**ELISHA NETANYAHU
INSTITUTE OF ADVANCED STUDIES IN MATHEMATICS**

**SPECIAL LECTURE SERIES **

by

**Professor Noam Elkies**

Harvard University, Cambridge

Lecture I: Monday, 3 May, 2004 at 15:30

Room 232 Amado, Mathematics Building

**Belyi Functions in Arithmetic Geometry**

A Belyi function on a compact Riemann surface S is a rational function f: S -> P1 whose ramification locus is contained in the preimage of {0,1,infinity}. Thus S admits such a function f if and only if S is the closure of a finite unramified cover of C-{0,1}. These are named after Belyi due to his remarkable theorem (1979) that S admits such a function if and only if S is isomorphic to an algebraic curve defined over a number field. Belyi functions occur surprisingly often in parts of modern number theory; we give examples ranging from Fermat's Last Theorem and the ABC conjecture for polynomials, to modular curves, to several open research questions.

Lecture II: Wednesday, 5 May, 2004 at 15:30

Butler Auditorium, Samuel Neaman Building

**Some Applications of Belyi Functions**

We explain some of the uses of Belyi functions hinted at in the Colloquium talk: connections with the ABC and related conjectures (Hall, Szpiro); field extensions with prescribed Galois groups; if time permits, further topics (some invariants of Belyi covers beyond the Galois group, or hypergeometric identities).

Lecture III: Wednesday, 12 May at 15:30

Room 232 Amado, Mathematics Building

**Modular Belyi Functions**

We outline the recent determination of some Shimura-Belyi functions (Belyi functions arising from maps between Shimura curves that correspond to congruence subgroups of arithmetic triangle groups) that were previously inaccessible to explicit computation. We outline some applications of such explicit formulas to efficient error-correcting codes, computation of coordinates of complex-multiplication points, or exotic isogenies between elliptic curves.