Summary: In 1944, Freeman Dyson [Eureka 8, 10–15 (1944)] conjectured the existence of a “crank” function for partitions that would provide a combinatorial proof of Ramanujan’s congruence modulo 11. Forty years later, G. E. Andrews and F. G. Garvan [Bull. Am. Math. Soc., New Ser. 18, 167–171 (1988; Zbl 0646.10008)] successfully found such a function and proved the celebrated result that the crank simultaneously “explains” the three Ramanujan congruences modulo 5, 7, and 11. This note announces the proof of a conjecture of Ono, which essentially asserts that the elusive crank satisfies exactly the same types of general congruences as the partition function.
This announcement begins with a review of modular forms of half-integral weight in Section 2. Following that, Section 3 explains how to write the generating function of the crank in terms of Klein forms. A condensed proof of the main Theorem is found in Section 4.
See also the commentary paper by George E. Andrews and Ken Ono in the same issue 102, No. 43, 15277 (2005; Zbl 1155.11349).