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Point counts and the cohomology of moduli spaces of curves
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2006 (English)Doctoral thesis, comprehensive summary (Other scientific)
##### Abstract [en]

n this thesis we count the number of points defined over finite fields of certain moduli spaces of pointed curves. The aim is primarily to gain cohomological information.

Paper I is joint work with Orsola Tommasi. Here we present details of the method of finding cohomological information on moduli spaces of curves by counting points. Another method of determining the cohomology of moduli spaces of curves is also presented. It is by stratifying them into pieces that are quotients of complements of discriminants in complex vector spaces. Results obtained by these two methods allow us to compute the Hodge structure of the cohomology of $\overline{\mathcal{M}}_4$.

In Paper II we consider the moduli space $\mathcal{H}_{g,n}$ of $n$-pointed smooth hyper-elliptic curves of genus $g$. We find that there are recursion formulas in the genus that the numbers of points of $\mathcal{H}_{g,n}$ fulfill. Thus, if we can make $\mathbb{S}_n$-equivariant counts of $\mathcal{H}_{g,n}$ for low genus, then we can do this for every genus. Information about curves of genus zero and one is then found to be sufficient to compute the answers for hyperelliptic curves of all genera and with up to seven points. These results are applied to $\overline{\mathcal{M}}_{2,n}$ for $n$ up to seven, and give us the $\mathbb{S}_n$-equivariant Hodge structure of their cohomology. Moreover, we find that the $\mathbb{S}_n$-equivariant counts of $\mathcal{H}_{g,n}$ depend upon whether the characteristic is even or odd, where the first instance of this dependence is for six-pointed curves of genus three.

In Paper III we consider the moduli space $\mathcal{Q}_{n}$ of smooth $n$-pointed nonhyperelliptic curves of genus three. Using the canonical embedding of these curves as plane quartics, we make $\mathbb{S}_n$-equivariant counts of the numbers of points of $\mathcal{Q}_{n}$ for $n$ up to seven. We also count pointed plane cubics. This gives us $\mathbb{S}_n$-equivariant counts of the moduli space $\mathcal{M}_{1,n}$ for $n$ up to ten. We can then determine the $\mathbb{S}_n$-equivariant Hodge structure of the cohomology of $\overline{\mathcal{M}}_{3,n}$ for $n$ up to five.

n this thesis we count the number of points defined over finite fields of certain moduli spaces of pointed curves. The aim is primarily to gain cohomological information.

Paper I is joint work with Orsola Tommasi. Here we present details of the method of finding cohomological information on moduli spaces of curves by counting points. Another method of determining the cohomology of moduli spaces of curves is also presented. It is by stratifying them into pieces that are quotients of complements of discriminants in complex vector spaces. Results obtained by these two methods allow us to compute the Hodge structure of the cohomology of $\overline{\mathcal{M}}_4$.

In Paper II we consider the moduli space $\mathcal{H}_{g,n}$ of $n$-pointed smooth hyper-elliptic curves of genus $g$. We find that there are recursion formulas in the genus that the numbers of points of $\mathcal{H}_{g,n}$ fulfill. Thus, if we can make $\mathbb{S}_n$-equivariant counts of $\mathcal{H}_{g,n}$ for low genus, then we can do this for every genus. Information about curves of genus zero and one is then found to be sufficient to compute the answers for hyperelliptic curves of all genera and with up to seven points. These results are applied to $\overline{\mathcal{M}}_{2,n}$ for $n$ up to seven, and give us the $\mathbb{S}_n$-equivariant Hodge structure of their cohomology. Moreover, we find that the $\mathbb{S}_n$-equivariant counts of $\mathcal{H}_{g,n}$ depend upon whether the characteristic is even or odd, where the first instance of this dependence is for six-pointed curves of genus three.

In Paper III we consider the moduli space $\mathcal{Q}_{n}$ of smooth $n$-pointed nonhyperelliptic curves of genus three. Using the canonical embedding of these curves as plane quartics, we make $\mathbb{S}_n$-equivariant counts of the numbers of points of $\mathcal{Q}_{n}$ for $n$ up to seven. We also count pointed plane cubics. This gives us $\mathbb{S}_n$-equivariant counts of the moduli space $\mathcal{M}_{1,n}$ for $n$ up to ten. We can then determine the $\mathbb{S}_n$-equivariant Hodge structure of the cohomology of $\overline{\mathcal{M}}_{3,n}$ for $n$ up to five.

n this thesis we count the number of points defined over finite fields of certain moduli spaces of pointed curves. The aim is primarily to gain cohomological information.

Paper I is joint work with Orsola Tommasi. Here we present details of the method of finding cohomological information on moduli spaces of curves by counting points. Another method of determining the cohomology of moduli spaces of curves is also presented. It is by stratifying them into pieces that are quotients of complements of discriminants in complex vector spaces. Results obtained by these two methods allow us to compute the Hodge structure of the cohomology of $\overline{\mathcal{M}}_4$.

