BUNTES A gallimaufry of applications (of Gross-Zagier) I (Alex)

Section 6.11 A gallimaufry of applications (of Gross-Zagier) I (Alex)

Subsection 6.11.1 Heegner points on rank 1 curves

The fun of the subject seems to me to be in the examples.
―Gross - Letter to Birch 1982

So let's do some examples of Heegner point computations, and see how Gross-Zagier gives us important information in a few ways.

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Following the algorithm in Cohen, Number theory part I.

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Fix an elliptic curve

$E(\QQ)$ of conductor

$N\text{,}$ we are interested in finding

$E(\QQ)\text{.}$ All elliptic curves over

$\QQ$ are now known to be modular and hence we may make use of the parameterisation

$\begin{equation*} \phi_N \colon X_0(N) \hookrightarrow J_0(N) \twoheadrightarrow E\text{.} \end{equation*}$

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Over

$\CC$ the modular curve is classically

$\begin{equation*} \mathcal H / \Gamma_0(N) \end{equation*}$

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and if

$E = E_f$ for

$f= \sum a_n q^n$ we have

$\Phi_w \colon \CC/\Lambda_E \to E(\CC)\text{.}$ Then the modular parameterisation comes down to

$\begin{equation*} \phi(\tau) = \phi_w(z_\tau) = \phi_w\underbrace{\left( c\int_{i\infty}^\tau 2\pi i f(z) \diff z\right)}_{c\sum_{n=1}^\infty \frac{a_n}{n} q^n} \end{equation*}$

$\begin{equation*} \phi \colon X_0(N) \to \CC/\Lambda \text{.} \end{equation*}$

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So integrating the

$q$ -expansion of a modular form and plugging in

$\tau$ gives us the corresponding point in the complex uniformization of the curve because the Abel-Jacobi map is defined by integration.

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Definition 6.11.1.

We have $\tau \in \HH$ CM points if they satisfy an equation

$\begin{equation*} A \tau ^2 + B'\tau + C = 0 \end{equation*}$

$\begin{equation*} A,B,C \in \ZZ \end{equation*}$

$\begin{equation*} \Delta (\tau ) = B^2 - 4AC \lt 0 \end{equation*}$

when we choose

$\begin{equation*} A \gt 0 \end{equation*}$

$\begin{equation*} (A,B,C) = 1 \end{equation*}$

then

$\begin{equation*} Ax^2 + Bxy + Cy^2 \end{equation*}$

is the associated quadratic form. A Heegner point of level $N$ is one for which

$\begin{equation*} \Delta (N\tau ) = \Delta (\tau )\text{.} \end{equation*}$

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Proposition

$8.6 .3 .$ Let

$\tau \in \mathcal{H}$ be a quadratic irrationality and let

$(A, B, C)$ be the quadratic form with discriminant

$D$ associated with

$\tau\text{.}$ Then

$\tau$ is a Heegner point of level

$N$ if and only if

$N | A$ and one of the following equivalent conditions is satisfied:

$\begin{equation*} \operatorname{gcd}(A / N, B, C N)=1 \end{equation*}$
$\begin{equation*} \operatorname{gcd}(N, B, A C / N)=1 \end{equation*}$
There exists $F \in \mathbb{Z}$ such that $B^{2}-4 N F=D$ with $\operatorname{gcd}(N, B, F)=1$

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Corollary 6.11.2.

If $\tau$ is heegner level $N$ disc $D$ so is

$\begin{equation*} W(\tau ) = -1/(N\tau )\text{.} \end{equation*}$

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Proposition 8.6.6. There is a one-to-one correspondence between on the one hand classes modulo

$\Gamma_{0}(N)$ of Heegner points of discriminant

$D$ and level

$N,$ and on the other hand, pairs

$(\beta,[\mathfrak{a}$ )] where

$\beta \in \mathbb{Z} / 2 N \mathbb{Z}$ is such that

$b^{2} \equiv D(\bmod 4 N)$ for any lift

$b$ of

$\beta$ to

$\mathbb{Z}\text{,}$ and

$[\mathfrak{a}$ \in \Cl(K)] is an ideal class. The correspondence is as follows: if

$(\beta,[\mathfrak{a}$ )] is as above, there exists a primitive quadratic form

$(A, B, C)$ whose class is equal to [a] and such that

$N | A$ and

$B \equiv \beta(\bmod 2 N),$ and the corresponding Heegner point is

$\tau=(-B+\sqrt{D}) /(2 A) .$ Conversely, if

$(A, B, C)$ is the quadratic form associated with a Heegner point

$\tau$ we take

$\beta=B$ mod

$2 N$ and

$\mathfrak{a}=\mathbb{Z}+\tau \mathbb{Z}\text{.}$

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The action of Galois (via the main theorem of CM) shows that the image

