The integer partition function p(n) counts the number of ways to write a positive integer n as a sum of other positive integers (e.g., p(4)=5 since we have: 4, 3+1, 2+2, 2+1+1, 1+1+1+1). In 1918 Ramanujan discovered that whenever 24n-1 is divisible by a power of 5, so too is p(n). Sets of divisibility properties like this are called "congruence families," and they exist for many integer sequences which are enumerated by modular forms. Such families all tend to look similar; however, some much more difficult to prove than others. We have discovered that some of the hardest families can somehow manifest in the coefficients of many different modular forms. We call this property "congruence multiplicity," and it is associated with a remarkably rich algebraic structure. In this talk we will give an example of congruence multiplicity. This is joint work with Frank Garvan and James Sellers.
Multiplicities in Congruences Between Modular Forms
29.04.2025 13:15 - 14:45
Organiser:
H. Grobner, A. Minguez-Espallargas, A. Mellit
Location: