the set of all $z\in\mathbb{H}$ satisfying the following conditions

[TABLE]

[TABLE]

Then $\Theta$ is invariant under the action of $\Gamma_{0}(N)$ and $\Theta$ has only a $h_{K}$ $\Gamma_{0}(N)$-orbits, where $h_{K}$ is the class number of $K$. Let $z_{1},\cdots,z_{h_{K}}$ be the representatives for these $\Gamma_{0}(N)$-orbits. Then $\phi_{E}(z_{1}),\cdots,\phi_{E}(z_{h_{K}})$ are defined over the Hilbert class field $H(K)$ of $K$, i.e., the maximal everywhere unramified extension of $K$. We define the Heegner point $P_{D,r}$ of $E$ by

[TABLE]

We observe that $\epsilon=1$, then $P_{D,r}\in E(\mathbb{Q})$.

Let $E^{(D)}$ be the elliptic curve (twisted from $E$) given by

[TABLE]

Then one knows that the $L$-series of $E$ over $K$ is equal to $L(E,s)\,L(E^{(D)},s)$ and that $L(E^{(D)},s)$ is the twist of $L(E,s)$ by the quadratic character of $K/\mathbb{Q}$.

Theorem 6.1** (Gross-Zagier [19, 10]).**

Let $E$ be an elliptic curve over $\mathbb{Q}$ of conductor $N$ such that $\epsilon=-1.$ Assume that $D$ is odd. Then

[TABLE]

where $c_{E}$ is a positive constant not depending on $D$ and $r,\ u$ is a half of the number of units of $K$ and ${\hat{h}}$ denotes the canonical height of $E$.

Since $E$ is modular by [10], there is a cusp form of weight 2 with respect to $\Gamma_{0}(N)$ such that $L(f,s)=L(E,s).$ Let $\phi(z,w)$ be the Jacobi form of weight 2 and index $N$ which corresponds to $f$ via the Skoruppa-Zagier correspondence. Then $\phi(z,w)$ has a Fourier series of the form (6.2).

B. Gross, W. Kohnen and D. Zagier [20] obtained the following result.

Theorem 6.2** (Gross-Kohnen-Zagier [20, 10]).**

Let $E$ be an elliptic curve over $\mathbb{Q}$ with conductor $N$ and suppose that $\epsilon=-1,\ r=1.$ Suppose that $(D_{1},D_{2})=1$ and $D_{i}\equiv r_{i}^{2}\,(\text{mod}\,4N)\,(i=1,2).$ Then

[TABLE]

where $c_{E}^{\prime}$ is a positive constant not depending on $D_{1},\,r_{1}$ and $D_{2},\,r_{2}$ and $<\ ,\ >$ is the height pairing induced from the Néron-Tate height function ${\hat{h}}$, that is, ${\hat{h}}(P_{D,r})=<P_{D,r},P_{D,r}>$.

We see from the above theorem that the value of $<P_{D_{1},r_{1}},P_{D_{2},r_{2}}>$ of two distinct Heegner points is related to the product of the Fourier coefficients $c((r_{1}^{2}-D_{1})/(4N),r_{1})\,c((r_{2}^{2}-D_{2})/(4N),r_{2})$ of the Jacobi forms of weight 2 and index $N$ corresponded to the eigenform $f$ of weight 2 associated to an elliptic curve $E$. We refer to [45] and [47] for more details.

Corollary. There is a point $P_{0}\in E(\mathbb{Q})\otimes_{\mathbb{Z}}\mathbb{R}$ such that

[TABLE]

for all $D$ and $r\,(D\equiv r^{2}\,(\text{mod}\,4N))$ with $(D,2N)=1.$

The corollary is obtained by combining Theorem H and Theorem I with the Cauchy-Schwarz inequality in the case of equality.

Remark 6.3**.**

R. Borcherds  [9] generalized the Gross-Kohnen-Zagier theorem to certain more general quotients of Hermitian symmetric spaces of high dimension, namely to quotients of the space associated to an orthogonal group of signature $(2,b)$ by the unit group of a lattice. Indeed he relates the Heegner divisors on the given quotient space to the Fourier coefficients of vector-valued holomorphic modular forms of weight $1+{\frac{b}{2}}$.

