Let $G$ be a reductive group, and let ${Bun}_G$ denote the moduli stack of $G$-bundles on a smooth projective curve. We begin the study of the singularities of a canonical compactification of ${Bun}_G$ due to Drinfeld (unpublished), which we refer to as the Drinfeld–Lafforgue–Vinberg compactification ${\overline{Bun}}_G$. For $G={GL}_2$ and $G={GL}_n$, certain smooth open substacks of this compactification have already appeared in the work of Drinfeld and Lafforgue on the Langlands correspondence for function fields. The stack ${\overline{Bun}}_G$ is, however, already singular for $G={SL}_2$; questions about its singularities arise naturally in the geometric Langlands program, and form the topic of the present article. Drinfeld’s definition of ${\overline{Bun}}_G$ for a general reductive group $G$ relies on the Vinberg semigroup of $G$ (we will study this case in a forthcoming article). In the present article we focus on the case $G={SL}_2$, where the compactification can alternatively be viewed as a canonical one-parameter degeneration of the moduli space of ${SL}_2$-bundles. We study the singularities of this one-parameter degeneration via the weight-monodromy theory of its nearby cycles. We give an explicit description of the nearby cycles sheaf together with its monodromy action in terms of certain novel perverse sheaves which we call Picard–Lefschetz oscillators and which govern the singularities of ${\overline{Bun}}_G$. We then use this description to determine the intersection cohomology sheaf of ${\overline{Bun}}_G$ and other invariants of its singularities. Our proofs rely on the construction of certain local models which themselves form one-parameter families of spaces which are factorizable in the sense of Beilinson and Drinfeld. We also briefly discuss the relationship of our results for $G={SL}_2$ with the miraculous duality of Drinfeld and Gaitsgory in the geometric Langlands program, as well as two applications of our results to the classical theory on the level of functions: to Drinfeld’s and Wang’s strange invariant bilinear form on the space of automorphic forms, and to the categorification of the Bernstein asymptotics map studied by Bezrukavnikov and Kazhdan as well as by Sakellaridis and Venkatesh.

We prove a conjecture of Shokurov which characterizes toric varieties using log pairs.

Under the Ornstein–Uhlenbeck semigroup $\left\{U_t\right\}$, any nonnegative measurable $f:{\mathbb{R}}^n\to {\mathbb{R}}_{+}$ exhibits a uniform tail bound better than that implied by Markov’s inequality and conservation of mass. For every $\alpha \ge e^3$, and $t>0$,

$${\gamma }_n\left(\right\{x\in {\mathbb{R}}^n:U_{t}f\left(x\right)>\alpha \int fd{\gamma }_n\left\}\right)\le C\left(t\right)\frac{1}{\alpha }\sqrt{\frac{loglog\alpha }{log\alpha }},$$ where ${\gamma }_n$ is the $n$-dimensional Gaussian measure and $C\left(t\right)$ is a constant depending only on $t$. This confirms positively the Gaussian limiting case of Talagrand’s convolution conjecture (1989). This is shown to follow from a more general phenomenon. Suppose that $f:{\mathbb{R}}^n\to {\mathbb{R}}_{+}$ is semi-log-convex in the sense that for some $\beta >0$, for all $x\in {\mathbb{R}}^n$, the eigenvalues of ${\nabla }^{2}logf\left(x\right)$ are at least $-\beta$. Then $f$ satisfies a tail bound asymptotically better than that implied by Markov’s inequality.

For every odd integer $n\ge 3$, we prove that there exist infinitely many number fields of degree $n$ and associated Galois group $S_n$ whose class number is odd. To do so, we study the class groups of families of number fields of degree $n$ whose rings of integers arise as the coordinate rings of the subschemes of ${\mathbb{P}}^1$ cut out by integral binary $n$-ic forms. By obtaining upper bounds on the mean number of $2$-torsion elements in the class groups of fields in these families, we prove that a positive proportion (tending to $1$ as $n$ tends to $\infty$) of such fields have trivial $2$-torsion subgroups in their class groups and narrow class groups. Conditional on a tail estimate, we also prove the corresponding lower bounds and obtain the exact values of these averages, which are consistent with the heuristics of Cohen and Lenstra, Cohen and Martinet, Malle, and Dummit and Voight. Additionally, for any order ${\mathcal{O}}_f$ of degree $n$ arising from an integral binary $n$-ic form $f$, we compare the sizes of ${Cl}_2\left({\mathcal{O}}_f\right)$, the $2$-torsion subgroup of ideal classes in ${\mathcal{O}}_f$, and of ${\mathcal{I}}_2\left({\mathcal{O}}_f\right)$, the $2$-torsion subgroup of ideals in ${\mathcal{O}}_f$. For the family of orders arising from integral binary $n$-ic forms and contained in fields with fixed signature $(r_1,r_2)$, we prove that the mean value of the difference $\left|{Cl}_2\right({\mathcal{O}}_f\left)\right|-2^{1-r_1-r_2}\left|{\mathcal{I}}_2\right({\mathcal{O}}_f\left)\right|$ is equal to $1$, generalizing a result of Bhargava and the third-named author for cubic fields. Conditional on certain tail estimates, we also prove that the mean value of $\left|{Cl}_2\right({\mathcal{O}}_f\left)\right|-2^{1-r_1-r_2}\left|{\mathcal{I}}_2\right({\mathcal{O}}_f\left)\right|$ remains $1$ for certain families obtained by imposing local splitting and maximality conditions.