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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.

Following the algorithm in Cohen, Number theory part I.

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*}

Over \(\CC\) the modular curve is classically

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

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*}

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.

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*}

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:

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

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

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*}

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*}

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*}

so by Galois theory once again we deduce that \(P \in E(\mathbb{Q})\)

Similarly if \(\epsilon =1\) then \(P+ \overline P\) is torsion.

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*}

which tells us the height of Heegner

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*}

To compute we evaluations of \(\phi ((-B+ D)/(2A))\) for the \(|Cl(K)|\) classes of quadratic forms \((A, B, C)\text{.}\)

the convergence of the series for \(\phi(\tau)\) is essentially that of a geometric series with ratio \(\exp (-2 \pi \Im(\tau))=\exp (-2 \pi \sqrt{|D|} /(2 A))\)

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*}

to halve the work we do.

So the heegner point method is

  1. 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*}
  2. find \(HB\) the height difference bound between canonical and naive heights
    \begin{equation*} HB = h(j(E)) / 12+\mu(E)+1.946 \end{equation*}
  3. \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
  4. 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.
  5. 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.
  6. \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 \(\phi \text{.}\)
  7. 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.

Algorithm choice of D

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*}

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*}

need 67 decimal digits.

Up to \(D = -40\) we have \(D =-31,-39\) are squares modulo \(4N\text{.}\)

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 \(m=4\text{.}\)

A square root \(b\) of \(D\) mod \(4N\) is

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

The class group of

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

is

\begin{equation*} \ZZ/4\text{.} \end{equation*}
\begin{equation*} z = 2\Re(\phi (x_1) + \phi (x_2)) \end{equation*}

for

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

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*}

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*}

applying this we get

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

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

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

we find that

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

so that

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

this we can recognise as

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

(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*}

which is quite a big triangle. This is saturated and of height \(54.6008892940170\text{.}\)

Remark 6.11.3.

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