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Hecke Operators and Hall Algebras for Global Function Fields

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Hecke Operators and Hall Algebras for Global Function Fields Instituto Nacional de Matemática Pura e Aplicada Hecke operators and Hall algebras for global function fields Roberto Alvarenga Advisor: Oliver Lorscheid Thesis presented to the Post-graduate Program in Mathematics at Instituto Nacional de Matemática Pura e Aplicada as partial fulfillment of the requirements for the degree of Doctor in Mathematics. Rio de Janeiro April 11, 2018 Abstract The subject of this thesis is arithmetic geometry. In particular, we aim to describe explicitly some graphs, which we call graphs of Hecke operators, for function fields of curves of small genus. The first two chapters are dedicated to the introduction the graphs of Hecke operators and its first properties. The Chapter 1 is dedicated to introduce the prerequisites from number theory and algebraic geometry necessaries to define the graphs of Hecke operators. In the Chapter 2 we give the number theoretic interpretation of graphs of Hecke operators, which were originally introduced by Lorscheid in his Phd thesis [30]. These graphs encode the complete data of the action of Hecke operators on automorphic forms over a global function field. After giving the number theoretic definition of graphs of Hecke operators we translate that definition in terms of algebra- geometric tools. We dedicate a section of Chapter 2 to show a necessary condition for two vertices of these graphs to be connected by an edge and end the chapter with the solution of how to obtain these graphs in the case of the function field of the projective line. Most of the content of this first part appears in the following pre-print: • [1] R. Alvarenga. On graphs of Hecke operators. http://front.math.ucdavis. edu/1709.09243, 2017. In the chapters 3 and 4 we aim to solve our problem of giving a description for the graphs of Hecke operators in the case of an elliptic function field. The theory of Hall algebras plays an important role in the elliptic case, for that we dedicate Chapter 3 to give a friendly introduction to that theory. We develop in Chapter 4 an algorithm to describe the graphs of Hecke operators using the elliptic Hall algebra. Namely, the edges in the graphs of Hecke operators can be recovered from certain products in the Hall algebra, from that our algorithm uses such information to calculate these products, returning the edges of the graphs. Most of the content of this two last chapters appears in the following pre-print: • [2] R. Alvarenga. Hall algebra and graphs of Hecke operators for elliptic curves. In preparation, 2018. Keywords: Hecke operators, automorphic forms, elliptic curves, arithmetic geom- etry, vector bundles, symmetric functions, Hall algebras. 3 Resumo O tópico dessa tese é geometria aritmética. Em particular, nosso objetivo é descrever explicitamente certos grafos, denominados grafos de operadores de Hecke, para curvas de gênero pequeno. Os primeiros dois capítulos são dedicados a introduzir os grafos de operadores de Hecke e suas primeiras propriedades. No Capítulo 1 introduzimos os pré-equisitos necessários da teoria dos números e da geometria algébrica. No Capítulo 2 começamos com a interpretação dos grafos de operadores de Hecke na linguagem da teoria dos números, que é originalmente introduzida por Lorscheid em sua tese de doutorado [30]. Esse grafos codificam todas as informações da ação dos operadores de Hecke nas formas automórficas definidas sobre um corpo de funções global. Depois da abordagem teórico numérica dos grafos de operadores de Hecke, traduzimos aquela definição em termos de objetos da geometria algébrica. Dedicamos uma seção do Capítulo 2 para provar uma condição necessária para dois vertices desses grafos serem conectados por uma aresta e terminamos o capítulo com a solução de como obter tais grafos no caso em que o corpo de funções é o corpo de funções da reta projetiva. Quase todo o conteúdo dessa primeira parte aparece no seguinte preprint: • [1] R. Alvarenga. On graphs of Hecke operators. http://front.math.ucdavis. edu/1709.09243, 2017. Nos capítulos 3 e 4 objetivamos a solução do nosso problema de descrever os grafos no caso do corpo de funções elíptico. A teoria de álgebras de Hall desempenha um papel fundamental no caso elíptico, por isso dedicamos o Capítulo 3 a uma amigavel introdução a esta teoria. Desenvolvemos no Capítulo 4 um algoritmo para descrever os grafos de operadores de Hecke usando a álgebra de Hall elíptica. A saber, as arestas nos grafos de operatodes de Hecke podem ser determinadas de certos produtos na álgebra de Hall, sabendo disto, nosso algoritmo calcula esses produtos que por sua vez nos retornam as arestas dos grafos. Boa parte destes dois últimos capítulos pode ser encontrada no seguinte