The prime factors of the monster group's order provided the basis for a mathematical challenge that has remained open since 1975. By identifying that these specific primes linked the normalizer of the Hecke congruence subgroup to Riemann surfaces of genus zero, Andrew Pollard Ogg initiated a line of inquiry now fundamental to the understanding of monstrous moonshine.
Academic Foundation
Born in Bowling Green, Ohio, on April 9, 1934, Andrew Ogg pursued his initial higher education at Bowling Green State University during the middle of the 1950s. He later attended Harvard University to continue his research, completing his doctoral degree in 1961 under the guidance of John Tate. This training formed the basis for his subsequent work in algebra, group theory, and the arithmetic of elliptic and modular curves.
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Ogg established several key formulas that remain standard in the field. The Grothendieck–Ogg–Shafarevich formula defines how étale cohomology reacts to ramification, while the Néron–Ogg–Shafarevich criterion provides a test for the good reduction of elliptic curves at specific primes. Additionally, he produced the Ogg formula for the conductor of an elliptic curve and authored the 1969 textbook Modular forms and Dirichlet series. In 1973, he formulated the torsion conjecture regarding uniform bounds on torsion points, which was later resolved by Barry Mazur for the rationals and Loïc Merel in broader contexts.
The Jack Daniel's Problem
While attending a lecture by Jacques Tits at the Collège de France in 1975, Ogg observed a numerical coincidence between the prime factors of the monster group order and the primes p for which the Hecke congruence subgroup Γ0(p)+ possesses a Riemann surface of genus zero. He proposed a public challenge, offering a bottle of Jack Daniel's whiskey to anyone who could provide an explanation for this connection. These 15 numbers—including 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, and 71—are now formally termed the supersingular primes. Although Richard Borcherds eventually proved the monstrous moonshine conjecture in 1992, Ogg's specific inquiry regarding the reverse direction of this property remains a subject of ongoing study, and the prize has not been claimed.
Fast facts
- Born: 9 April 1934, Bowling Green, Ohio
- Citizenship: United States
- Doctoral Advisor: John Tate
- Ph.D. Year: 1961
- Primary Employer: University of California, Berkeley
- Notable Publication: Modular forms and Dirichlet series
- Field of Study: Algebra and Number Theory
Questions readers ask
What is the Jack Daniel's Problem?
It is a mathematical challenge posed by Ogg concerning why the prime factors of the monster group order correspond to specific primes that yield a Riemann surface of genus zero.
Has the Jack Daniel's bottle been claimed?
No, the bottle remains unclaimed as of 2026.
Achievements
- Held posts at University of California, Berkeley
- Fields: group theory
