# The use of embedding a curve into its Jacobian

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.

## 5 Answers

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.

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.

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

• Nice. I guess the surjectivity of the map isn't so easy to prove. xD – Harry Mar 26 '12 at 13:09

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 $k\geq 2$.

• ... 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
• 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
• ... viva Verdi! – Filippo Alberto Edoardo Mar 26 '12 at 18:10

One can deduce the Weil conjectures for curves from the case of Abelian varieties. (See Milne's Abelian Varieties lecture notes http://jmilne.org/math/CourseNotes/AV.pdf section 11 or his article in Cornell-Silverman.)