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Section 5.1 Modular curves (Aash)

Definition 5.1.1. Lattices.

A lattice is a free abelian group of rank 2

\begin{equation*} \Lambda \otimes \RR \to \CC \end{equation*}

is an isomorphism

\begin{equation*} \Lambda = \ZZ\lb \alpha \rb \oplus \ZZ\lb \beta \rb \end{equation*}

if

\begin{equation*} \Lambda = \gamma \Lambda ',\,\gamma \in \CC \end{equation*}

then we say the two lattices are homothetic.

Any lattice is homothetic to one of the form

\begin{equation*} \Lambda = \langle 1 , \tau \rangle \end{equation*}

as we can take a positively oriented basis we have that all such are equivalent to

\begin{equation*} \tau \in \HH = \{ z\in \CC: \Im(z) \gt 0 \}\text{.} \end{equation*}

So there is a bijection between \(\HH\) and ordered bases of lattices.

\(\SL_2(\ZZ)\) acts on \(\HH\) and the action corresponds to changing bases.

The action of \(\PSL_2(\ZZ)\) is faithful. \(i,\rho = e^{\pi i /3}\) have non-trivial stabilisers

\begin{equation*} \Stab_i = \langle S\rangle = \begin{pmatrix} 0\amp1 \\ -1 \amp 0 \end{pmatrix} \end{equation*}
\begin{equation*} \Stab_\rho = \langle TS\rangle,\,T = \begin{pmatrix} 1\amp1 \\ 0 \amp 1 \end{pmatrix} \end{equation*}

We can determine the order of elements by looking at the characteristic polynomials.

We then have

\begin{equation*} Y(1) =\SL_2(\ZZ) \backslash \HH \end{equation*}

a complex manifold and

\begin{equation*} j\colon Y(1) \to \CC \end{equation*}

is an isomorphism.

We have a fundamental domain for this action

\begin{equation*} D = \{ z\in \CC : |z| \ge 1,\, |\Re(z)| \le \frac 12 \} \end{equation*}

\(Y(1)\) is Hausdorff because the action is properly discontinuous.

Care must be taken around the elliptic points (those with larger stabiliser), to define the complex structure.

The extended upper half plane

\begin{equation*} \HH^* = \HH \cup \PP^1 (\QQ) \end{equation*}

also has an \(\SL_2(\ZZ)\) action via fractional linear transformations, which is proper.

We can define a basis of neighbourhoods around the cusps by transforming them to the cusp \(\infty\) where we can use the basis of neighbourhoods given by

\begin{equation*} \HH_N = \{ z\in \HH : | \Im(z)| \gt N\}\text{.} \end{equation*}

The parameter \(q\) around \(\infty\) is defined as \(e^{2\pi i z/N}\) for some \(N\in \ZZ\text{,}\) \(q\) is fixed by \(T\text{.}\)

We can quotient by the action of \(\SL_2(\ZZ) \) on \(\HH^*\) to get

\begin{equation*} X(1) = \SL_2(\ZZ) \backslash \HH^* \end{equation*}

which is now compact, genus 0, which matches up with \(Y(1)\) having \(\CC\) points \(\CC\) earlier.

If \(X\) is a projective curve then \(X(\CC)\) has the structure of a compact Riemann surface. If \(S\) is such a surface then there exists a unique up to isomorphism \(X\) with \(X(\CC) = S\text{.}\)

The meromorphic functions on \(S\) are the function field of \(X\) and there is a correspondence

\begin{equation*} \text{Compact Riemann surfaces} \leftrightarrow \text{Smooth proj. curves} \end{equation*}

Given a finite index subgroup of \(\SL_2(\ZZ)\) we can do something similar to obtain

\begin{equation*} \Gamma\backslash \HH\text{.} \end{equation*}

One of the most prominent examples of such a subgroup is

\begin{equation*} \Gamma(N) = \left\{ \gamma \in \SL_2(\ZZ) : \gamma \equiv \begin{pmatrix} 1\amp0 \\ 0 \amp 1 \end{pmatrix} \pmod N \right\} \end{equation*}

along with

\begin{equation*} \Gamma_1(N) = \left\{ \gamma \in \SL_2(\ZZ) : \gamma \equiv \begin{pmatrix} 1\amp\ast \\ 0 \amp 1 \end{pmatrix} \pmod N \right\} \end{equation*}
\begin{equation*} \Gamma_0(N) = \left\{ \gamma \in \SL_2(\ZZ) : \gamma \equiv \begin{pmatrix} \ast\amp\ast \\ 0 \amp \ast \end{pmatrix} \pmod N \right\}\text{.} \end{equation*}

