The use of embedding a curve into its Jacobian - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T10:05:34Z http://mathoverflow.net/feeds/question/92147 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/92147/the-use-of-embedding-a-curve-into-its-jacobian The use of embedding a curve into its Jacobian Harry 2012-03-25T09:08:42Z 2012-03-27T01:49:55Z <p>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. </p> <p>I know of </p> <p>Chabauty's method of proving a special case of the Mordell conjecture.</p> <p>Faltings' use of the Torelli map in his proof of the Shafarevich conjecture for curves.</p> <p>Raynaud's theorem (previously Manin-Mumford conjecture).</p> <p>The Bogomologov conjecture (proven by Ullmo and Zhang).</p> <p>The Mordell-Lang theorem.</p> <p>In some of these examples the embedding of X into its Jacobian is simply part of the statement. I also consider this as "useful".</p> <p>Are there any other nice examples? They don't have to be as difficult as the ones mentioned above.</p> http://mathoverflow.net/questions/92147/the-use-of-embedding-a-curve-into-its-jacobian/92157#92157 Answer by Niels for The use of embedding a curve into its Jacobian Niels 2012-03-25T11:56:43Z 2012-03-25T12:02:05Z <p>In the <em>section conjecture</em> for a number field $k$: the proof of the <em>injectivity</em> of the map</p> <p>$$X(k)\to \mathrm{HomExt}_{G_k}(G_k,\pi_1(X,\overline{x}))$$</p> <p>that attributes to a rational point a section of the fundamental exact sequence</p> <p>$$1\to \pi_1(\overline{X},\overline{x})) \to \pi_1(X,\overline{x})\to G_k \to 1$$</p> <p>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.</p> <p>See </p> <p>Jakob Stix</p> <p>On cuspidal sections of algebraic fundamental groups </p> <p><a href="http://arxiv.org/abs/0809.0017" rel="nofollow">http://arxiv.org/abs/0809.0017</a></p> <p>appendix B</p> <p>for details. This was known to Grothendieck back in 1983, see </p> <p>Grothendieck, Alexander</p> <p>Brief an G. Faltings. (German) [Letter to G. Faltings] </p> <p><a href="http://www.math.jussieu.fr/~leila/grothendieckcircle/GtoF.pdf" rel="nofollow">http://www.math.jussieu.fr/~leila/grothendieckcircle/GtoF.pdf</a></p> http://mathoverflow.net/questions/92147/the-use-of-embedding-a-curve-into-its-jacobian/92230#92230 Answer by Filippo Alberto Edoardo for The use of embedding a curve into its Jacobian Filippo Alberto Edoardo 2012-03-26T05:56:00Z 2012-03-26T05:56:00Z <p>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 $k\geq 2$.</p> http://mathoverflow.net/questions/92147/the-use-of-embedding-a-curve-into-its-jacobian/92299#92299 Answer by ACL for The use of embedding a curve into its Jacobian ACL 2012-03-26T19:07:28Z 2012-03-26T19:07:28Z <p>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).</p> <p>Faltings's presentation is explicitly about subvarieties of Abelian varieties.</p> <p>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.</p> http://mathoverflow.net/questions/92147/the-use-of-embedding-a-curve-into-its-jacobian/92336#92336 Answer by Michael Stoll for The use of embedding a curve into its Jacobian Michael Stoll 2012-03-27T01:44:30Z 2012-03-27T01:49:55Z <p>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 $p \in S$.</p> <p>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 <code>$X({\mathbb Q})$</code> is empty in this way (if it is empty). See <a href="http://journals.cambridge.org/action/displayAbstract?fromPage=online&amp;aid=7881094&amp;fulltextType=RA&amp;fileId=S1461157009000187" rel="nofollow">this paper</a>.</p>