PhD defense Desiree Gijon Gomez
Title: Diophantine estimates on modular structures; Modular polynomials and transcendence questions on genus 2 curve.
Abstract: The first part of the thesis contains preliminaries and consists of 4 Chapters, recapitulating the theory of invariants of genus two curves in Chapter 1, Hilbert modular surfaces in Chapter 2, Humbert singular relations and Kani’s refined Humbert invariant in Chapter 3, and finally Shimura curves in Chapter 4.
The second part concerns new results. Chapter 5 is an article with Florian Breuer and Fabien Pazuki: Explicit bounds on the coefficients of modular polynomials and the size of $X_0(N)$ published at the Proceedings of the London Mathematical Society. Chapter 6 concerns the single-authored article: On the CM exception to a generalization of the Stéphanois theorem. Chapter 7 is an adaptation of the proof of the Stéphanois theorem, using only modular polynomials $\phi_N$ where $N = p^a$ is a prime power. Finally, Chapter 8 adapts three of the four steps of the proof from Chapter 7 to the case of dimension 2 and addresses how to adapt the missing step.
Join Zoom Meeting: https://ucph-ku.zoom.us/j/4030387200
Supervisor:
Professor Fabien Pazuki
University of Copenhagen
Assessment Committee:
Professor Elisenda Feliu (chair)
University of Copenhagen
Professor Javier Fresan
Sorbonne University Paris
Directeur de Recherche Damien Robert
INRIA University of Bordeaux