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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. Bohr sets in sumsets II: countable abelian groups

   https://doi.org/10.1017/fms.2023.49  

   Griesmer, John T.; Le, Anh N.; Lê, Thái Hoàng (July 2023, Forum of Mathematics, Sigma)

   Abstract We prove three results concerning the existence of Bohr sets in threefold sumsets. More precisely, lettingGbe a countable discrete abelian group and$$\phi _1, \phi _2, \phi _3: G \to G$$be commuting endomorphisms whose images have finite indices, we show that(1)If$$A \subset G$$has positive upper Banach density and$$\phi _1 + \phi _2 + \phi _3 = 0$$, then$$\phi _1(A) + \phi _2(A) + \phi _3(A)$$contains a Bohr set. This generalizes a theorem of Bergelson and Ruzsa in$$\mathbb {Z}$$and a recent result of the first author.(2)For any partition$$G = \bigcup _{i=1}^r A_i$$, there exists an$$i \in \{1, \ldots , r\}$$such that$$\phi _1(A_i) + \phi _2(A_i) - \phi _2(A_i)$$contains a Bohr set. This generalizes a result of the second and third authors from$$\mathbb {Z}$$to countable abelian groups.(3)If$$B, C \subset G$$have positive upper Banach density and$$G = \bigcup _{i=1}^r A_i$$is a partition,$$B + C + A_i$$contains a Bohr set for some$$i \in \{1, \ldots , r\}$$. This is a strengthening of a theorem of Bergelson, Furstenberg and Weiss. All results are quantitative in the sense that the radius and rank of the Bohr set obtained depends only on the indices$$[G:\phi _j(G)]$$, the upper Banach density ofA(in (1)), or the number of sets in the given partition (in (2) and (3)). 

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3. PETERZIL–STEINHORN SUBGROUPS AND -STABILIZERS IN ACF

   https://doi.org/10.1017/S147474802100030X  

   Kamensky, Moshe; Starchenko, Sergei; Ye, Jinhe (May 2023, Journal of the Institute of Mathematics of Jussieu)

   Abstract We considerG, a linear algebraic group defined over$$\Bbbk $$, an algebraically closed field (ACF). By considering$$\Bbbk $$as an embedded residue field of an algebraically closed valued fieldK, we can associate to it a compactG-space$$S^\mu _G(\Bbbk )$$consisting of$$\mu $$-types onG. We show that for each$$p_\mu \in S^\mu _G(\Bbbk )$$,$$\mathrm {Stab}^\mu (p)=\mathrm {Stab}\left (p_\mu \right )$$is a solvable infinite algebraic group when$$p_\mu $$is centered at infinity and residually algebraic. Moreover, we give a description of the dimension of$$\mathrm {Stab}\left (p_\mu \right )$$in terms of the dimension ofp. 

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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)

   Abstract 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. 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)

   Abstract 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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