7. Brief Reviews on the Works of Bhargava’s School

In this section, we briefly describe the recent works done by Bhargava’s School. First we review the Selmer group. Let $A=A(\overline{K})$ and $B=B(\overline{K})$ be abelian varieties over number field $K$ and let $f:A\rightarrow B$ be a nonzero isogeny with finite kernel

[TABLE]

Then we get a short exact sequence:

[TABLE]

Let $G_{K}=$Gal$(\overline{K}/K)$ be the Galois group of $\overline{K}$ over $K$. Then we have the following long exact sequence of Galois cohomology groups :

[TABLE]

From (7.1) we obtain the following short exact sequence:

[TABLE]

We let $K_{v}$ and $G_{v}$ the completion at $v$ and the decomposition group of $K_{v}$ respectively. We put $A_{v}:=A(\overline{K}_{v})$. Since $G_{v}$ acts on $A_{v},\;B_{v}$ and $A_{v}[f]$, we get the short exact sequence:

[TABLE]

From the above short exact sequences (7.2) and (7.3), we have the following commutative diagram:

[TABLE]

Here $\mathrm{Res}_{v}(\xi)=\xi|_{G_{v}}$ for $\xi\in H^{1}(G_{K},A[f])$.

Definition 7.1**.**

With the above notations, the $f$-Selmer group of $A/K$ is defined by

[TABLE]

Definition 7.2**.**

The Shafarevich-Tate group of $A/K$ is defined by

[TABLE]

Theorem 7.3**.**

With the above notations, we get the following facts:

• (a)

There is an exact sequence

[TABLE]

• (b)

The Selmer group $\mathrm{Sel}^{(f)}(A/K)$ is finite.

Example 7.1**.**

Let $A=B=E$ be an elliptic curve over $K=\mathbb{Q}$ and let $f=[m]$ be the multiplication by $m$ endomorphism. Then we obtain the following exact sequence :

[TABLE]

$\mathrm{Sel}_{m}:=\mathrm{Sel}^{[m]}(E/\mathbb{Q})$ is called the $m$-Selmer group.

Any elliptic curve $E$ over $\mathbb{Q}$ is isomorphic to the unique cubic curve $E_{A,B}$ in the plane of the form

[TABLE]

where $A,B\in\mathbb{Z}$ and for all primes $p:p^{6}\nmid A$ whenever $p^{4}\mid A$. Let $\mathscr{E}$ be the set of all such $E_{A,B}$. If $E=E_{A,B}\in\mathscr{E}$, then we define the (naive) height of $E$ by

[TABLE]

For $X\in\mathbb{R}_{>0},$ we define $\mathscr{E}_{<X}:=\{E\in\mathscr{E}\;|\;H(E)<X\}.$

Definition 7.4**.**

For any $\phi:\mathscr{E}\rightarrow\mathbb{R}$, we define

[TABLE]

if the limit exists. Define $\overline{\text{Aver}}_{\mathscr{E}}(\phi)$ and $\underline{\text{Aver}}_{\mathscr{E}}(\phi)$ using limsup or liminf. If the property $P$ can be identified with characteristic fucntion $\chi_{P}:\mathscr{E}\longrightarrow\{0,1\}$,

[TABLE]

Similarly we define $\overline{\text{Prob}}_{\mathscr{E}}(P)$ and $\underline{\text{Prob}}_{\mathscr{E}}(P)$.

Bhargava and Shanker [1, 2, 3, 4] proved the follwing results:

Theorem 7.5** (Bhargava and Shankar).**

1. (1)

$\text{Aver}_{\mathscr{E}}(\#\mathrm{Sel}_{2})=3.$ (cf. [1]) 2. (2)

$\text{Aver}_{\mathscr{E}}(\#\mathrm{Sel}_{3})=4.$ (cf. [2]) 3. (3)

$\text{Aver}_{\mathscr{E}}(\#\mathrm{Sel}_{4})=7.$ (cf. [3]) 4. (4)

$\text{Aver}_{\mathscr{E}}(\#\mathrm{Sel}_{5})=6.$ (cf. [4])

Corollary 7.6** (Bhargava and Shankar [4], 2013).**

[TABLE]

Theorem 7.7** (T. Dokchitser - V. Dokchitser [17], 2010).**

Let $E$ be an elliptic curve over $\mathbb{Q}$ and let $p$ be any prime. Let $s_{p}(E)$ and $t_{p}(E)$ denote the rank of the $p$-Selmer group of $E$ and the rank of $E(\mathbb{Q})[p]$, respectively. Then the quantity $r_{p}(E):=s_{p}(E)-t_{p}(E)$ is even if and only if the root number of $E$ is $+1$. Here we recall that $s_{p}(E)$ and $t_{p}(E)$ are defined as

[TABLE]

respectively.

