Umbral Moonshine

Understand

We describe surprising relationships between automorphic forms of various kinds, imaginary quadratic number fields and a certain system of six finite groups that are parameterised naturally by the divisors of twelve.

• The Mathieu group correspondence recently discovered by Eguchi-Ooguri-Tachikawa is recovered as a special case.
• We introduce a notion of extremal Jacobi form and prove that it characterises the Jacobi forms arising by establishing a connection to critical values of Dirichlet series attached to modular forms of weight two.
• These extremal Jacobi forms are closely related to certain vector-valued mock modular forms studied recently by Dabholkar-Murthy-Zagier in connection with the physics of quantum black holes in string theory.