Algebraic geometric codes and spaces of modular forms

12/04/2012 - 12:15pm
12/04/2012 - 1:10pm
Hiren Maharaj (Clemson University)

Elkies' modularity conjecture states that every recursively defined sequence of curves over a finite field GF(q^2) whose limit of the ratio of the number of rational points to the genus is $q-1$ must come from a sequence of modular curves by reduction. Elkies' came to this conjecture as a result of work of coding theorists who explicitly constructed such sequences for the purpose of explicitly constructing algebraic geometric codes. It is natural to ask what implications does modularity have on algebraic geometric codes. In this talk I will give an overview of the history of this conjecture and I will also talk about recent results on modularity and algebraic geometric codes: for m = 2, 3, 5, the space of modular forms M_{2k}( Gamma_0(m^n) ), n>=1 is naturally isomorphic to one point Riemann-Roch spaces which arise from the modular curve X_0(m^n). Riemann-Roch spaces are used to construct algebraic geometric codes. Thus, in principle, algorithms to construct explicit bases for the above mentioned spaces of modular forms can also be used to construct explicit bases for the corresponding algebraic geometric codes.

Millikan 208 (Pomona College)