ARITHMETIC OF THE PARTITION FUNCTION KEN ONO Department of Mathematics University of Wisconsin at Madison Madison, Wisconsin 53706 USA 1. Introduction Here we describe some recent advances that have been made regarding the arithmetic of the unrestricted partition function p(n). A partition of a non- negative integer n is any nonincreasing sequence of positive integers whose sum is n. As usual, we let p(n) denote the number of partitions of n. For example, it is easy to see that p(4) = 5 since the partitions of 4 are: 4, 3+1, 2+2, 2+1+1, 1+1+1+1. Partitions have played an important role in many aspects of combina- torics, Lie theory, physics, and representation theory. Here we describe some of the recent discoveries regarding the arithmetic of the partition function, including a relationship between partitions and Tate-Shafarevich groups of modular motives in arithmetic algebraic geometry. Euler [4] showed that the generating function for p(n) is given by the convenient infinite product ∞ X n=0 p(n)q n = ∞ Y n=1 1 1 - q n =1+ q +2q 2 +3q 3 +5q 4 + ··· , (1) and his Pentagonal Number Theorem asserts that ∞ Y n=1 (1 - q n )= ∞ X n=-∞ (-1) n q (3n 2 +n)/2 . (2) It is easy to see that (1) and (2) together imply, for every positive n, that p(n)= ∞ X k=1  (-1) k+1 p(n - 3(k 2 + k)/2) + (-1) k+1 p(n - (3k 2 - k)/2)  . (3)