\(\Gamma(N)\) is normal inside \(\SL_2(\ZZ)\) and \(\Gamma_1(N)\) is normal inside \(\Gamma_0(N)\text{.}\)

The aforementioned equivalence of categories gives us a smooth projective curve for each of these examples.

In fact one can find a smooth projective curve with \(\QQ\)-coefficients realising each of these Riemann surfaces.

For

\begin{equation*} \Gamma_0(N) \backslash \HH^* \end{equation*}

we have the function \(j(z)\) from before, but also \(j(Nz)\) which is still a function on the quotient now as

\begin{equation*} j(N \gamma z) = j\left( N\frac{az + b}{cz+d} \right) \end{equation*}
\begin{equation*} = j\left( N\frac{az + b}{c'Nz+d} \right) \end{equation*}
\begin{equation*} = j\left( \frac{aNz + bN}{c'Nz+d} \right) \end{equation*}
\begin{equation*} = j\left( \gamma ' Nz\right) \end{equation*}
\begin{equation*} = j\left( Nz\right) \end{equation*}

We can therefore let

\begin{equation*} g = \prod_{\gamma} (Y - j(\gamma Nz)) \end{equation*}

the product over the cosets of \(\Gamma_0(N) \subseteq \SL_2(\ZZ)\text{.}\)

The coefficients of \(g\) are meromorphic functions on \(X(1) = \CC\lb j \rb\text{.}\) So we have

\begin{equation*} g(Y) = F(j(z), Y) \end{equation*}

and

\begin{equation*} g (j(Nz)) = F(j(z), j(Nz)) = 0 \end{equation*}

then \(F(X,Y)\) is irreducible and has integer coefficients.

Then the curve \(X_0(N)\) whose function field is

\begin{equation*} \QQ \lb X,Y\rb/ F(X,Y) \end{equation*}

so \(U \subseteq X_0(N)\) is isomorphic to an affine variety defined by

\begin{equation*} F(X,Y) = 0 \smallsetminus \text{singular pts} \end{equation*}
\begin{equation*} \Gamma_0(N) \backslash \HH \to U(\CC) \end{equation*}
\begin{equation*} z \mapsto (j(z), j(Nz)) \end{equation*}

\(j(\gamma z) = z\,\forall z\) iff \(\gamma \in \SL_2(\ZZ)\text{.}\)

If for \(z= z_1,z_2\) have \((j(z),j(Nz))\) equal then \(z_1,z_2\) are in the same \(\Gamma_0(N)\) orbit.

We can do similar for \(\Gamma_1\) but only over \(\QQ(\zeta_N)\text{.}\)

Elliptic curves.

Several definitions:

  1. Smooth proj. curve genus 1 with a rational point.
  2. smooth curve given by Weierstrass eqn.
    \begin{equation*} y^2 + a_1xy + a_3 y = x^3 + a_2x^2 + a_4 x + a_6\text{.} \end{equation*}
  3. Complex torus of dimension 1.

Over \(\CC\) at least all are equivalent.

To get the weierstrass equation from the curve we use Riemann-Roch to see that

\begin{equation*} H(1[0]) = 1,\, H(2[0]) = 2,\,H(3[0]) =3 \end{equation*}

So we call a generator of \(H(2\lb 0 \rb) \smallsetminus H(\lb 0 \rb)\) the function \(x\) same for \(y\) and \(H(3\lb 0 \rb)\text{,}\) now in \(H(6 \lb 0 \rb)\) we have

\begin{equation*} 1,x,y,x^2 ,xy,y^2,x^3 \end{equation*}

so there is a linear relation among these, giving the Weierstrass equation.

To get the equation for a torus we use the Weierstrass \(\wp\) function.