preprint • [2] R. Alvarenga. Hall algebra and graphs of Hecke operators for elliptic curves. In preparation, 2018. Palavras-chave: Operadores de Hecke, formas automórficas, curvas elípticas, ge- ometria aritmética, fibrados vetoriais, funções simétricas, álgebras de Hall. Dedicated to my family. Acknowledgment First of all, I would like to express my highest gratitude to my advisor Oliver Lorscheid, without him this thesis would not be possible. His advising, about life or math, was ever very helpful for me. Moreover, I would to tank Oliver for suggesting the main problem of this thesis and for many stimulating conversations. I also wish to express my thank to my parents Elza and Roberto, which ever given me all the necessary support and their love. Even if I continuous to say ”thank you” for ever, it will not be enough. I thank my brother Weiner and my sister Isabella for all, hoping that this thesis serves of inspiration. I would like to thank my beloved (future) wife Natália, for her comprehension, support and love. I thank all my family. I thank all my IMPA’s friends, it was a pleasure to share these years with you. I also would like to thank my friend Victor Hugo. For many reasons I thank Parham Salehyan and Alex Massarenti. I would like to express my thank to professor Olivier Schiffmann for hosting me for a term in Paris and Dragos Fratila for fruitful discussions. I wish express my gratitude to all IMPA’s staff, in particular Antônio Carlos, Kênia, Isabel, Andréia and Josenildo. Also my gratitude to the IMPA’s researches, in par- ticular Carolina Araujo, Eduardo Esteves, Karl-Otto Störn, Reimundo Heluani, Misha Belolipetsky and Henrique Bursztyn. To the thesis defence committee Eduardo Esteves, Reimundo Heluani, Karl-Otto Störn, Cecília Salgado and Olivier Schiffmann, thank you. For the financial support I would like to thank Faperj and Capes. ”I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.” Isaac Newton Contents Introduction 9 1 Background 16 1.1 Notation . 16 1.2 Hecke operators . 19 1.3 Automorphic forms . 20 1.4 Coherent sheaves . 22 2 Graphs of Hecke operators for GLn 26 2.1 Graphs of Hecke operators . 26 2.2 Graphs of unramified Hecke operators . 29 2.3 Geometry of graphs of unramified Hecke operators . 34 2.4 The δ−invariant . 37 2.5 Graphs for the projective line . 44 3 Hall algebras 58 3.1 The classical Hall algebras . 59 3.2 The Hall algebras of a finitary category . 64 3.3 The Hall algebra of a curve . 69 4 Graphs of Hecke operators for elliptic curves 72 4.1 Introduction . 72 4.2 Hall algebras and graphs of Hecke operators . 77 4.3 Graphs of Hecke operators for elliptic curves . 79 4.4 The algorithm . 87 4.5 Calculating structure constants . 93 4.6 The case of rank 2 . 96 8 Introduction Hecke operators have been proved to be a powerful tool in number theory. Before Hecke had published his theory in 1937 about these operators in [20] and [21], they were used by Mordel in 1917 to show that the Ramanujan tau function is multi- plicative, cf. [34]. Hecke operators play an important role in the theory of modular forms (in the classical context) and more generally in the study of automorphic forms and automorphic representations. Autormorphic forms in their turn, are a central ingredient of the Langlands program and play an important role in modern number theory. Zagier develops in [52] the theory of toroidal automorphic forms over Q and ties it up with the Riemann hypothesis. In the end of this article, he asks questions about an analogous theory for global function fields and the potential connections to the Hasse-Weil theorem. Lorscheid introduces in [30] graphs of Hecke operators as a computational device to determine the spaces of toroidal automorphic forms for function fields. In this thesis, we continue the study of graphs of Hecke operators and generalize some results of Lorscheid in [32] from PGL2 to GLn. In particular, we describe how to determine these graphs for curves of small genus, namely: for the projective line and elliptic curves. We summarize our main results in the following. General theory The first two chapters are dedicated to generalizing the original definition of those graphs, given by Lorscheid in [32], from PGL2 to GLn and give its first properties. Moreover, we prove some structure results about those graphs and end Chapter 2 with a solution of how to obtain the graphs of Hecke operators for the rational function field. Let us define the central object of our investigation, the graph of Hecke operator. Let X be a smooth projective and geometrically irreducible curve whose function field is F . We say that two exact sequences of coherent sheaves 0 0 0 −! F1 −! F −! F2 −! 0 and 0 −! F1 −! F −! F2 −! 0 9 0 0 are isomorphic with fixed F if there are isomorphism F1 !F1 and F2 !F2 such that 0 / F1 / F / F2 / 0 =∼ =∼ 0 0 0 / F1 / F / F2 / 0 commutes.
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