User xavier roulleau - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T09:44:22Z http://mathoverflow.net/feeds/user/21030 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95729/what-are-supersingular-varieties/95828#95828 Answer by Xavier Roulleau for What are supersingular varieties? Xavier Roulleau 2012-05-03T06:53:09Z 2012-05-03T06:53:09Z <p>For a surface $S$, supersingular means that the étale cohomology group $H^{2}(S,\mathbb{Q}_\ell)$ ($\ell$ a prime, prime to the characteristic $p$) is generated by divisors on $S$ (thus the Picard number equals to the second Betti number). Supersingularity is useful if one wishes to compute the Zeta function of $S$.</p> <p>Shioda worked on supersingular surfaces, see e.g. "An Example of Unirational Surfaces in Characteristic $p$" or "On Unirationality of Supersingular Surfaces". Supersingularity is a necessary condition for a surface to be unirational (be it is not sufficient). He gives examples of Fermat surfaces of degree $>4$ that are unirational.</p> http://mathoverflow.net/questions/88619/how-to-compute-the-etale-cohomology-of-the-quotient-of-a-surface-by-a-finite-grou How to compute the étale cohomology of the quotient of a surface by a finite group of automorphisms ? Xavier Roulleau 2012-02-16T10:17:34Z 2012-02-17T16:06:19Z <p>Let $S$ be a smooth surface defined over a finite field $K$ of char. $p$. Let $G$ be a finite group of automorphisms of $S$. Let $Z\to S/G$ be the minimal resolution of the quotient of $S$ by $G$. Suppose that the fixed point set of every elements of $G$ is defined over $K$.</p> <p>Let $\ell$ be a prime, $\ell\not=p$. Is it true that $$H^{i}(Z,\mathbb Q_\ell) \simeq H^{i}(S,\mathbb Q_\ell)^G$$ for $i=1,3$ and $$H^{2}(Z,\mathbb Q_\ell)\simeq H^{2}(S,\mathbb Q_\ell)^G+\mathbb Q_\ell C_1+\dots +\mathbb Q_\ell C_k$$where the $C_i$ are the exceptional curves of the resolution $Z\to S/G$ ?</p> <p>This question is an echo of the question "Are there any known formulas about the Hodge-Deligne structure of quotients by action of groups ?" formulated in this forum 2 or 3 weeks ago.</p> http://mathoverflow.net/questions/88520/can-we-decide-if-an-abelian-variety-is-simple-by-knowing-its-zeta-function Can we decide if an abelian variety is simple by knowing its Zeta function ? Xavier Roulleau 2012-02-15T14:21:05Z 2012-02-16T01:52:14Z <p>Let $A$ be an Abelian variety defined over the finite field with $q$ elements. Let $P_i(T)$ be the characteristic polynomial of the action of the Frobenius on the $i^{th}$ étale cohomology group. </p> <p>Is the following assertion true:</p> <p>"The Abelian variety $A$ is simple over the finite field with $q$ elements if and only if $P_1(T)$ is irreducible over $\mathbb Q$" ?</p> <p>One implication is obvious, what about the other one ?</p> http://mathoverflow.net/questions/88619/how-to-compute-the-etale-cohomology-of-the-quotient-of-a-surface-by-a-finite-grou/88724#88724 Comment by Xavier Roulleau Xavier Roulleau 2012-02-17T15:03:35Z 2012-02-17T15:03:35Z Great, thank you very much. http://mathoverflow.net/questions/88520/can-we-decide-if-an-abelian-variety-is-simple-by-knowing-its-zeta-function/88521#88521 Comment by Xavier Roulleau Xavier Roulleau 2012-02-15T16:21:16Z 2012-02-15T16:21:16Z Nice example, thank you. The implication I was thinking about was not so obvious. http://mathoverflow.net/questions/88520/can-we-decide-if-an-abelian-variety-is-simple-by-knowing-its-zeta-function/88523#88523 Comment by Xavier Roulleau Xavier Roulleau 2012-02-15T16:12:30Z 2012-02-15T16:12:30Z I will study that.Thanks a lot David. http://mathoverflow.net/questions/88520/can-we-decide-if-an-abelian-variety-is-simple-by-knowing-its-zeta-function Comment by Xavier Roulleau Xavier Roulleau 2012-02-15T14:48:23Z 2012-02-15T14:48:23Z To Ulrich : what means ordinary ?