We study the spaces of twisted conformal blocks attached to a
Suppose
 Award ID(s):
 1952707
 NSFPAR ID:
 10531459
 Publisher / Repository:
 Compositio Mathematica
 Date Published:
 Journal Name:
 Compositio Mathematica
 Volume:
 159
 Issue:
 9
 ISSN:
 0010437X
 Page Range / eLocation ID:
 1898 to 1915
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this

curve$\Gamma$ with marked$\Sigma$ orbits and an action of$\Gamma$ on a simple Lie algebra$\Gamma$ , where$\mathfrak {g}$ is a finite group. We prove that if$\Gamma$ stabilizes a Borel subalgebra of$\Gamma$ , then the propagation theorem and factorization theorem hold. We endow a flat projective connection on the sheaf of twisted conformal blocks attached to a smooth family of pointed$\mathfrak {g}$ curves; in particular, it is locally free. We also prove that the sheaf of twisted conformal blocks on the stable compactification of Hurwitz stack is locally free. Let$\Gamma$ be the parahoric Bruhat–Tits group scheme on the quotient curve$\mathscr {G}$ obtained via the$\Sigma /\Gamma$ invariance of Weil restriction associated to$\Gamma$ and the simply connected simple algebraic group$\Sigma$ with Lie algebra$G$ . We prove that the space of twisted conformal blocks can be identified with the space of generalized theta functions on the moduli stack of quasiparabolic$\mathfrak {g}$ torsors on$\mathscr {G}$ when the level$\Sigma /\Gamma$ is divisible by$c$ (establishing a conjecture due to Pappas and Rapoport).$\Gamma $ 
Abstract The wellstudied moduli space of complex cubic surfaces has three different, but isomorphic, compact realizations: as a GIT quotient
, as a Baily–Borel compactification of a ball quotient${\mathcal {M}}^{\operatorname {GIT}}$ , and as a compactified${(\mathcal {B}_4/\Gamma )^*}$ K moduli space. From all three perspectives, there is a unique boundary point corresponding to nonstable surfaces. From the GIT point of view, to deal with this point, it is natural to consider the Kirwan blowup , whereas from the ball quotient point of view, it is natural to consider the toroidal compactification${\mathcal {M}}^{\operatorname {K}}\rightarrow {\mathcal {M}}^{\operatorname {GIT}}$ . The spaces${\overline {\mathcal {B}_4/\Gamma }}\rightarrow {(\mathcal {B}_4/\Gamma )^*}$ and${\mathcal {M}}^{\operatorname {K}}$ have the same cohomology, and it is therefore natural to ask whether they are isomorphic. Here, we show that this is in fact${\overline {\mathcal {B}_4/\Gamma }}$ not the case. Indeed, we show the more refined statement that and${\mathcal {M}}^{\operatorname {K}}$ are equivalent in the Grothendieck ring, but not${\overline {\mathcal {B}_4/\Gamma }}$ K equivalent. Along the way, we establish a number of results and techniques for dealing with singularities and canonical classes of Kirwan blowups and toroidal compactifications of ball quotients. 
Abstract We study higher uniformity properties of the Möbius function
, the von Mangoldt function$\mu $ , and the divisor functions$\Lambda $ on short intervals$d_k$ with$(X,X+H]$ for a fixed constant$X^{\theta +\varepsilon } \leq H \leq X^{1\varepsilon }$ and any$0 \leq \theta < 1$ .$\varepsilon>0$ More precisely, letting
and$\Lambda ^\sharp $ be suitable approximants of$d_k^\sharp $ and$\Lambda $ and$d_k$ , we show for instance that, for any nilsequence$\mu ^\sharp = 0$ , we have$F(g(n)\Gamma )$ $$\begin{align*}\sum_{X < n \leq X+H} (f(n)f^\sharp(n)) F(g(n) \Gamma) \ll H \log^{A} X \end{align*}$$ when
and$\theta = 5/8$ or$f \in \{\Lambda , \mu , d_k\}$ and$\theta = 1/3$ .$f = d_2$ As a consequence, we show that the short interval Gowers norms
are also asymptotically small for any fixed$\ff^\sharp \_{U^s(X,X+H]}$ s for these choices of . As applications, we prove an asymptotic formula for the number of solutions to linear equations in primes in short intervals and show that multiple ergodic averages along primes in short intervals converge in$f,\theta $ .$L^2$ Our innovations include the use of multiparameter nilsequence equidistribution theorems to control type
sums and an elementary decomposition of the neighborhood of a hyperbola into arithmetic progressions to control type$II$ sums.$I_2$ 
Abstract Let
f be an normalized holomorphic newform of weight$L^2$ k on with$\Gamma _0(N) \backslash \mathbb {H}$ N squarefree or, more generally, on any hyperbolic surface attached to an Eichler order of squarefree level in an indefinite quaternion algebra over$\Gamma \backslash \mathbb {H}$ . Denote by$\mathbb {Q}$ V the hyperbolic volume of said surface. We prove the supnorm estimate$$\begin{align*}\ \Im(\cdot)^{\frac{k}{2}} f \_{\infty} \ll_{\varepsilon} (k V)^{\frac{1}{4}+\varepsilon} \end{align*}$$ with absolute implied constant. For a cuspidal Maaß newform
of eigenvalue$\varphi $ on such a surface, we prove that$\lambda $ $$\begin{align*}\\varphi \_{\infty} \ll_{\lambda,\varepsilon} V^{\frac{1}{4}+\varepsilon}. \end{align*}$$ We establish analogous estimates in the setting of definite quaternion algebras.

Abstract We prove that the rational cohomology group
vanishes unless$H^{11}(\overline {\mathcal {M}}_{g,n})$ and$g = 1$ . We show furthermore that$n \geq 11$ is pure Hodge–Tate for all even$H^k(\overline {\mathcal {M}}_{g,n})$ and deduce that$k \leq 12$ is surprisingly well approximated by a polynomial in$\# \overline {\mathcal {M}}_{g,n}(\mathbb {F}_q)$ q . In addition, we use and its image under Gysin pushforward for tautological maps to produce many new examples of moduli spaces of stable curves with nonvanishing odd cohomology and nontautological algebraic cycle classes in Chow cohomology.$H^{11}(\overline {\mathcal {M}}_{1,11})$