BUNTES Modular Curves and Heegner Points (Ricky)

Section 6.3 Modular Curves and Heegner Points (Ricky)

Subsection 6.3.1 CM theory

Let \(E/\CC\) be an elliptic curve so \(E(\CC) = \CC/\Lambda_\tau \text{,}\) where \(\Lambda = \ZZ + \ZZ\tau \text{,}\) writing \(E = E+\tau \text{,}\) \(\tau \in \HH\text{.}\)

Recall that

\begin{equation*} \End(E) = \begin{cases} \ZZ \\ \ints \subseteq K ,\,K/\QQ \text{ im. quad.}\end{cases} \end{equation*}

Lemma 6.3.1.

\begin{equation*} \Hom(\CC/\Lambda ,\CC/\Lambda')= \{\alpha \in \CC : \alpha \Lambda \subseteq \Lambda '\} \end{equation*}

Proof.

Lift \(\phi \colon \CC/\Lambda \to \CC/\Lambda '\) to

\begin{equation*} \phi \colon \CC \to \CC \end{equation*}

to see \(\phi (z) = \alpha z\) for some \(\alpha \in \CC^\times\text{.}\)

So \(\End(E_\tau ) = \{\alpha \in \CC : \alpha \Lambda _\tau \subseteq \Lambda _\tau \}\text{.}\) If \(\alpha \cdot 1 = m_1 + m_2 \tau \) and

\begin{equation*} \alpha \cdot \tau = n_1 + n_2 \tau \end{equation*}

then

\begin{equation*} m_2 \tau ^2 + (m_1 - n_1) \tau - n_1 = 0 \end{equation*}

call these coefficients \(A,B,C \in \ZZ\text{.}\) And \(\Delta = B^2 - 4AC\text{,}\) so \(\Delta = -f^2 d \lt 0\) where \(f\) is the conductor of \(\tau \) and \(d\) is the discriminant of \(\tau \text{.}\)

Then if \(\End(E_\tau ) \ne \ZZ\) we have

\begin{equation*} \End(E_\tau ) = \ZZ \oplus f \ZZ \left\lb \frac {-d + \sqrt{-d}}{2} \right \rb = \ints_\Delta \subseteq \ints_{\QQ(\sqrt{-d})} \end{equation*}

We say \(E\) has CM by \(\ints_\Delta \text{.}\)

Remark 6.3.2.

We can create elliptic curves with CM by \(\ints\) by creating \(\CC/\ints\text{.}\) In fact, all elliptic curves with CM by \(\ints\) are isomorphic to \(\CC/\ideal a\) for \(\ideal a\) a fractional ideal of \(\ints\text{.}\)

Theorem 6.3.3.

Let \(E/\CC\) be an elliptic curve with CM by \(\ints_K\) for \(K/\QQ\) imaginary quadratic. Then \(j(E) \in \ints_{H_K}\text{.}\) Where \(H_K\) is the hilbert class field of \(K\text{,}\) so \(E\) admits a model over a number field.

Theorem 6.3.4.

Let \(G = \Gal{H_K}{K}\) then we have an isomorphism

\begin{equation*} s\colon \Pic(\ints_K) \to G \end{equation*}

\begin{equation*} \ideal b\mapsto s(\ideal b) \end{equation*}

\begin{equation*} j(\ideal a)^{s(\ideal b)} = j(\ideal b \inv \ideal a)\text{.} \end{equation*}

(The \(j\)-invariants generate \(\ints_H\text{,}\) this characterises \(G\) as a Galois group).

Subsection 6.3.2 Modular curves

Let

\begin{equation*} \Gamma_0 (N) = \left\{\begin{pmatrix} a \amp b \\ c \amp d \end{pmatrix}\in \SL_2(\ZZ) : \begin{pmatrix} a \amp b \\ c \amp d \end{pmatrix} \equiv \begin{pmatrix} \ast \amp \ast \\ 0 \amp \ast \end{pmatrix} \pmod N\right\} \end{equation*}

degine \(Y_0(N) = \Gamma _0(N)\backslash \HH\) , \(X_0(N) = \Gamma _0(N)\backslash \overline \HH\text{.}\) Then \(X_0(N)\) can be given the structure of a projective algebraic variety \(/\QQ\text{.}\)

The for \(L/\QQ\) a field we have the modular interpretation,

\begin{equation*} Y_0(N)(L) = \{ (E,E',\phi ) : E,E' / L\text{ ell. curves},\, \phi \colon E \to E' / L\text{ cyclic isog. degree } N\} \end{equation*}

i.e. \(\phi \) for which \(\ker \phi \simeq \ZZ/N\text{.}\)

Atkin-Lehner involutions.

Let \(d|N\text{,}\) \((d, N/d) = 1\text{.}\) We get an involution

\begin{equation*} w_d \colon X_0(N) \to X_0(N) \end{equation*}

such that

\begin{equation*} w_N(\epsilon \colon E \to E') = (\hat \phi \colon E' \to E) \end{equation*}

(and it swaps the two cusps.????????????)

These generate a group \(W \subseteq \Aut(X_0(N))\) with the relation

\begin{equation*} w_{d}w_{d'} = w_{dd'/(d,d')^2}\text{.} \end{equation*}

So \(W \simeq (\ZZ/2)^s\) where \(s\) is the number of primes dividing \(N\text{.}\)