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Admissible covers, modular operads and modular forms
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2011 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis contains three articles related to operads and moduli spaces of admissible covers of curves. In Paper A we isolate cohomology classes coming from modular forms inside a certain space of admissible covers, thereby showing that this moduli space can be used as a substitute for a Kuga–Sato variety. Paper B contains a combinatorial proof of Ezra Getzler’s semiclassical approximation for modular operads, and a proof of a formula needed in Paper A. In Paper C we explain in what sense spaces of admissible covers form a modular operad, by introducing the notion of an operad colored by a groupoid.

Abstract [sv]

Denna avhandling innehåller tre artiklar relaterade till operader och modulirum för godtagbara övertäckningar av kurvor. I artikel A isoleras kohomologiklasser associerade till modulära former inuti ett visst rum av godtag- bara övertäckningar, vilket visar att detta modulirum kan användas som ett substitut för en Kuga–Sato-varietet. Artikel B innehåller ett kombinatoriskt bevis av Ezra Getzlers semiklassiska approximation för modulära operader, och beviset av en formel som behövs i artikel A. I artikel C förklaras i vilken mening rum av tillåtbara övertäckningar utgör en modulär operad, nämligen en operad färgad av en gruppoid.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology , 2011. , vii, 20 p.
Series
Trita-MAT. MA, ISSN 1401-2278 ; 2011:09
National Category
Algebra and Logic
Identifiers
URN: urn:nbn:se:kth:diva-48923ISBN: 978-91-7501-179-0 (print)OAI: oai:DiVA.org:kth-48923DiVA: diva2:458869
Presentation
2011-12-01, 3721, KTH, Lindstedtsvägen 25, Stockholm, 14:00 (English)
Opponent
Supervisors
Note
QC 20111124Available from: 2011-11-24 Created: 2011-11-24 Last updated: 2011-12-02Bibliographically approved
List of papers
1. Cusp form motives and admissible G-covers
Open this publication in new window or tab >>Cusp form motives and admissible G-covers
2012 (English)In: Algebra & Number Theory, ISSN 1937-0652, E-ISSN 1944-7833, Vol. 6, no 6, 1199-1221 p.Article in journal (Refereed) Published
Abstract [en]

There is a natural S-n-action on the moduli space (M) over bar (1,n) (B(Z/mZ)(2)) of twisted stable maps into the stack B(Z/mZ)(2), and so its cohomology may be decomposed into irreducible S-n-representations. Working over Spec Z [1/m] we show that the alternating part of the cohomology of one of its connected components is exactly the cohomology associated to cusp forms for Gamma(m). In particular this offers an alternative to Scholl's construction of the Chow motive associated to such cusp forms. This answers in the affirmative a question of Manin on whether one can replace the Kuga-Sato varieties used by Scholl with some moduli space of pointed stable curves.

Keyword
Chow motive, cusp form, admissible cover, twisted curve, level structure
National Category
Algebra and Logic
Identifiers
urn:nbn:se:kth:diva-48905 (URN)10.2140/ant.2012.6.1199 (DOI)000307648000005 ()2-s2.0-84865483777 (Scopus ID)
Note

QC 20120924. Updated from submitted to published.

Available from: 2011-11-24 Created: 2011-11-24 Last updated: 2017-12-08Bibliographically approved
2. A remark on Getzler's semi-classical approximation
Open this publication in new window or tab >>A remark on Getzler's semi-classical approximation
2012 (English)In: Geometry And Arithmetic, EMS Publishing House , 2012, 309-316 p.Conference paper, Published paper (Refereed)
Abstract [en]

Ezra Getzler notes in the proof of the main theorem of [Get98] that ”A proof of the theorem could no doubt be given using [a combinatorial interpretation in terms of a sum over necklaces]; however, we prefer to derive it directly from Theorem 2.2”. In this note we give such a direct combinatorial proof using wreath product symmetric functions.

Place, publisher, year, edition, pages
EMS Publishing House, 2012
Series
EMS Series of Congress Reports
Keyword
Modular operads, graphical enumeration, tensor species
National Category
Algebra and Logic
Identifiers
urn:nbn:se:kth:diva-48904 (URN)10.4171/119 (DOI)000317792500018 ()978-3-03719-119-4 (ISBN)
Conference
Conference on Geometry and Arithmetic Location: Schiermonnikoog Isl, Netherlands Date: SEP 20-24, 2010
Note

QC 20130527

Available from: 2011-11-24 Created: 2011-11-24 Last updated: 2013-05-27Bibliographically approved
3. On the operad structure of admissible G-covers
Open this publication in new window or tab >>On the operad structure of admissible G-covers
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We describe the modular operad structure on the moduli spaces of pointed stable curves equipped with an admissible G-cover. To do this we are forced to introduce the notion of an operad colored not by a set but by the objects of a groupoid. This construction interpolates in some sense between “framed” and “colored” versions of operads; we hope that it will be of independent interest. An algebra over this operad is the same thing as a G-equivariant CohFT. Our main theorem is an extension of the symmetric function formalism for modular operads to this setting; we prove an analogue of the formula of Getzler and Kapranov describing the effect of the “free modular operad” functor on the level of symmetric functions.

National Category
Algebra and Logic
Identifiers
urn:nbn:se:kth:diva-48902 (URN)
Note

QS 2011

Available from: 2011-11-24 Created: 2011-11-24 Last updated: 2013-05-20Bibliographically approved

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