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
- Smooth projective curve of genus 1 with a marked rational point.
- A smooth projective curve with a group law
- 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*}
- if \characteristic k \ne 2,3 A smooth projective curve specified by\begin{equation*} y^2= x^3+ ax + b\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.
- In this definition proper is equivalent to projective.
-
The rigidity theorem tells us:
- Any morphism of abelian varieties that preserves the identity is a homomorphism.
- Abelian varieties are abelian
Subsection 5.3.2 Ablelian varieties over \CC
Proposition 5.3.4.
Let A/k\subseteq \CC then
where g = \dim A and \Lambda \subseteq \CC^g is a rank 2g lattice.
Proof.
The lie algebra \(\Lie(A(\CC))\) is a complex vector space of dimension \(g\text{.}\) We have the exponential
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
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
which is \CC-linear in the first component, \CC-antilinear in the second (i.e. a sesquilinear form). And satisfies
A Riemann form on (V,\Lambda) is a positive definite Hermitian form on V s.t. \im(H|_V) \colon \Lambda \to \ZZ\text{.}
Proposition 5.3.6.
We have a bijection
Proof.
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
Subsection 5.3.3 Jacobian varieties
Definition 5.3.8.
Given X a curve
this is some abelian group.
Theorem 5.3.9.
Let X be a genus g curve /k\text{.} Then there exists an abelian variety \Jac(X)/k of \dim = g s.t.
Remark 5.3.10.
This is false as stated unless X(k) \ne \emptyset\text{.}
Proof.
Idea: Pick \(P_0 \in X(k)\) we have a bijection
we have a map
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 thenTheorem 5.3.11.
J(X) is the \CC points of an abelian variety over \CC\text{.} Further the map
is an isomorphism of abelian groups.
Proof.
For the first claim we need a Riemann form on
we have
Remark 5.3.12.
In this case we see
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{.}Remark 5.3.13.
Jacobian varieties always admit principal polarizations.
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
if f is finite then we have a pullback
(with multiplicity).
Definition 5.3.15.
A correspondence between X,Y is a curve Z and a pair of finite morphisms.
then we get induced maps
Modular jacobians and Hecke correspondences.
Consider p\nmid N we haveDefinition 5.3.16.
the Hecke correspondence T_p on X_0(N) is
Theorem 5.3.17. Eichler-Shimura.
T_{p*} = \Frob_p + p \Frob_p\inv \in \End(J_0(N)_{\FF_p})\text{.}