9 problems

Let $A_i$ and $B_j$ be Sakai's $W(E_8)$-invariant Jacobi forms, let $E_6$ be the classical modular form, and let $P_{16,5}$ be the exceptional polynomial … A Jacobi form expressed…

Let $W(E_8)$ be the Weyl group of $E_8$, and consider all $W(E_8)$-invariant weak Jacobi forms of integral weight and integral index. Let $M_*(\operatorname{SL}_2(\mathbb Z))$ deno…

Let $H(t)$ be the dimension of the space of $W(E_8)$-invariant holomorphic Jacobi forms of weight $4$ and positive index $t$. Let $N(t)$ be the number of distinct Weyl orbits of ve…

For each index $t$, let $d_{k,t}$ be the number of weight-$k$ generators of the free module $J_{*,E_8,t}^{\operatorname{w},W(E_8)}$ of $W(E_8)$-invariant weak Jacobi forms. Stabili…

Let $t$ be an index, and let $W(E_8)$ denote the Weyl group of the $E_8$ root system. A weak Jacobi form is allowed to have Fourier terms with negative discriminant, and its weight…

Holomorphicity conjecture. The quotient $P_{16,5}/E_4$ is holomorphic. The conjecture arose from numerical verification that $P_{16,5}$ vanishes at the zero points of $E_4$ for gen…

Let $F_{(n,g,m)}$ be the coefficient in the refined free-energy expansion of $m$ $E$-strings, and let $W(E_8)$ act on the lattice variable $\mathfrak{z}$ encoding the $E_8$ flavor…

Let $\phi$ be a $W(E_8)$-invariant Jacobi form, and let $A_1,A_2,A_3,A_4,A_5,B_2,B_3,B_4,B_6$ be Sakai's nine $W(E_8)$-invariant holomorphic Jacobi forms; the $A_i$ have weight $4$…

Let $F_{(n,g,m)}$ denote the refined free energy of $m$ $E$-strings in the massive case, with $W(E_8)$ acting on the $E_8$ flavor variable. A quasi Jacobi form is a Jacobi form all…