Bhargava and Shanker [4, 5] proved the follwing:

Theorem 7.8**.**

1. (1)

$\underline{\text{Prob}}_{\mathscr{E}}(r=0)\geqq 0.2062$. (cf. [4]) 2. (2)

$\underline{\text{Prob}}_{\mathscr{E}}(r=1)\geqq 0.2612$. (cf. [4]) 3. (3)

$\underline{\text{Prob}}_{\mathscr{E}}(r=0\text{ or }r=1)\geqq 0.8375$. (cf. [4]) 4. (4)

$\underline{\text{Prob}}_{\mathscr{E}}(r=r_{\text{an}}=0)$ is positive. (cf. [2]) 5. (5)

$\underline{\text{Prob}}_{\mathscr{E}}(r=r_{\text{an}}=1)$ is positive. (cf. [5])

The results (4) and (5) in Theorem 7.8 imply that a positive proportion of $\mathscr{E}$ satisfies the BSD conjecture.

Quite recently Bhargava, Elkies and Shnidman [6] computed the average size of the $\phi_{k}$-Selmer group as $k$ varies over the integers, where

[TABLE]

is a natural 3-isogeny.

8. Adjoint Orbits of $SL(2,\mathbb{R})$

In this section, we describe the adjoint orbits of the special linear group $SL(2,\mathbb{R})$ explicitly.

For brevity, we write $G=SL(2,\mathbb{R})$ and let $K=SO(2)$ be a maximal compact subgroup of $G$. The Lie algebra $\mathfrak{g}$ of $G$ is given by

[TABLE]

We put

[TABLE]

Then the set $\left\{X,Y,Z\right\}$ forms a basis for $\mathfrak{g}$. We define an element $F(x,y,z)\in\mathfrak{g}$ by

[TABLE]

Then we have the relations

[TABLE]

It is easy to see that $X$ and $Y$ are hyperbolic elements and $Z$ is an elliptic element. For a nonzero real number $\alpha$, the $G$-orbit of $\alpha X$ is represented by the one-sheeted hyperboloid

[TABLE]

The $G$-orbit of $\alpha Y\,(\alpha\in\mathbb{R}^{\times})$ is also represented by the hyperboloid (8.2). The $G$-orbit of $\alpha Z\,(\alpha\in\mathbb{R}^{\times})$ is represented by two-sheeted hyperboloids

[TABLE]

Since

[TABLE]

we have for any $k\in\mathbb{Z}^{+},$

[TABLE]

Thus we see that $F(x,y,z)$ is nilpotent if and only if $x^{2}+y^{2}-z^{2}=0.$ Therefore the set ${\mathcal{N}}_{\mathbb{R}}$ of all nilpotent elements in $\mathfrak{g}$ is given by

[TABLE]

We put

[TABLE]

Obviously $S$ and $T$ are nilpotent elements in ${\mathcal{N}}_{\mathbb{R}}$ and they satisfy

[TABLE]

and

[TABLE]

where $\theta$ is the Cartan involution defined by $\theta(g)=-{}^{t}g$ for $g$ in $\mathfrak{g}$.

According to equation (8.6) and (8.7), $\{X,\,S,\,T\}$ and $\{-X,\,-S,\,-T\}$ are KS-triples in $\mathfrak{g}$.

The $G$-orbit of $\alpha S\,(\alpha\in\mathbb{R}^{\times})$ is represented by the cone

[TABLE]

depending on the sign of $\alpha$. If $\alpha>0,$ the $G$-orbit of $\alpha S$ is characterized by the one-sheeted cone

[TABLE]

If $\alpha<0,$ the $G$-orbit of $\alpha S$ is characterized by the one-sheeted cone

[TABLE]

The $G$-orbits of $\alpha T\,(\alpha>0)$ are characterized by the one-sheeted cone (8.10) and the $G$-orbits of $\alpha T\,(\alpha<0)$ are characterized by the one-sheeted cone (8.9).

