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1. Conformal blocks for Galois covers of algebraic curves

   https://doi.org/10.1112/S0010437X23007418  

   Hong, Jiuzu ; Kumar, Shrawan ( October 2023 , Compositio Mathematica)

   We study the spaces of twisted conformal blocks attached to a$\Gamma$-curve$\Sigma$with marked$\Gamma$-orbits and an action of$\Gamma$on a simple Lie algebra$\mathfrak {g}$, where$\Gamma$is a finite group. We prove that if$\Gamma$stabilizes a Borel subalgebra of$\mathfrak {g}$, 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$\Gamma$-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$\mathscr {G}$be the parahoric Bruhat–Tits group scheme on the quotient curve$\Sigma /\Gamma$obtained via the$\Gamma$-invariance of Weil restriction associated to$\Sigma$and the simply connected simple algebraic group$G$with Lie algebra$\mathfrak {g}$. We prove that the space of twisted conformal blocks can be identified with the space of generalized theta functions on the moduli stack of quasi-parabolic$\mathscr {G}$-torsors on$\Sigma /\Gamma$when the level$c$is divisible by$|\Gamma |$(establishing a conjecture due to Pappas and Rapoport).

    

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2. Non-isomorphic smooth compactifications of the moduli space of cubic surfaces

   https://doi.org/10.1017/nmj.2023.27  

   Casalaina-Martin, Sebastian ; Grushevsky, Samuel ; Hulek, Klaus ; Laza, Radu ( June 2024 , Nagoya Mathematical Journal)

   The well-studied moduli space of complex cubic surfaces has three different, but isomorphic, compact realizations: as a GIT quotient${\mathcal {M}}^{\operatorname {GIT}}$, as a Baily–Borel compactification of a ball quotient${(\mathcal {B}_4/\Gamma )^*}$, and as a compactifiedK-moduli space. From all three perspectives, there is a unique boundary point corresponding to non-stable surfaces. From the GIT point of view, to deal with this point, it is natural to consider the Kirwan blowup${\mathcal {M}}^{\operatorname {K}}\rightarrow {\mathcal {M}}^{\operatorname {GIT}}$, whereas from the ball quotient point of view, it is natural to consider the toroidal compactification${\overline {\mathcal {B}_4/\Gamma }}\rightarrow {(\mathcal {B}_4/\Gamma )^*}$. The spaces${\mathcal {M}}^{\operatorname {K}}$and${\overline {\mathcal {B}_4/\Gamma }}$have the same cohomology, and it is therefore natural to ask whether they are isomorphic. Here, we show that this is in factnotthe case. Indeed, we show the more refined statement that${\mathcal {M}}^{\operatorname {K}}$and${\overline {\mathcal {B}_4/\Gamma }}$are equivalent in the Grothendieck ring, but notK-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.

    

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3. Higher uniformity of arithmetic functions in short intervals I. All intervals

   https://doi.org/10.1017/fmp.2023.28  

   Matomäki, Kaisa ; Shao, Xuancheng ; Tao, Terence ; Teräväinen, Joni ( January 2023 , Forum of Mathematics, Pi)

   We study higher uniformity properties of the Möbius function$\mu $, the von Mangoldt function$\Lambda $, and the divisor functions$d_k$on short intervals$(X,X+H]$with$X^{\theta +\varepsilon } \leq H \leq X^{1-\varepsilon }$for a fixed constant$0 \leq \theta < 1$and any$\varepsilon>0$.

   More precisely, letting$\Lambda ^\sharp $and$d_k^\sharp $be suitable approximants of$\Lambda $and$d_k$and$\mu ^\sharp = 0$, we show for instance that, for any nilsequence$F(g(n)\Gamma )$, we have$$\begin{align*}\sum_{X < n \leq X+H} (f(n)-f^\sharp(n)) F(g(n) \Gamma) \ll H \log^{-A} X \end{align*}$$

   when$\theta = 5/8$and$f \in \{\Lambda , \mu , d_k\}$or$\theta = 1/3$and$f = d_2$.

   As a consequence, we show that the short interval Gowers norms$\|f-f^\sharp \|_{U^s(X,X+H]}$are also asymptotically small for any fixedsfor these choices of$f,\theta $. As applications, we prove an asymptotic formula for the number of solutions to linear equations in primes in short intervals and show that multiple ergodic averages along primes in short intervals converge in$L^2$.

   Our innovations include the use of multiparameter nilsequence equidistribution theorems to control type$II$sums and an elementary decomposition of the neighborhood of a hyperbola into arithmetic progressions to control type$I_2$sums.

    

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4. Theta functions, fourth moments of eigenforms and the sup-norm problem II

   https://doi.org/10.1017/fmp.2024.9  

   Khayutin, Ilya ; Nelson, Paul D ; Steiner, Raphael S ( January 2024 , Forum of Mathematics, Pi)

   Letfbe an$L^2$-normalized holomorphic newform of weightkon$\Gamma _0(N) \backslash \mathbb {H}$withNsquarefree or, more generally, on any hyperbolic surface$\Gamma \backslash \mathbb {H}$attached to an Eichler order of squarefree level in an indefinite quaternion algebra over$\mathbb {Q}$. Denote byVthe hyperbolic volume of said surface. We prove the sup-norm 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$\varphi $of eigenvalue$\lambda $on such a surface, we prove that$$\begin{align*}\|\varphi \|_{\infty} \ll_{\lambda,\varepsilon} V^{\frac{1}{4}+\varepsilon}. \end{align*}$$

   We establish analogous estimates in the setting of definite quaternion algebras.

    

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5. A TOPOLOGICAL APPROACH TO UNDEFINABILITY IN ALGEBRAIC EXTENSIONS OF Q

   https://doi.org/10.1017/bsl.2023.37  

   EISENTRÄGER, KIRSTEN ; MILLER, RUSSELL ; SPRINGER, CALEB ; WESTRICK, LINDA ( December 2023 , The Bulletin of Symbolic Logic)

   For any subset$Z \subseteq {\mathbb {Q}}$, consider the set$S_Z$of subfields$L\subseteq {\overline {\mathbb {Q}}}$which contain a co-infinite subset$C \subseteq L$that is universally definable inLsuch that$C \cap {\mathbb {Q}}=Z$. Placing a natural topology on the set${\operatorname {Sub}({\overline {\mathbb {Q}}})}$of subfields of${\overline {\mathbb {Q}}}$, we show that ifZis not thin in${\mathbb {Q}}$, then$S_Z$is meager in${\operatorname {Sub}({\overline {\mathbb {Q}}})}$. Here,thinandmeagerboth mean “small”, in terms of arithmetic geometry and topology, respectively. For example, this implies that only a meager set of fieldsLhave the property that the ring of algebraic integers$\mathcal {O}_L$is universally definable inL. The main tools are Hilbert’s Irreducibility Theorem and a new normal form theorem for existential definitions. The normal form theorem, which may be of independent interest, says roughly that every$\exists $-definable subset of an algebraic extension of${\mathbb Q}$is a finite union of single points and projections of hypersurfaces defined by absolutely irreducible polynomials.

    

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