Geometry Vs Arithmetic of schemes - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T18:52:15Z http://mathoverflow.net/feeds/question/7373 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/7373/geometry-vs-arithmetic-of-schemes Geometry Vs Arithmetic of schemes Csar Lozano Huerta 2009-12-01T04:46:28Z 2010-01-02T21:22:37Z <p>Let's suppose we have a Scheme $X$ over the the field $k$, where such a field can be though to be either $\mathbb{C}$ or a finite field $\mathbb{F}_q$. Then having this in mind, Where do we find some representative examples where Geometry governs arithmetic? That is to say, examples where the geometry (or topology) of $X$ over $\mathbb{C}$ dictates the arithmetic behavior over $\mathbb{F}_q$.</p> <p>Answers along with references would be highly appreciated.</p> http://mathoverflow.net/questions/7373/geometry-vs-arithmetic-of-schemes/7384#7384 Answer by David Lehavi for Geometry Vs Arithmetic of schemes David Lehavi 2009-12-01T06:23:35Z 2009-12-01T06:23:35Z <p>Look at Dan Abramovich's <a href="http://www.math.brown.edu/~abrmovic/GOTTINGEN/claynotes.pdf" rel="nofollow">Birational geometry for number theorists</a> </p> http://mathoverflow.net/questions/7373/geometry-vs-arithmetic-of-schemes/7421#7421 Answer by Thomas Riepe for Geometry Vs Arithmetic of schemes Thomas Riepe 2009-12-01T12:29:56Z 2009-12-01T12:29:56Z <p>E. Kowalski just published a very <a href="http://www.math.ethz.ch/~kowalski/exponential-sums-rism.pdf" rel="nofollow" title="pdf">beautifull survey</a> on a related issue: "My main emphasis has been to try to present some of the theory and applications surrounding the Deligne Equidistribution Theorem, for non-specialists (in particular, for readers with little experience in algebraic geometry)". Deligne's theorem and the work of Katz and others later on it are tough to enter, this survey provides a kind of bridge. </p> http://mathoverflow.net/questions/7373/geometry-vs-arithmetic-of-schemes/7501#7501 Answer by Ilya Nikokoshev for Geometry Vs Arithmetic of schemes Ilya Nikokoshev 2009-12-01T20:32:50Z 2009-12-02T23:25:22Z <p>Let's start with the most elementary example: <strong>projective space</strong> $\mathbb P^n$. It's not hard to see that that the number of points on it is always $q^n + q^{n-1} + \dots + q + 1.$</p> <p>Note that this is because $\mathbb P^n$ can be always decomposed into simpler pieces: $\mathbb A^n \cup \mathbb A^{n-1}\cup\dots\cup \mathbb A^0$. Interestingly, something similar applies to <strong>all $\mathbb F_q$-varieties</strong>. Specifically, the Lefschetz fixed points formula from topology applied to arithmetics gives the following statement for a variety $X/\mathbb F_q:$</p> <blockquote> <p>There exist some algebraic numbers $\alpha_i$ with $|\alpha_i| = q^{n_i/2}$ for some $(n_i)$ such that the number of points $$\# X(\mathbb F_{q^l}) = \sum_i (-1)^{n_i}\alpha^l_i\quad \text{for}\ l > 0 .$$</p> </blockquote> <p>Numbers $\alpha_i$ in fact come from geometry: they are eigenvalues of some operators acting on etale cohomology groups $H_{et}(X)$. In particular, the numbers $n_i$ can only occupy an interval between 0 and $\text{dim}\, X$ and there are as many of them as the dimension of this group.</p> <p>These groups can directly compared to the case of $\mathbb C$ whenever you construct your variety in a geometric way. To see how, consider the example of curves. Over $\mathbb C$ the cohomology have the form $\mathbb C \oplus \mathbb C^{2g} \oplus \mathbb C\$ for some $g$ called <em>genus</em>; the same holds over $\mathbb F_q$:</p> <ul> <li><strong>projective line</strong> $\mathbb P^1$ has genus 0, so it always has $n+1$ points</li> <li><strong>elliptic curves</strong> $x^2 = y^3 + ay +b$ have genus 1, so they must have exactly $n + 1 + \alpha + \bar\alpha$ points for some $\alpha\in \mathbb C$ with $|\alpha| = \sqrt q.$ This is exactly the <strong><a href="http://en.wikipedia.org/wiki/Hasse%5Fbound" rel="nofollow">Hasse bound</a></strong> mentioned in another post. </li> </ul> <p>These theorems, which provided an unexpected connecion between topology and arithmetics some half-century ago, were just the beginning of studying varieties over $\mathbb F_q$ using the geometric intuition that comes from the complex case. </p> <p>You can read more at any decent introduction to <a href="http://en.wikipedia.org/wiki/Arithmetic%5Fgeometry" rel="nofollow">arithmetic geometry</a> or <a href="http://en.wikipedia.org/wiki/Etale%5Fcohomology" rel="nofollow">étale cohomology</a>. There are also some questions here about <a href="http://mathoverflow.net/questions/tagged/motives" rel="nofollow">motives</a> which are a somewhat more abstract version of the above picture.</p> <p><hr /></p> <p>As a reply to Ben's comment above about reconstructing the genus if you know <code>$X_n = \#X(F_{q^n})$</code>:</p> <ul> <li><p>You know with certainty that $1 + q^n - X_n = \sum \alpha_i^n\$ for some algebraic numbers $\alpha_i, i = 1, 2, \dots$ having property $|\alpha_i| = \sqrt q.$</p></li> <li><p>There cannot be two different solutions $(\alpha_i)$ and $(\beta_i)$ for a given sequence of $X_n$ because if $N$ is a number such that both $\alpha_i = \beta_i = 0$ for $i>N$ then both $\alpha$ and $\beta$ are uniquely determined from the first $N+1$ terms of the sequence.</p></li> <li><p>So a given sequence uniquely determines the genus.</p></li> </ul> <p>I don't know, however, if a constructive algorithm that guarantees to terminate and return genus for a sequence $X_n$ is possible. The first idea is to loop over natural numbers testing the conjecture that genus is less then $N$, but there seem to be some nuances. </p> http://mathoverflow.net/questions/7373/geometry-vs-arithmetic-of-schemes/7599#7599 Answer by Mikhail Bondarko for Geometry Vs Arithmetic of schemes Mikhail Bondarko 2009-12-02T17:59:10Z 2009-12-02T17:59:10Z <p>A brief answer is: in order to relate a variety over a finite field with a one over complex numbers, a common 'nice' model for them over some number field is needed. If such a model exists, then the varieties in question have isomorphic etale cohomology groups. Probably they also have isomorphic etale homotopy types; then l-completions of their homotopy groups are isomorphic. Note here: etale cohomology 'almost computes' singular cohomology of complex varieties, and completely computes the number of point of a variety over a finite field.</p> http://mathoverflow.net/questions/7373/geometry-vs-arithmetic-of-schemes/10527#10527 Answer by Anweshi for Geometry Vs Arithmetic of schemes Anweshi 2010-01-02T21:22:37Z 2010-01-02T21:22:37Z <p>Look at Darmon's article on "<a href="http://www.math.mcgill.ca/darmon/pub/Articles/Expository/12.Clay/paper.pdf" rel="nofollow">Arithmetic of Curves</a>".</p>