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Section 6.1 An Overview of Gross-Zagier and Related Objects / Formulas of interest (Sachi)

Goal today is to motivate and give some high level overview of the objects in Gross-Zagier. It involves many things \(L\)-functions, elliptic curves, modular forms.

Main reference: [106].

Subsection 6.1.1 A big example

Today we will study

\begin{equation*} E\colon y^2 + y= x^3 + x^2 \end{equation*}

LMFDB label 43.a1, http://lmfdb.xyz/EllipticCurve/Q/43.a1/.

Figure 6.1.1.

One fundamental invariant we can compute is the conductor, in this case 43, we only have bad reduction at 43 and no other prime.

To compute the real period we can transform to short Weierstrass form.

\begin{equation*} y^2 = x^3 - 432 x + 15120 \end{equation*}

then we have invariant differential

\begin{equation*} \frac{\diff x}{2y } = \frac{\diff x}{2\sqrt{x^3 - 432 x + 15120}}\text{.} \end{equation*}

Real period is then

\begin{equation*} \omega _1 = \int_{E(\RR)} \frac{\diff x}{2y} \approx 5.4687\ldots\text{.} \end{equation*}

For \(E/\CC\) fix a complex conjugate root of \(E\text{,}\) \(\alpha \text{,}\) and \(\beta =\) real root.

\begin{equation*} \omega _2 = \int_\alpha ^\beta \frac{\diff x}{2y} \end{equation*}
\begin{equation*} = 2.73434476498379 + 1.36318241817043i \end{equation*}

We can look at \(E/\FF_p\) for various \(p\text{.}\) Obtained by looking at the equation \(y^2 + y =x^3 +x^2 \pmod p\) for various \(p\text{.}\)

At 43 we have non-split multiplicative reduction, which means that we have a singular curve with tangent slopes not defined over \(\FF_{43}\text{.}\)

Figure 6.1.2.
\begin{equation*} N_p = \# E(\FF_p) \end{equation*}
\begin{equation*} L_E = \left(\frac 1 {1+43^{-s}} \right)\prod_{p\ne 43} \frac{1}{1- (N_p - p - 1)p^{-s} + p p^{-2s}} \end{equation*}
\begin{equation*} = \sum_{n\ge 1} \frac{a_n}{n^{s}} \end{equation*}

We can tabulate the \(a_n\)

\(n\) 1 2 3 4 5 6 7 8 9
\(a_n\) 1 -2 -2 2 -4 4 0 0 1
Table 6.1.3. \(a_n\)s

As we have \(E/\QQ\) we can determine that

\begin{equation*} E(\QQ) \simeq \ZZ \cdot \underbrace{P}_{=(0,0)}\text{.} \end{equation*}

Next up the Néron-Tate canonical height:

\begin{equation*} \hat h(P) = \lim_{n\to \infty } \frac{\log(h_{\text{naive}}(2^nP))}{4^n} \end{equation*}

naive height is the max of the absolute values of the numerator and denominator of the \(x\)-coordinate. In our case this is

\begin{equation*} \hat h(P) \approx 0.0628165070875\text{.} \end{equation*}

We have the Hasse-Weil bound:

\begin{equation*} | N_p - p + 1 | \lt 2 \sqrt p \end{equation*}

so the \(L\)-function converges for \(\Re (s ) \gt 3/2\text{.}\) So modularity implies that \(L_E(s)\) extends to an entire function \(\widetilde L_E\) satisfying a functional equation

\begin{equation*} \widetilde L_E(s) = - \widetilde L_E(2-s) \end{equation*}

in particular \(\widetilde L_E(s)\) vanishes at \(s=1\text{.}\)

BSD for rank 1 then says:

  1. \begin{equation*} \ord_{s=1} \widetilde L_E(s) = \rank E(\QQ) = 1 \end{equation*}
  2. \begin{equation*} \frac{\diff}{\diff s} \widetilde L_E(s)|_{s=1} = \underbrace{\hat h ( P) \omega _1}_{\approx 0.34352397} |\Sha| \end{equation*}
    \(|\Sha|\) is predicted to be finite (in which case the order is a square). the LHS can be computed using
    \begin{equation*} 2\sum_{n=1}^\infty a_n \int_1^\infty \log t \exp\left(- \frac{-2 n \pi t }{\sqrt{43}}\right) \diff t\text{.} \end{equation*}
Modularity.