In Paper II we consider the moduli space $\mathcal{H}_{g,n}$ of $n$-pointed smooth hyper-elliptic curves of genus $g$. We find that there are recursion formulas in the genus that the numbers of points of $\mathcal{H}_{g,n}$ fulfill. Thus, if we can make $\mathbb{S}_n$-equivariant counts of $\mathcal{H}_{g,n}$ for low genus, then we can do this for every genus. Information about curves of genus zero and one is then found to be sufficient to compute the answers for hyperelliptic curves of all genera and with up to seven points. These results are applied to $\overline{\mathcal{M}}_{2,n}$ for $n$ up to seven, and give us the $\mathbb{S}_n$-equivariant Hodge structure of their cohomology. Moreover, we find that the $\mathbb{S}_n$-equivariant counts of $\mathcal{H}_{g,n}$ depend upon whether the characteristic is even or odd, where the first instance of this dependence is for six-pointed curves of genus three.

In Paper III we consider the moduli space $\mathcal{Q}_{n}$ of smooth $n$-pointed nonhyperelliptic curves of genus three. Using the canonical embedding of these curves as plane quartics, we make $\mathbb{S}_n$-equivariant counts of the numbers of points of $\mathcal{Q}_{n}$ for $n$ up to seven. We also count pointed plane cubics. This gives us $\mathbb{S}_n$-equivariant counts of the moduli space $\mathcal{M}_{1,n}$ for $n$ up to ten. We can then determine the $\mathbb{S}_n$-equivariant Hodge structure of the cohomology of $\overline{\mathcal{M}}_{3,n}$ for $n$ up to five.

##### Place, publisher, year, edition, pages
Stockholm: KTH , 2006.
##### Series
Trita-MAT, ISSN 1401-2286
Mathematics
##### Identifiers
ISBN: 91-7178-447-0OAI: oai:DiVA.org:kth-4105DiVA: diva2:10754
##### Public defence
2006-09-29, F3, Lindstedtsvägen 26, Stockholm, 14:00
##### Note
QC 20100701Available from: 2006-09-18 Created: 2006-09-18 Last updated: 2010-07-01Bibliographically approved
##### List of papers
1. The rational cohomology of $\overline{\mathcal{M}}_4$
Open this publication in new window or tab >>The rational cohomology of $\overline{\mathcal{M}}_4$
Mathematics
##### Identifiers
urn:nbn:se:kth:diva-6128 (URN)
##### Note
QC 20100701Available from: 2006-09-18 Created: 2006-09-18 Last updated: 2010-07-01Bibliographically approved
2. Equivariant counts of points of the moduli spaces of pointed hyperelliptic curves
Open this publication in new window or tab >>Equivariant counts of points of the moduli spaces of pointed hyperelliptic curves
2009 (English)In: Documenta Mathematica, ISSN 1431-0643, E-ISSN 1431-0635, Vol. 14, 259-296 p.Article in journal (Refereed) Published
##### Abstract [en]

We consider the moduli space H-g,H-n of n-pointed smooth hyperelliptic curves of genus g. In order to get cohomological information we wish to make S-n-equivariant counts of the numbers of points defined over finite fields of this moduli space. We find recurrence relations in the genus that these numbers fulfill. Thus, if we can make S-n-equivariant counts of H-g,H-n for low genus, then we can do this for every genus. Information about curves of genus 0 and 1 is then found to be sufficient to compute the answers for H-g,H-n for all g and for n <= 7. These results are applied to the moduli spaces of stable curves of genus 2 with up to 7 points, and this gives us the S-n-equivariant Galois (resp. Hodge) structure of their l-adic (resp. Betti) cohomology.

##### Keyword
Cohomology of moduli spaces of curves; curves over finite fields
Mathematics
##### Identifiers
urn:nbn:se:kth:diva-6129 (URN)000268811200001 ()
##### Note
QC 20100701Available from: 2006-09-18 Created: 2006-09-18 Last updated: 2010-07-01Bibliographically approved
3. Cohomology of moduli spaces of curves of genus three via point counts
Open this publication in new window or tab >>Cohomology of moduli spaces of curves of genus three via point counts
2008 (English)In: Journal für die Reine und Angewandte Mathematik, ISSN 0075-4102, E-ISSN 1435-5345, Vol. 622, 155-187 p.Article in journal (Refereed) Published
##### Abstract [en]

In this article we consider the moduli space of smooth n-pointed non-hyperelliptic curves of genus 3. In the pursuit of cohomological information about this space, we make S-n-equivariant counts of its numbers of points defined over finite fields for n <= 7. Combining this with results on the moduli spaces of smooth pointed curves of genus 0, 1 and 2, and the moduli space of smooth hyperelliptic curves of genus 3, we can determine the S-n-equivariant Galois and Hodge structure of the (l-adic respectively Betti) cohomology of the moduli space of stable curves of genus 3 for n <= 5 ( to obtain n <= 7 we would need counts of "8-pointed curves of genus 2'').

##### Keyword
FINITE-FIELDS; ABELIAN SURFACES; LOCAL SYSTEMS
Mathematics
##### Identifiers
urn:nbn:se:kth:diva-6130 (URN)10.1515/CRELLE.2008.068 (DOI)000260245900005 ()2-s2.0-46649104448 (ScopusID)
##### Note
QC 20100701Available from: 2006-09-18 Created: 2006-09-18 Last updated: 2012-04-14Bibliographically approved

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