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Section 5.3 Abelian varieties and Jacobians (Angus)

Subsection 5.3.1 Background

Definition 5.3.1.

An elliptic curve is any one of the following

  1. Smooth projective curve of genus 1 with a marked rational point.
  2. A smooth projective curve with a group law
  3. if \(k \subseteq \CC\) we have
    \begin{equation*} E(\CC) = \CC/ \Lambda \end{equation*}
    \begin{equation*} \Lambda = \ZZ \omega_1 \oplus \ZZ \omega_2,\, \omega_1/\omega_2 \not\in \RR \end{equation*}
  4. if \(\characteristic k \ne 2,3\) A smooth projective curve specified by
    \begin{equation*} y^2= x^3+ ax + b\text{.} \end{equation*}

Aash showed that 1 implies 4 and 3 implies 1.

One can view the group law on \(E\) either via the chord-tangent method (Bezout's theorem). Or via the isomorphism

\begin{equation*} E \to \Pic^0(E) \end{equation*}
\begin{equation*} P \mapsto [P] - [0]\text{.} \end{equation*}
Definition 5.3.2.

An abelian variety is a proper irreducible variety with a group law given by regular functions.

Remark 5.3.3.
  1. In this definition proper is equivalent to projective.

  2. The rigidity theorem tells us:

    1. Any morphism of abelian varieties that preserves the identity is a homomorphism.
    2. Abelian varieties are abelian

Subsection 5.3.2 Ablelian varieties over \(\CC\)

The lie algebra \(\Lie(A(\CC))\) is a complex vector space of dimension \(g\text{.}\) We have the exponential

\begin{equation*} \exp \colon \Lie(A(\CC)) \to A(\CC) \end{equation*}

which is surjective onto the connected component of the identity, and locally at \(0\) a diffeomorphism. So \(\exp\) surjects. Since its locally isomorphic at \(0\) we have \(\ker(\exp)\) discrete and hence a lattice. \(A \) proper means \(A(\CC)\) is compact so

\begin{equation*} \rank \ker (\exp) = 2g\text{.} \end{equation*}

We have a map

\begin{equation*} \{\text{AVs}/\CC\} \to \{\text{complex tori}\} \end{equation*}

but this is not surjective. Which lattices give AVs?

Definition 5.3.5. Hermitian forms.

Let \(V\) be a \(\CC\)-vector space and \(\Lambda \subseteq V\) be a full lattice. A Hermitian form on \(V\) is a function

\begin{equation*} H \colon V\times V \to \CC \end{equation*}

which is \(\CC\)-linear in the first component, \(\CC\)-antilinear in the second (i.e. a sesquilinear form). And satisfies

\begin{equation*} H(u,v) = \overline{H(v,u)} \end{equation*}

A Riemann form on \((V,\Lambda)\) is a positive definite Hermitian form on \(V\) s.t. \(\im(H|_V) \colon \Lambda \to \ZZ\text{.}\)

Swinnerton-Dyer analytic theory of AVs ch.2.

Example 5.3.7.

For an elliptic curve \(E(\CC) = \CC/ \ZZ \omega_1 + \ZZ \omega_2\)

\begin{equation*} H(u,v) = u\bar v/ \im(\omega_1 \bar \omega_2)\text{.} \end{equation*}

Subsection 5.3.3 Jacobian varieties

Definition 5.3.8.

Given \(X\) a curve

\begin{equation*} \Pic^0(X) = \divisors^0(X)/\{(f) : f \in K(X)\} \end{equation*}

this is some abelian group.

Remark 5.3.10.

This is false as stated unless \(X(k) \ne \emptyset\text{.}\)

Idea: Pick \(P_0 \in X(k)\) we have a bijection

\begin{equation*} \divisors^0(X) \to \divisors^r(X) \end{equation*}
\begin{equation*} D \mapsto D + r[P_0] \end{equation*}

we have a map

\begin{equation*} X^r \to X^r/ S_r = X^{(r)} \to \divisors^r(X) \end{equation*}

we can construct \(\Jac(X)\) as a quotient of \(X^{(r)}\) full details Milne AVs ch. 2.

Jacobians over \(\CC\).

Given \(X\) a compact Riemann surface of genus \(g\) then

\begin{equation*} H^0(X, \Omega_X^1) \simeq \CC^g \end{equation*}

one might wish to consider, for \(P,Q \in X\text{,}\) \(\omega\in H^0(X, \Omega_X^1)\)

\begin{equation*} \int_P^Q \omega \end{equation*}

this is not well defined as there are choices of path \(P\to Q\text{.}\)

\begin{equation*} H_1(X,\ZZ) = \ZZ^{2g} \end{equation*}

have a map

\begin{equation*} H_1(X,\ZZ) \to H^0(X, \Omega_X^1) ^\vee \end{equation*}
\begin{equation*} \gamma \mapsto (\omega \mapsto \int_\gamma \omega) \end{equation*}

Let

\begin{equation*} J(X) = H^0(X, \Omega_X^1) ^\vee/H_1(X,\ZZ) \end{equation*}

For the first claim we need a Riemann form on

\begin{equation*} (H^0(X, \Omega_X^1)^\vee , H_1(X,\ZZ)) \end{equation*}

we have

\begin{equation*} H_1(X, \ZZ) \times H_1(X, \ZZ) \to \ZZ \end{equation*}
\begin{equation*} (\gamma_1, \gamma_2) \mapsto -(\gamma_1\cap \gamma_2)\text{.} \end{equation*}
Remark 5.3.12.

