12
$\begingroup$

The precise form of Torelli's theorem is as follows (translated from Serre's appendix to Lauter - Geometric methods for improving the upper bounds on the number of rational points on algebraic curves over finite fields):

Let $k$ be a field, and let $X_{/k}$ be a nice (= smooth, projective and geometrically integral) curve over $k$ of genus $g > 1$. Let $(\operatorname{Jac}(X),a)$ denote the Jacobian of $X$ together with the Jacobian's canonical principal polarization $a$, which is of degree 1. Let $X'_{/k}$ be another nice curve. Any isomorphism $f: X \to X'$ defines by transport of structure an isomorphism $f_J: (J, a) \to (J, a')$.

Torelli's theorem says that we get almost all of the isomorphisms: $(J, a) \to (J', a')$ in this way. More precisely:

Theorem 1. Suppose $X$ is hyperelliptic. Then, for every isomorphism of polarized abelian varieties $F: (\operatorname{Jac}(X),a) \stackrel{\sim}{\rightarrow} (\operatorname{Jac}(X'),a')$, there exists a unique isomorphism $f: X \stackrel{\sim}{\rightarrow} X'$ such that $F = \operatorname{Jac} f$.

Theorem 2. Suppose $X$ is not hyperelliptic. Then, for every isomorphism of polarized varieties $F: (\operatorname{Jac}(X),a) \stackrel{\sim}{\rightarrow} (\operatorname{Jac}(X'),a')$, there exists an isomorphism $f: X \stackrel{\sim}{\rightarrow} X'$ and $e \in \{ \pm 1\}$ such that $F = e \cdot \operatorname{Jac} f$. Moreover, the pair $(f,e)$ is uniquely determined by $F$.

So, we conclude:

$$\operatorname{Aut}(\operatorname{Jac}(X), a) \simeq \begin{cases} \operatorname{Aut}(X) & \text{if $X$ is hyperelliptic} \\ \operatorname{Aut}(X) \oplus \mathbb{Z}/2 & \text{if $X$ is not hyperelliptic} \end{cases} $$

Serre also states there that he does not know of a place where the precise Torelli theorem is proved in its full glory. In the algebraically closed field case, it is in Weil's Zum Beweis des Torellischen Satzes.

Is there a source that proves it over general fields?

$\endgroup$
  • 3
    $\begingroup$ Possible duplicate from mathoverflow.net/questions/23848/…. See in particular Dan Petersen's answer and the comments below. $\endgroup$ – Francesco Polizzi Jun 30 '18 at 8:12
  • 4
    $\begingroup$ There are proofs by Torelli, Weil, Matsusaka, Andreotti, ... The precise formulation you give follows from the theorem, p. 109, of Martens, H., A new proof of Torelli's theorem, Annals of Math. 78, no. 1 (1963), pp. 107--111, jstor.org/stable/1970505?seq=1#page_scan_tab_contents In the hyperelliptic case, since the reflection of $W^1$ equals a translation of $W^1$, the general theorem reduces to your Theorem 1. $\endgroup$ – Jason Starr Jun 30 '18 at 17:33

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.