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Graphs of Hecke Operators and Rational Points

Grant number: 17/21259-3
Support type:Scholarships in Brazil - Post-Doctorate
Effective date (Start): May 01, 2018
Effective date (End): April 30, 2022
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Algebra
Principal researcher:Herivelto Martins Borges Filho
Grantee:Roberto Carlos Alvarenga da Silva Junior
Home Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
Associated scholarship(s):18/18805-9 - L-Functions and Rational points, BE.EP.PD


This concern about an abstract for an application for a FAPESP postdoc position at Instituto de Ciências Matemáticas e Computação - ICMC in São Carlos under supervision of professor Herivelto Martins Borges Filho from May 2018 till April 2020. \\My research plan concerns, among other things, questions that follow from my doctoral thesis and research in rational points theory and Hall algebras. We outline these topics in brevity. More details can be found in the research plan.Motivated by Zagier's work, Lorscheid develops in his doctoral thesis the theory of "graphs of Hecke operators", which plays an important role in the proofs of his main theorems. In my Ph.D thesis, we extend this theory from $\mathrm{PGL}_2$ to $\mathrm{GL}_n$, generalize some results of Lorscheid and describe these graphs for the projective line and elliptic curves. In the case of elliptic curves, we use the connection with the Hall algebra of the category of coherent sheaves.Our first project concerns an application of our results to explicit calculations with automorphic forms. Our second project is to describe these graphs for the weighted projective line and curves of large genus. A third project is dedicated to investigate the Hall algebra of an elliptic curve in more detail. Besides proceeding with follow-up projects from my doctoral research, I am interested in extending my research horizon towards problems in rational points, like height functions, ranks and the torsion points of elliptic curves, Zariski density, Manin's conjecture, and related topics. I am also interested in bounds for the number of rational points of curves over finite fields, justifying the importance of professor Herivelto Borges, and their applications to coding theory.

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Scientific publications
(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)
ALVARENGA, ROBERTO. Hall algebras and graphs of Hecke operators for elliptic curves. Israel Journal of Mathematics, SEP 2020. Web of Science Citations: 0.
ALVARENGA, ROBERTO. On graphs of Hecke operators. JOURNAL OF NUMBER THEORY, v. 199, p. 192-228, JUN 2019. Web of Science Citations: 0.

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