$\phi (\tau )$ is defined over

$H$ the hilbert class field of

$K\text{.}$ To get back down to

$K$ we take traces

$\begin{equation*} P=\sum_{\sigma \in \operatorname{Gal}(H / K)} \varphi((\beta,[\mathfrak{a}]))^{\sigma}=\sum_{[\mathfrak{b}] \in \Cl(K)} \varphi\left(\left(\beta,\left[\mathfrak{a} \mathfrak{b}^{-1}\right]\right)\right)=\sum_{[\mathfrak{b}] \in \Cl(K)} \varphi((\beta,[\mathfrak{b}])) \end{equation*}$

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Lemma 8.6.8. If

$\varepsilon=-1\text{,}$ then in fact

$P \in E(\mathbb{Q})$ Proof. Indeed, it is easy to see that

$\varepsilon=-1$ is equivalent to saying that

$\varphi \circ W=\varphi,$ so that

$\begin{equation*} \varphi((\beta,[\mathfrak{b}]))=\overline{\varphi(W(\beta,[\mathfrak{b}]))}=\overline{\varphi\left(\left(-\beta,\left[\ideal b \mathfrak{n}^{-1}\right]\right)\right)}=\varphi\left(\left(\beta,\left[\mathfrak{b}^{-1} \mathfrak{n}\right]\right)\right) \end{equation*}$

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hence

$\begin{equation*} \begin{aligned} \bar{P}=\amp \sum_{[\mathfrak{b}] \in \Cl(K)} \varphi\left(\left(\beta,\left[\mathfrak{b}^{-1} \mathfrak{n}\right]\right)\right)=\sum_{[\mathfrak{b}] \in Cl(K)} \varphi((\beta,[\mathfrak{b}]))=P \end{aligned} \end{equation*}$

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so by Galois theory once again we deduce that

$P \in E(\mathbb{Q})$

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Similarly if

$\epsilon =1$ then

$P+ \overline P$ is torsion.

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We have the Gross-Zagier formula

$\begin{equation*} \widehat{h}(P)=\frac{\sqrt{|D|}}{4 \operatorname{Vol}(E)} L^{\prime}(E, 1) L\left(E_{D}, 1\right) \end{equation*}$

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which tells us the height of Heegner

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In rank 1

$P = \ell G$ for some generator

$G$ of mordell-weil then GZ + BSD

$\begin{equation*} \frac{\ell^{2}}{|\operatorname{III}(E)|}=\omega_{1}(E) \frac{c(E) \sqrt{|D|}}{4 \operatorname{Vol}(E)\left|E_{t}(\mathbb{Q})\right|^{2}} L\left(E_{D}, 1\right) \end{equation*}$

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To compute we evaluations of

$\phi ((-B+ D)/(2A))$ for the

$|Cl(K)|$ classes of quadratic forms

$(A, B, C)\text{.}$

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the convergence of the series for

$\phi(\tau)$ is essentially that of a geometric series with ratio

$\exp (-2 \pi \Im(\tau))=\exp (-2 \pi \sqrt{|D|} /(2 A))$

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We can use

$\begin{equation*} \overline{\varphi((\beta,[\mathfrak{a}]))}=\varphi\left(\left(\beta,\left[\mathfrak{a}^{-1} \mathfrak{n}\right]\right)\right) \end{equation*}$

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to halve the work we do.

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So the heegner point method is

via BSD find
$\begin{equation*} |\operatorname{III}(E)| R(E)=\frac{\left|E_{t}(\mathbb{Q})\right|^{2} L^{\prime}(E, 1)}{c(E) \omega_{1}(E)} \end{equation*}$
find $HB$ the height difference bound between canonical and naive heights
$\begin{equation*} HB = h(j(E)) / 12+\mu(E)+1.946 \end{equation*}$
$\begin{equation*} d=2(|\operatorname{III}(E)| R(E)+H B) \end{equation*}$

