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Cusp form motives and admissible G-covers
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
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.

Place, publisher, year, edition, pages
2012. Vol. 6, no 6, 1199-1221 p.
Keyword [en]
Chow motive, cusp form, admissible cover, twisted curve, level structure
National Category
Algebra and Logic
Identifiers
URN: urn:nbn:se:kth:diva-48905DOI: 10.2140/ant.2012.6.1199ISI: 000307648000005Scopus ID: 2-s2.0-84865483777OAI: oai:DiVA.org:kth-48905DiVA: diva2:458842
Note

QC 20120924. Updated from submitted to published.

Available from: 2011-11-24 Created: 2011-11-24 Last updated: 2017-12-08Bibliographically approved
In thesis
1. Admissible covers, modular operads and modular forms
Open this publication in new window or tab >>Admissible covers, modular operads and modular forms
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:nbn:se:kth:diva-48923 (URN)978-91-7501-179-0 (ISBN)
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

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