We define the $G$-orbits ${\mathcal{N}}_{\mathbb{R}}^{+}$ and ${\mathcal{N}}_{\mathbb{R}}^{-}$ by

[TABLE]

and

[TABLE]

Then we obtain

[TABLE]

According to (8.2), (8.3) and (8.13), we see that there are infinitely many hyperbolic orbits and elliptic orbits, and on the other hand there are only three nilpotent orbits in $\mathfrak{g}$.

9. Final Remarks

In the final section, we describe the relation between the nilpotent orbits of $SL(2,\mathbb{R})$ and CM points, and propose several conjectures relating to the BSD conjecture.

Let $G=SL(2,\mathbb{R})$ and let $\mathfrak{g}$ be the Lie algebra of $G$. Let $\mathcal{E}_{\mathbb{R}},\ \mathcal{H}_{\mathbb{R}}$ and $\mathcal{N}_{\mathbb{R}}$ be the set of all $G$-elliptic orbits in $\mathfrak{g}$, the set of all $G$-hyperbolic orbits in $\mathfrak{g}$, and the set of all $G$-nilpotent orbits in $\mathfrak{g}$ respectively. Let $\exp:\mathfrak{g}\longrightarrow G$ be the exponential map and let $\phi:G\longrightarrow\mathbb{H}$ be the map defined by

[TABLE]

We define the composition map $\omega:=\phi\circ\exp:\mathfrak{g}\longrightarrow\mathbb{H}.$

Proposition 9.1**.**

[TABLE]

Proof. The proof can found in [25]. $\Box$

First we recall the thirteen CM points

[TABLE]

We put $q:=e^{2\pi i\tau}\ (\tau\in\mathbb{H}).$ The $q$-expansion of the modular invariant $j(\tau)$ is given by

[TABLE]

It is known that

[TABLE]

For any element $\tau\in\mathbb{H}$, we let

[TABLE]

be the elliptic curve over $\mathbb{C}$, where $\Lambda_{\tau}:=\mathbb{Z}+\mathbb{Z}\tau$ is the lattice in $\mathbb{C}$.

For any $g=\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in G$, we write

[TABLE]

Let

[TABLE]

be the $K$-orbit of $\alpha_{j}$. Clearly $\mathcal{O}_{2}=\{i\}.$

We propose the following conjecture.

Conjecture D**.**

Let $\tau$ be an element of $\mathcal{O}_{j}\ (1\leq j\leq 13)$ such that $E_{\tau}$ is defined over $\mathbb{Q}$. Then the BSD conjecture for $E_{\tau}$ holds.

For any $\alpha_{j}\ (j\neq 2,\ 1\leq j\leq 13)$, by Proposition 9.1, we can choose $F(a_{j},b_{j},c_{j})\in\mathcal{N}_{\mathbb{R}}$ with $c_{j}\neq 0.$ We define

[TABLE]

and

[TABLE]

We also define

[TABLE]

and

[TABLE]

Finally we define

[TABLE]

and

[TABLE]

We propose the following conjectures.

Conjecture E**.**

Let $\tau$ be an element of $\mathfrak{X}_{j}\,(j\neq 2,\,1\leq j\leq 13)$ such that $E_{\tau}$ is defined over $\mathbb{Q}$. Then the BSD conjecture for $E_{\tau}$ holds.

Conjecture F**.**

Let $\tau$ be an element of $\mathfrak{Y}_{j}\,(j\neq 2,\,1\leq j\leq 13)$ such that $E_{\tau}$ is defined over $\mathbb{Q}$. Then the BSD conjecture for $E_{\tau}$ holds.

Conjecture G**.**

Let $\tau$ be an element of $\mathfrak{Z}_{j}\,(j\neq 2,\,1\leq j\leq 13)$ such that $E_{\tau}$ is defined over $\mathbb{Q}$. Then the BSD conjecture for $E_{\tau}$ holds.