$\begin{equation*} d d=\lceil d / \log (10)\rceil+10 \end{equation*}$
this is the number of decimal digits we will work with
Run through fundamental discs $D$ for each. Check $D$ square mod $4N$ all primes split and
$\begin{equation*} L\left(E_{D}, 1\right)=2 \sum_{n \geq 1} \frac{a_{n}}{n}\left(\frac{D}{n}\right) \exp \left(\frac{-2 \pi n}{\sqrt{N D^{2} / \operatorname{gcd}(D, N)}}\right) \end{equation*}$
not too close to zero if this is not satisfied, choose the next fundamental discriminant. Otherwise fix $\beta \in$ $\mathbb{Z} /(2 N) \mathbb{Z}$ such that $D \equiv \beta^{2}(\bmod 4 N)$ and compute $m>0$ such that
$\begin{equation*} m^{2}=\omega_{1}(E) \frac{c(E) \sqrt{|D|}(w(D) / 2)^{2}}{4 \operatorname{Vol}(E)\left|E_{t}(\mathbb{Q})\right|^{2}} 2^{\omega(\operatorname{gcd}(D, N))} L\left(E_{D}, 1\right) \end{equation*}$
This m should be very close to an integer, or at least to a rational number with small denominator.
Find List of Forms below, compute a list $L$ of $|\Cl(K)|$ representatives $(A, B, C)$ of classes of positive definite quadratic forms of discriminant $D\text{,}$ where $A$ must be chosen divisible by $N$ and minimal, and $B \equiv \beta(\bmod 2 N)$ (this is always possible). Whenever possible pair elements $(A, B, C)$ and $\left(A^{\prime}, B^{\prime}, C^{\prime}\right)$ of this list such that $\left(A^{\prime}, B^{\prime}, C^{\prime}\right)$ is equivalent to $(C N, B, A / N)$ by computing the unique canonical reduced form equivalent to each.
$\begin{equation*} z=\sum_{(A, B, C) \in \mathcal{L}} \phi\left(\frac{-B+\sqrt{D}}{2 A}\right) \in \mathbb{C} \end{equation*}$
taking a few more than $A d /(\pi \sqrt{|D|})$ terms for $\phi \text{.}$
Find Rational Point Let $e$ be the exponent of the group $E_{t}(\mathbb{Q}),$ let $\ell=$ $\operatorname{gcd}\left(e, m^{\infty}\right)=\operatorname{gcd}\left(e, m^{3}\right),$ and $m^{\prime}=m \ell .$ For each pair $(u, v) \in\left[0, m^{\prime}-\right.$ $1$ ^{2},] set $z_{u, v}=\left(\ell z+u \omega_{1}(E)+v \omega_{2}(E)\right) / m^{\prime}\text{.}$ Compute $x=\wp\left(z_{u, v}\right),$ where $\left(\wp, \wp^{\prime}\right)$ is the isomorphism from $\mathbb{C} / \Lambda$ to $E(\mathbb{C})\text{.}$ For each $(u, v)$ such that the corresponding point $(x, y) \in E(\mathbb{C})$ has real coordinates.

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Algorithm choice of D

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Recall a congruent number is a number which appears as the area of a right triangle with rational side lengths. this reduces to finding non-torsion points on the congruent number curves

$\begin{equation*} E_n \colon y^2 = x^3 - n^2 x\text{.} \end{equation*}$

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E.g. for

$n = 157$ BSD predicts rank 1, but how do we find the point? Using standard techniques can compute real period, period volume (

$0.209262974439979^2$ ) and torsion order (4), conductor (788768 outside LMFDB range) and Tamagawa product (8). Together we get

$\begin{equation*} |\Sha(E)|R(E) \approx 54.6 \end{equation*}$

$\begin{equation*} HB = 10.6 \end{equation*}$

$\begin{equation*} d \approx 130.4 \end{equation*}$

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need 67 decimal digits.

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Up to

$D = -40$ we have

$D =-31,-39$ are squares modulo

$4N\text{.}$

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For both of these

$D$ we try to compute

$m^2(D)\text{.}$ When we take

$-31$ we get a number close to 0. For

$-39$ we get

$\approx 16$ so fix

$D = -39$ and

$m=4\text{.}$

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A square root

$b$ of

$D$ mod

$4N$ is

$\begin{equation*} b = 1275547\text{.} \end{equation*}$

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The class group of

$\begin{equation*} \QQ(\sqrt{-39}) \end{equation*}$

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$\begin{equation*} \ZZ/4\text{.} \end{equation*}$

$\begin{equation*} z = 2\Re(\phi (x_1) + \phi (x_2)) \end{equation*}$

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for

$\begin{equation*} x_i = (-b + \sqrt{-39})/(2j N) \end{equation*}$

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So we have four classes of quadratic forms, of these the largest value of

$A$ is

$2N\text{.}$ So we need

$\begin{equation*} \approx 10 500 000 \end{equation*}$

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terms of the series

$\begin{equation*} \phi (\tau ) = \sum_{n=1}^\infty \frac{a_n}{n} q^n,\,q = \exp(2\pi i \tau ) \end{equation*}$

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applying this we get

$\begin{equation*} z = -5.63911127500831766007696166307316036323562406574706\ldots \end{equation*}$

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we can add multiples of the period lattice to make it smaller, as

$\begin{equation*} z/\omega \approx -26.9469552131277 \end{equation*}$

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we find that

$\begin{equation*} z' = z + 27 \omega \approx 0.0111003098794358 \end{equation*}$

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so that

$\begin{equation*} \wp(\Lambda , (2 z' + 2 \omega )/8) \approx 344.99665832468973990799841297983141563953148876481 \end{equation*}$

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this we can recognise as

$\begin{equation*} \frac{95732359354501581258364453}{526771095761^2} \end{equation*}$

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(using the fact we are looking for something with square denominator) and compute the point

$\begin{equation*} \left(\frac{95732359354501581258364453}{526771095761^2} : \frac{834062764128948944072857085701103222940}{526771095761^3} : 1\right) \end{equation*}$

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which is quite a big triangle. This is saturated and of height

$54.6008892940170\text{.}$

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Remark 6.11.3.

Calling the sage function gens() fails on this example!