14 problems

Let $p$ be a prime and let $n\geq 1$. Consider non-CM newform Galois orbits of level $p^n$, weight $k$, and trivial Nebentypus. For an inertial local type $\tau$ of conductor $p^n$…

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

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 $x$,…

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

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

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 $(G,X)$ 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 $K$.…

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…

Let $A_{g,1}$ be the coarse moduli space of principally polarized Abelian varieties. A $g$-dimensional Abelian variety $B$ is CM if…