Tópicos Avançados em Teoria de Números

This is a reading course and the students will be assigned to read part of papers or books. We are going to cover: Gross-Zagier theorem which is the Birch-Swinnerton-Dyer conjecture for elliptic curves of rank zero or one. Conductor of elliptic curves. Chebotarev density theorem. Gauss-Manin connection over finite fields, its relation with Hodge filtration. 

Referências:
J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), 33-52. Lecture Notes in Math., Vol. 476, Springer, Berlin, 1975.
J.H. Silverman, Advanced topics in the arithmetic of elliptic curves, GTM 151, Springer-Verlag, New York, 1994
Gross, Benedict H. Lectures on the conjecture of {Birch} and Swinnerton-Dyer, 2011,
Gross, Benedict H. and Zagier, Don B., Heegner points and derivatives of L-series, Invent. Math., 1986.
Chebotarev, N. G., Die Bestimmung der Dichtigkeit einer Menge von Primzahlen, welche zu einer gegebenen Substitutionsklasse geh”oren., Math. Ann. 1925,
N. M. Katz, Nilpotent connections and the monodromy theorem: Applications of a result of turrittin, Publications mathématiques de l’IHES 39 (1970), 175–232
Katz, Nicholas M., Algebraic solutions of differential equations ({$p$}-curvature and the {H}odge filtration)}, Invent. Math., Inventiones Mathematicae, 1972.