Goal: Verify \(E\) is modular. Two definitions today:

  1. There exists a newform \(f \in S_2(\Gamma _0(N))\) with fourier coefficients the same as the \(L\)-series:
    \begin{equation*} a_p(f) = a_p(E) \end{equation*}
    for all \(p\nmid N\text{.}\)
  2. There exists \(X_0(N) \to E\) finite defined over \(\QQ\text{.}\)

Consider

\begin{equation*} X_0(43) \end{equation*}

the modular curve for the congruence subgroup generated by \(\Gamma _0(43) = \begin{pmatrix} a \amp b \\ c \amp d \end{pmatrix}\) with \(c \equiv 0 \pmod{43}\text{.}\)

\begin{equation*} w_{43} = \begin{pmatrix} 0 \amp \frac{-1}{\sqrt{43}} \\ \sqrt{43} \amp 0\end{pmatrix} \end{equation*}
\begin{equation*} \Gamma _0(43)^+ \backslash \HH \simeq \text{genus 1 curve} \end{equation*}

which is potentially equal to \(E\text{.}\)

Strategy: Find \(\eta ,\xi \) s.t.

\begin{equation*} \eta(\tau )^2 + \eta(\tau ) = \xi (\tau )^3 + \xi (\tau )^2 \end{equation*}

and

\begin{equation*} \frac{\diff \eta}{2\xi + 1} = f(q) \frac{\diff q}{q} \end{equation*}
\begin{equation*} \xi f^2, \eta f^3 \end{equation*}

should be holomorphic modular forms in \(M_4(\Gamma _0(43))\) and \(M_6(\Gamma _0(43))\text{.}\) we can compute \(q\)-series expansions and use modular symbols to prove they exist.

Quadratic twists of \(E\).

Let \(\Delta \lt 0\) be a fundamental discriminant.

\begin{equation*} E\colon y^2 = f(x) \end{equation*}

then

\begin{equation*} E_{\Delta }\colon \Delta y^2 = f(x) \end{equation*}

these are not isomorphic over \(\QQ\text{.}\)

The \(L\)-function of \(E_\Delta \text{.}\) For any \(\Delta \) coprime to 43

\begin{equation*} L_{E_\Delta } (s) = \sum_{n\ge 1} \legendre \Delta n \frac{a_n}{n^s} \end{equation*}

can prove that for \(p\nmid 6\cdot 43 \cdot \Delta \text{.}\)

\begin{equation*} a_p(E) \leftrightarrow a_p(E_\Delta ) \end{equation*}

are related by considering

\begin{equation*} E\colon y^2 = f(x) \end{equation*}
\begin{equation*} E_\Delta \colon \Delta y^2= f(x) \end{equation*}

mod \(p\text{,}\) so if \(\Delta \) is a square we have isomorphisms locally and the \(a_p\) are equal, otherwise all non-square and squares are swapped.

BSD says

\begin{equation*} L_{E,\Delta } (1) = \Omega _{E,\Delta }^+ \prod_p c_p A_\Delta \end{equation*}

if \(\rank = 0\text{.}\)

Waldspurger's implies that \(A_\Delta \) is a square.

Adding in Waldspurger we get

\begin{equation*} A(\Delta ) = c_\Delta ^2 \end{equation*}
\begin{equation*} \hat h(P_\Delta )= \hat h(b_\Delta P) = b_\Delta ^2\hat h(P) \end{equation*}

but also

\begin{equation*} \hat h(P_\Delta ) = \frac{\sqrt{|\Delta |}}{8\pi ^2 \| f\|} L_E'(1) \Omega _{E,\Delta }^+ \prod_p c_p c_\Delta^2 \end{equation*}

as \(\Omega _{E,\Delta }^+ = \Omega _E^- / \sqrt \Delta \) we have cancellation and \(c_\Delta ^2 = b_\Delta ^2\) for all \(\Delta \text{.}\)