In this case we see

\begin{equation*} \Lie(\Jac(X)) = H^0(X, \Omega_X^1) \end{equation*}

this is true in general.

Subsection 5.3.4 Some constructions/properties of AVs

Let \(A,B\) be AVs\(/k\text{.}\) Any identity preserving morphism \(\phi \colon A \to B\) is a homomorphism. Such a homomorphism is called an isogeny if it surjective with finite kernel. i.e. \(\lb n \rb \colon A \to A\) is an isogeny and for \(\characteristic(k) \nmid n\text{.}\)

\begin{equation*} A[n] \simeq (\ZZ/n)^{2g} \end{equation*}

then we have the Tate module for \(l\) prime

\begin{equation*} T_lA = \varprojlim_n A[l^n] \simeq \ZZ_l^{2g} \end{equation*}

in fact

\begin{equation*} H^1_{\et} (A, \ZZ_l) \simeq T_lA ^\vee \end{equation*}

we can also consider \(\Pic^0(A)\text{.}\) There exists an abelian variety

\begin{equation*} \hat A/l \end{equation*}

s.t.

\begin{equation*} \hat A (L) = \Pic^0(A \otimes L) \end{equation*}

this is called the dual abelian variety. So earlier we saw \(\hat E \simeq E\text{.}\) in general \(\hat A \not \simeq A\text{.}\)

However for an ample divisor \(D\) we get an isog

\begin{equation*} \phi_D \colon A \to \hat A \end{equation*}
\begin{equation*} P \mapsto t_P^* D -D \end{equation*}

an isogeny \(\phi\colon A\to \hat A\) is a polarization if

\begin{equation*} \phi = \phi_D /\bar k \end{equation*}

over \(\CC\) a polarization is equivalent to a choice of Riemann form.

A principal polarization is a polarization which is an isomorphism. e.g.

\begin{equation*} \phi_{[0]} \colon E \to \hat E \end{equation*}
\begin{equation*} P \mapsto [P] - [0] \end{equation*}
Remark 5.3.13.

Jacobian varieties always admit principal polarizations.

On \(T_lA \) we have a Weil pairing

\begin{equation*} T_lA \times T_lA^\vee \to \ZZ_l \end{equation*}
Maps between Jacobians.

Let \(X,Y/k\) be curves and \(f\colon X \to Y\) a morphism.

Definition 5.3.14.

We have a pushforward map

\begin{equation*} f_*\colon \Pic^0(X) \to \Pic^0(Y) \end{equation*}
\begin{equation*} \sum n_x [x] \mapsto \sum n_x[f(x)] \end{equation*}

if \(f\) is finite then we have a pullback

\begin{equation*} f^* \colon \Pic^0(Y) \to \Pic^0(X) \end{equation*}
\begin{equation*} \sum n_y[y] \mapsto \sum n_y [f\inv (y)] \end{equation*}

(with multiplicity).

We want further maps between jacobians

Definition 5.3.15.

A correspondence between \(X,Y\) is a curve \(Z\) and a pair of finite morphisms.

\begin{equation*} X \leftarrow Z \to Y \end{equation*}

then we get induced maps

\begin{equation*} T_* = g_* f^* \colon \Pic^0(X) \to \Pic^0(Y) \end{equation*}
\begin{equation*} T^* = f_* g^* \colon \Pic^0(Y) \to \Pic^0(X) \end{equation*}
Modular jacobians and Hecke correspondences.

Consider \(p\nmid N\) we have

\begin{equation*} X_0(N) = \{ (E,C_N) : E\text{ e.c. }, C_N \text{ cyclic sub order } N\} \end{equation*}
\begin{equation*} X_0(pN) = \{(E,C_{pN})\} = \{(E,C_N,C_p)\} \end{equation*}

so we have

Definition 5.3.16.

the Hecke correspondence \(T_p\) on \(X_0(N)\) is

\begin{equation*} X_0(N) \mapsfrom X_0(pN) \to X_0(N) \end{equation*}
\begin{equation*} (E,C_N)\mapsfrom (E,C_N,C_p)\to (E/C_p, C_p + C_N)\text{.} \end{equation*}

We have the modular jacobian \(J_0(N)\) and the induced map

\begin{equation*} T_p \colon J_0(N) \to J_0(N) \end{equation*}
\begin{equation*} [E] \mapsto \sum_{C_p \subseteq E} [E/C_p] \end{equation*}

One can consider \(J_0(N)_{\FF_p}\)