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I'm looking for as many examples/applications as possible of the use of embedding a smooth projective geometrically connected curve $X$ over a number field $k$ with $X(k)\neq \emptyset$ into its Jacobian (via a rational point). Please do not take the meaning of "use" to seriously.

I know of

Chabauty's method of proving a special case of the Mordell conjecture.

Faltings' use of the Torelli map in his proof of the Shafarevich conjecture for curves.

Raynaud's theorem (previously Manin-Mumford conjecture).

The Bogomologov conjecture (proven by Ullmo and Zhang).

The Mordell-Lang theorem.

In some of these examples the embedding of X into its Jacobian is simply part of the statement. I also consider this as "useful".

Are there any other nice examples? They don't have to be as difficult as the ones mentioned above.

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4 Answers

You could also mention Vojta's proof of Mordell conjecture, as well as its generalization by Faltings (the proof of the so-called Mordell-Lang conjecture) and its simplification by Bombieri (Mordell conjecture revisited).

Faltings's presentation is explicitly about subvarieties of Abelian varieties.

In the presentation by Vojta-Bombieri, you work on a power of a curve and prove a certain height inequality; you then need to interprete this within the Jacobian, as a lower bound for the angle made (in the Mordell-Weil lattice) by two points of the curve.

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Another use of embedding a curve into its Jacobian is to apply the `Mordell-Weil Sieve'. Suppose $k = \mathbb Q$ for simplicity and that you don't know a rational point on $X$, but you know a rational divisor (class) $D$ of degree 1 on $X$. Then you can use $D$ to define an embedding $\iota$ of $X$ into its Jacobian $J$. Now assume in addition that $J(\mathbb Q)$ is known explicitly. Then for every prime $p$ (of good reduction, say), you can consider the images of $X(\mathbb F_p)$ and of $J(\mathbb Q)$ in $J(\mathbb F_p)$ (the first under $\iota$, the second by reduction mod $p$). Clearly, $X(\mathbb Q)$ has to map into the intersection of these two. Now instead of considering one prime, we can consider all primes in a finite set $S$ and look at the product of $\iota(X(\mathbb F_p))$ over all $p \in S$ and the image of $J(\mathbb Q)$ in the product of $J(\mathbb F_p)$ over all $p \in S$.

Now for a suitable choice of $S$ it may be the case that the two sets are disjoint, which then proves that $X$ has no rational points. There are good reasons to believe that it is always possible to prove that $X({\mathbb Q})$ is empty in this way (if it is empty). See this paper.

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In the section conjecture for a number field $k$: the proof of the injectivity of the map

$$X(k)\to \mathrm{HomExt}_{G_k}(G_k,\pi_1(X,\overline{x}))$$

that attributes to a rational point a section of the fundamental exact sequence

$$ 1\to \pi_1(\overline{X},\overline{x})) \to \pi_1(X,\overline{x})\to G_k \to 1 $$

uses an embedding of $X$ into its jacobian to reduce to an abelian variety $A$. The map above is then interpreted as limit of coboundary maps in étale cohomology for the Kummer exact sequences for $A$. One applies Mordell-Weil theorem ($A(k)$ is an abelian group of finite type) to conclude.

See

Jakob Stix

On cuspidal sections of algebraic fundamental groups

http://arxiv.org/abs/0809.0017

appendix B

for details. This was known to Grothendieck back in 1983, see

Grothendieck, Alexander

Brief an G. Faltings. (German) [Letter to G. Faltings]

http://www.math.jussieu.fr/~leila/grothendieckcircle/GtoF.pdf

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Nice. I guess the surjectivity of the map isn't so easy to prove. xD –  Harry Mar 26 '12 at 13:09
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This is not really about the embedding, rather about the identification of points on the Jacobian with degree-0-divisors. If you are given a cuspidal form $f\in S_2(\Gamma_0(N),\mathbb{Q})$, by definition you have a differential form on $X_0(11)$. In order to say that there is an elliptic curve corresponding to $f$ (Shimura-Taniyama-Weil conjecture – and Wiles proves – that every elliptic curve arises in this way, but the direction I am discussing is much older) you want to find a quotient of $J_0(N)=\mathrm{Jac}(X_0(N))$ corresponding to $f$ and to do this you need to transfer the action of endomorphisms of modular forms (i.e. of diferentials) to endomorphisms of the Jacobian. In this way you find an ideal $I_f\subseteq \mathrm{End}(J_0(N))$ such that the corresponding quotient $J_0(N)/I_f$ is the elliptic curve $E_f$. The way one ''transfers the action of endomorphisms of differentials on $X_0(N)$ to endomorphisms of $J_0(N)$'' is through the embedding $X_0(N)\hookrightarrow J_0(N)$. Much of what I said can be generalized to forms of weight $k\geq 2$.

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... viva verdi! –  diverietti Mar 26 '12 at 7:00
    
This might be a silly question, but why wouldn't this work for any congruence subgroup $\Gamma$ of SL$_2(\mathbf{Z})$? If I understood correctly, one starts by taking a cuspidal form $f\in S_2(\Gamma,\mathbf{Q})$. This gives some differential form and the action of endormorphisms of modular forms can be transported to endomorphisms of the Jacobian via the embedding, right? what do we get as a quotient of $J(\Gamma) = \mathrm{Jac}(X(\Gamma))$? –  Harry Mar 26 '12 at 13:07
    
Good question. Eichler-Shimura is much more than a transfer of endomorphisms from modular forms to the Jacobian. It's a very precise statement about the exact action of Hecke operators on $J_0(N)$. Moreover, even that only gives you the result that there is a one-dimensional quotient of $J_0(N)$. Indeed this quotient need not be unique and there are often quotients of $J_0(N)$ which are not one-dimensional. –  stankewicz Mar 26 '12 at 13:48
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I am not claiming it was the more general or the more efficient construction of $E_f$. I was just providing an example. Or do not K understand your question? You are anyhow right, you can play with all congruence subgroups. –  Filippo Alberto Edoardo Mar 26 '12 at 18:10
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... viva Verdi! –  Filippo Alberto Edoardo Mar 26 '12 at 18:10
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