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Section 1.2 Introduction (Angus)

Subsection 1.2.1 Definitions

Definition 1.2.1. Abelian varieties.

An abelian variety is a complete connected algebraic group.

Definition 1.2.2. Algebraic groups.

An algebraic group is an algebraic variety \(G\) along with regular maps \(m\colon G\times G\to G\text{,}\) \(e \colon * \to G\text{,}\) \(\operatorname{inv}\colon G\to G\) such that the following diagrams commute.

Identity

\begin{equation*} \xymatrix{ \ast\times G\ar[dr]_{\sim}\ar[r]^{e\times \id} & G\times G\ar[d]_m &G\times \ast\ar[l]^{\id\times e}\ar[dl]^\sim\\ & G & } \end{equation*}

Inverse

\begin{equation*} \xymatrix{ G\ar[d]\ar[r]^{\operatorname{inv}, \id} & G\times G\ar[d]_m &G\ar[l]^{\id, \operatorname{inv}}\ar[d]\\ \ast\ar[r]_e& G &\ast\ar[l]_e } \end{equation*}

Associativity

\begin{equation*} \xymatrix{ G \times G \times G \ar[d]_{m\times \id}\ar[r]^{\id\times m} & G\times G\ar[d]_m\\ G\times G\ar[r]_m & G } \end{equation*}
Definition 1.2.3. Complete varieties.

A variety \(X\) is complete if every projection map

\begin{equation*} X\times Y \to Y \end{equation*}

is closed.

Example 1.2.4. Abelian varieties.
  • Elliptic curves.
  • Weil restriction \(\Res_{K/\QQ} E\) of an elliptic curve \(E\text{.}\)
  • Jacobian varieties of curves.

Plan:

  • Some motivation via elliptic curves.
  • Gathering some material about “completeness”.
  • Prove that abelian varieties are abelian.

Subsection 1.2.2 Elliptic curves (\(\characteristic(k) \ne 2,3\))

Strategy: Item 1 \(\iff\) Item 2 \(\iff\)Item 3 and Item 2 \(\implies\) Item 4 \(\implies\) Item 1.

Item 1 \(\implies\) Item 2 is done.

Item 2 \(\implies\)Item 1: Riemann-Roch states that \(l(D) = l(K-D) + \deg(D) + 1 -g \) so here \(l(D) = l(K-D) + \deg (D)\) further is \(D\gt 0\) then \(l(K-D) = 0\) in which case \(l(D) = \deg(D)\text{.}\) Consider \(L(nP_0)\) for \(n \gt 0\) Riemann-Roch implies that \(l(nP_0) = n\) then it always contains the constants.

\begin{equation*} L(P_0) = k \end{equation*}
\begin{equation*} L(2P_0) = k \oplus kx \end{equation*}
\begin{equation*} L(3P_0) = k \oplus kx \oplus ky \end{equation*}
\begin{equation*} \vdots \end{equation*}
\begin{equation*} L(6P_0) = k \oplus kx \oplus ky \oplus k x^2 \oplus ky^2 \oplus kxy \oplus kx^3/\sim \end{equation*}

so we must have a relation which after manipulation is of the desired form. We get an embedding

\begin{equation*} E \hookrightarrow \PP^2 \end{equation*}
\begin{equation*} P\mapsto (x(P):y(P): 1) \,(P\ne P_0) \end{equation*}
\begin{equation*} P_0 \mapsto (0:1 : 0) \end{equation*}

and thus \(E\) is of the desired form.

Definition 1.2.6. Elliptic curves.

An elliptic curve over \(k\) is any/all of that 1.2.5.

Which of the above characterisations generalise to abelian varieties?

  1. No, in general we don't know that the equations look like.
  2. One could possibly replace “genus” with a condition on the dimension of cohomology groups.
  3. Yes, this is essentially the definition.
  4. Yes, stay tuned!

Subsection 1.2.3 Complete varieties

Idea: if \(X \times Y\) had product topology (instead of its Zariski topology) then complete is equivalent to compact.

We'd like to gather a few results about complete varieties we can use to access properties of abelian varieties (like abelianness).

Let \(\Gamma_\phi = \{(v, \phi(v))\} \subseteq V\times W\) be the graph of \(\phi\text{.}\) Its a closed subvariety of \(V\times W\text{.}\) Under the projection \(V\times W \to W\text{,}\) the image of \(\Gamma_\phi\) is \(\phi(V)\) and thus closed.

A regular function is a morphism \(f\colon V \to \aff^1\text{.}\) By the above \(f(V) \subseteq \aff^1\) is closed, and this is a finite set of points. But connected implies we just have one point.

We have an embedding \(W \hookrightarrow \aff^n\text{.}\) On \(\aff^n\) we have the coordinate functions \(\aff^n \xrightarrow{x_i} \aff^1\text{.}\) The composition

\begin{equation*} V \xrightarrow\phi W \hookrightarrow\aff^n \to \aff^1 \end{equation*}

be the above is constant. Thus the coordinates of \(\phi(V)\) are constant, so \(\phi(V) = \{\text{pt}\}\text{.}\)

A final result of interest that I won't prove today:

The main goal of this section is to prove the following theorem:

Since \(V\times W\) is geometrically irreducible, \(V\) must be connected. Denote the projection \(q\colon V\times W \to W\text{.}\) Let \(U_0 \ni u_0\) be an open neighborhood. We consider the set

\begin{equation*} Z = \{w\in W : \alpha((v,w)) \not\in U_0 \text{ for some } v\in V\} = q(\alpha^{-1}(U\smallsetminus U_0)) \end{equation*}

Since \(q\) is closed, \(Z\subseteq W\) is closed. Since \(w_0\in W\smallsetminus Z\text{,}\) \(W\smallsetminus Z\) is a nonempty open subset of \(W\text{.}\)

Consider \(w \in W\smallsetminus Z\text{.}\) Since \(V\times\{w\} \cong V\) it is complete and connected. Thus

\begin{equation*} \alpha(V\times \{w\}) = \{\text{pt}\} = \alpha((v_0,w)) = \{u_0\} \end{equation*}

which implies that

\begin{equation*} \alpha(V\times (W\smallsetminus Z)) = \{u_0\} \end{equation*}

Since \(V\times (W\smallsetminus Z) \subseteq V\times W\) is open and \(V\times W\) is irreducible, it is dense. So \(\alpha(V\times W) = \{u_0\}\text{.}\)

First compose by a translation on \(B\) such that \(\alpha(0) = 0\text{.}\) Consider the map

\begin{align*} \phi \colon A\times A\amp\to B\\ (a,a') \amp\mapsto \alpha(a+a') - \alpha(A) -\alpha(a') \end{align*}

Then

\begin{align*} \phi(A\times\{0\}) \amp = \alpha(a+ 0) - \alpha(a) - \alpha(0) = 0\\ \phi(\{0\}\times A) \amp = \alpha(0+ a) - \alpha(0) - \alpha(a) = 0\text{.} \end{align*}

By the rigidity theorem 1.2.11 \(\phi(A\times A) = \{0\}\) hence \(\alpha(a + a') = \alpha(a) + \alpha(a')\text{.}\)

The inversion map \(a \mapsto -a\) sends \(0\) to \(0\text{,}\) thus is a homomorphism. Therefore

\begin{equation*} a+ b - a -b = a+b -(a+b) = 0 \end{equation*}

and so

\begin{equation*} a+b=b+a\text{.} \end{equation*}