RUI: Arithmetic of Modular Forms and of Diagonal Hypersurfaces

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Gouvea This award funds the investigations of Prof. Fernando Gouvea under the Research in Undergraduate Institutions Program. Prof. Gouvea will study the arithmetic of modular forms and the arithmetic of diagonal hypersurfaces over finite fields. The central topic in the first part of the work is the study of two operators introduced by Atkin. In the second part the investigator will study the simplest cases of arithmetic on abelian varieties. This research falls under the general heading of Number Theory. Number Theory is the study of the properties of the whole numbers and is the oldest branch of mathematics. From the beginning problems in number theory have furnished the driving force to creation of new mathematics in almost all parts of the discipline. One of the philosophies of modern number theory is that many facts about numbers can be studied using calculus and geometry. This approach is very evident in the work of this project since modular forms are studied using calculus and abelian varieties are studied using geometry. This kind of number theory is very technical and deep, but it has had astonishing applications in areas like theoretical computer science and coding theory. ***