Random Matrix Theory and Elliptic Curves

: This grant focused on the use of random matrix theory (RMT) to understand the zeros of L-functions, in the context of the statistical properties of elliptic curves and arithmetic statistics. The rank of a curve (an integer describing the number of rational points on the curve) is relevant to cryptography; this work showed relations between RMT and classification of elliptic families based on L-functions. The second function of the grant is the application of RMT to determine statistical properties of the prime numbers, of great relevance to cryptography. The team was able to prove two long-standing conjectures (Hooley 1974, Goldston/Montgomery 1984) and computed formulae related to the intervals of prime numbers.