23 problems

Let $N$ be a fixed level, and let $\mathbf{NCM}(N,k)$ denote the number of non-CM Galois orbits of newforms of weight $k$ and level $N$. Tsaknias's generalized Maeda conjecture. Th…

Large Galois orbits conjecture. There exist positive constants $C_1$ and $C_2$ such that, for every $s\in C\cap\Sigma$,

Let $K$ be a global function field, let $M_K$ be the set of places of $K$, and let $\widehat h$ be the canonical height associated to the Drinfeld module $\phi$. For…

Let $\mathbf{NCM}(N,k)$ be the number of non-CM Galois orbits and let $\mathbf{LO}(N)$ be the lower bound obtained by counting compatible pairs of the local inertial-type and minim…

Large Galois Orbits conjecture. There exist positive constants $C_{\mathrm{mult}}$ and $C_{\mathrm{exp}}$ such that, for all $s\in C\cap\Sigma$,

Large Galois Orbits conjecture. There exist positive constants $C_{\mathrm{mult}}$ and $C_{\mathrm{exp}}$, depending only on $g$, $L$ and $C$, such that for every point…

Let $p$ be a prime, and consider the weight-$2$ newforms for $\Gamma_0(p)$ in each of the two Atkin–Lehner eigenspaces. Maeda-type conjecture. In each eigenspace, the newforms shou…

For each $n>1$, let $\mathcal{T}_n$ be the relevant three-dimensional $\mathcal{N}=4$ theory, and consider its A- and B-twists. A–B twist Galois-orbit conjecture. The B-twisted…

Let $\mathrm{SM}(3,5)$ denote the fermionic minimal model and let $\mathrm{Majorana}$ denote the Majorana theory, with the conjugate taken at the level of the bulk TQFT. Consider t…

Let $(A_1,A_2)$ denote the Argyres-Douglas theory, and label its four inequivalent higher sheets by $gamma=1,2,3,4$. The Lee-Yang TQFT, Fibonacci TQFT, and their conjugates form th…

Let $S$ be a Shimura variety, let $F/E$ be a finite type extension of the reflex field of $S$, let $s=[x,1]\in S(F)$, and let $\Sigma^g(s)$ be a geometric hybrid orbit. Let $w(x')$…

Let $S$ be a Shimura variety, let $F$ be a field of finite type over the reflex field of $S$, and let $s=[x,1]\in S(F)$. Let $\Sigma^g(x)$ denote the geometric hybrid orbit of $x$,…

Let $g\geq 1$, let $d\in\mathbb{Z}_{\geq 0}$, and let $V$ be a Hodge generic irreducible subvariety of $\mathcal{A}_g$ of dimension less than $\dim(\mathcal{A}_g)-d$, defined over…

Let $S$ be a Shimura variety, let $\boldsymbol{\Sigma}(d)$ denote the set of special subvarieties of $S$ of dimension at most $d$, and let $V\subset S$ be an irreducible algebraic…

For $N\in\mathrm{Sq}_{\mathrm{odd}}$, choose a maximal order $\mathcal O_N$ in the definite quaternion algebra $B_N$. For an eigenform $\varphi\in S(\mathcal O_N)$, its degree is t…

In the Shimura-datum setup, let $S_{\mathbf H,\gamma}$ be the special subvariety associated with $\gamma\in\Omega\subset\mathbf G(\mathbb Q)_+$ and let $N(\gamma)$ be the complexit…

Let $(\mathbf G,X)$ be a Shimura datum and let $S$ be an associated Shimura variety component. Let $(\mathbf H,X_{\mathbf H})\subset(\mathbf G,X)$ be a sub-Shimura datum, and for e…

Let $q$ be a positive integer, let $\xi$ be a primitive $\phi(q)$-th root of unity, and let $G=\operatorname{Gal}(\mathbb{Q}(\xi)/\mathbb{Q})$ act on the primitive Dirichlet charac…

Let $(G,X)$ be a Shimura datum, let $K$ be neat and a product of compact open subgroups $K_p$ of $G(\mathbb Q_p)$, and fix a connected component $S=\Gamma\backslash X^+$, where…

Let $K$ be a field finitely generated over $\mathbb{Q}$. Suppose $X$ is an abelian variety defined over $K$ or $X=Y(1)^n$, and let $V\subseteq X$ be a subvariety defined over $K$.…

Consider the space of newforms of weight $44$ and level $17$, with any nebentypus of order $8$ at $17$. Unique-orbit conjecture. This space contains a unique orbit of Galois-conjug…

Let $K$ be a CM field of degree $2g$, and let $S$ be a primitive CM type for $K$. Let $L$ be the normal closure of $K$, and let $r_K:L^{\times}\to K^{\times}$ be the reciprocity mo…

Let $A_{g,1}$ be the moduli space of principally polarized complex Abelian varieties of dimension $g$. Let $x$ be a CM point in $A_{g,1}$ corresponding to a principally polarized A…