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Realizing Hecke Actions on Modular Forms Via Cohomology of Dessins d'Enfants.

Record Type:	Electronic resources : Monograph/item
Title/Author:	Realizing Hecke Actions on Modular Forms Via Cohomology of Dessins d'Enfants./
Author:	Ho, Mingxue Zhu.
Published:	Ann Arbor : ProQuest Dissertations & Theses, : 2021,
Description:	208 p.
Notes:	Source: Dissertations Abstracts International, Volume: 83-04, Section: B.
Contained By:	Dissertations Abstracts International83-04B.
Subject:	Mathematics. -
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=28543724
ISBN:	9798460429653

Realizing Hecke Actions on Modular Forms Via Cohomology of Dessins d'Enfants.
Ho, Mingxue Zhu.

Realizing Hecke Actions on Modular Forms Via Cohomology of Dessins d'Enfants. - Ann Arbor : ProQuest Dissertations & Theses, 2021 - 208 p.

Source: Dissertations Abstracts International, Volume: 83-04, Section: B.

Thesis (Ph.D.)--Duke University, 2021.

This item must not be sold to any third party vendors.

A well known action on the space of Modular forms is done by Hecke operators, which also play an important role in modularity. This action can be further break down into steps, and form what we call a Hecke correspondence. On the other hand, through Belyi and Grothendieck there is a one to one correspondence between the equivalence classes of algebraic curves defined over Q equipped with Belyi functions and equivalence classes of dessins d'enfants. This applies in particular to modular curves. In my dissertation work, I will study the action of Hecke operators on certain dessins, namely, those that correspond to X0(N). This is done by defining a cohomology with coefficients in a local system on dessins, and have Hecke operators act on it. We will also construct a Hecke-equivalent isomorphism of the cohomology group with the space of cusp forms. We hope that this work can present the first step in studying the Hecke action on more general dessins.

ISBN: 9798460429653Subjects--Topical Terms:

515831
Mathematics.
Subjects--Index Terms:

Dessins d'enfants

Realizing Hecke Actions on Modular Forms Via Cohomology of Dessins d'Enfants.
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300 $a 208 p.
500 $a Source: Dissertations Abstracts International, Volume: 83-04, Section: B.
500 $a Advisor: Saper, Leslie.
502 $a Thesis (Ph.D.)--Duke University, 2021.
506 $a This item must not be sold to any third party vendors.
520 $a A well known action on the space of Modular forms is done by Hecke operators, which also play an important role in modularity. This action can be further break down into steps, and form what we call a Hecke correspondence. On the other hand, through Belyi and Grothendieck there is a one to one correspondence between the equivalence classes of algebraic curves defined over Q equipped with Belyi functions and equivalence classes of dessins d'enfants. This applies in particular to modular curves. In my dissertation work, I will study the action of Hecke operators on certain dessins, namely, those that correspond to X0(N). This is done by defining a cohomology with coefficients in a local system on dessins, and have Hecke operators act on it. We will also construct a Hecke-equivalent isomorphism of the cohomology group with the space of cusp forms. We hope that this work can present the first step in studying the Hecke action on more general dessins.
590 $a School code: 0066.
650 4 $a Mathematics. $3 515831
650 4 $a Theoretical mathematics. $3 3173530
650 4 $a Applied mathematics. $3 2122814
653 $a Dessins d'enfants
653 $a Hecke operators
653 $a Modular forms
690 $a 0405
690 $a 0642
690 $a 0364
710 2 $a Duke University. $b Mathematics. $3 1036449
773 0 $t Dissertations Abstracts International $g 83-04B.
790 $a 0066
791 $a Ph.D.
792 $a 2021
793 $a English
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=28543724