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1. 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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2. Cyclic base change of cuspidal automorphic representations over function fields

   https://doi.org/10.1112/S0010437X24007243  

   Böckle, Gebhard; Feng, Tony; Harris, Michael; Khare, Chandrashekhar B; Thorne, Jack A (September 2024, Compositio Mathematica)

   Let$$G$$be a split semisimple group over a global function field$$K$$. Given a cuspidal automorphic representation$$\Pi$$of$$G$$satisfying a technical hypothesis, we prove that for almost all primes$$\ell$$, there is a cyclic base change lifting of$$\Pi$$along any$$\mathbb {Z}/\ell \mathbb {Z}$$-extension of$$K$$. Our proof does not rely on any trace formulas; instead it is based on using modularity lifting theorems, together with a Smith theory argument, to obtain base change for residual representations. As an application, we also prove that for any split semisimple group$$G$$over a local function field$$F$$, and almost all primes$$\ell$$, any irreducible admissible representation of$$G(F)$$admits a base change along any$$\mathbb {Z}/\ell \mathbb {Z}$$-extension of$$F$$. Finally, we characterize local base change more explicitly for a class of toral representations considered in work of Chan and Oi. 

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3. Calibrated representations of the double Dyck path algebra

   https://doi.org/10.1007/s00208-024-02937-2  

   González, Nicolle; Gorsky, Eugene; Simental, José (August 2024, Mathematische Annalen)

   Abstract The double Dyck path algebra$$\mathbb {A}_{q,t}$$ $A_{q,t}$and its polynomial representation first arose as a key figure in the proof of the celebrated Shuffle Theorem of Carlsson and Mellit. A geometric formulation for an equivalent algebra$$\mathbb {B}_{q,t}$$ $B_{q,t}$was then given by the second author and Carlsson and Mellit using the K-theory of parabolic flag Hilbert schemes. In this article, we initiate the systematic study of the representation theory of the double Dyck path algebra$$\mathbb {B}_{q,t}$$ $B_{q,t}$. We define a natural extension of this algebra and study its calibrated representations. We show that the polynomial representation is calibrated, and place it into a large family of calibrated representations constructed from posets satisfying certain conditions. We also define tensor products and duals of these representations, thus proving (under suitable conditions) the category of calibrated representations is generically monoidal. As an application, we prove that tensor powers of the polynomial representation can be constructed from the equivariant K-theory of parabolic Gieseker moduli spaces. 

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4. Finding solutions with distinct variables to systems of linear equations over $$\mathbb {F}_p$$

   https://doi.org/10.1007/s00208-022-02391-y  

   Sauermann, Lisa (April 2022, Mathematische Annalen)

   Abstract Let us fix a primepand a homogeneous system ofmlinear equations$$a_{j,1}x_1+\dots +a_{j,k}x_k=0$$ $a_{j,1}{x}_1+\cdots +a_{j,k}{x}_k=0$for$$j=1,\dots ,m$$ $j=1,\cdots ,m$with coefficients$$a_{j,i}\in \mathbb {F}_p$$ $a_{j,i}\in F_p$. Suppose that$$k\ge 3m$$ $k\ge 3m$, that$$a_{j,1}+\dots +a_{j,k}=0$$ $a_{j,1}+\cdots +a_{j,k}=0$for$$j=1,\dots ,m$$ $j=1,\cdots ,m$and that every$$m\times m$$ $m\times m$minor of the$$m\times k$$ $m\times k$matrix$$(a_{j,i})_{j,i}$$ ${\left(a_{j,i}\right)}_{j,i}$is non-singular. Then we prove that for any (large)n, any subset$$A\subseteq \mathbb {F}_p^n$$ $A\subseteq F_p^n$of size$$|A|> C\cdot \Gamma ^n$$ $\left|A\right|>C·{\Gamma }^n$contains a solution$$(x_1,\dots ,x_k)\in A^k$$ $(x_1,\cdots ,x_k)\in A^k$to the given system of equations such that the vectors$$x_1,\dots ,x_k\in A$$ $x_1,\cdots ,x_k\in A$are all distinct. Here,Cand$$\Gamma $$ $\Gamma$are constants only depending onp,mandksuch that$$\Gamma $\Gamma <p$. The crucial point here is the condition for the vectors$$x_1,\dots ,x_k$$ $x_1,\cdots ,x_k$in the solution$$(x_1,\dots ,x_k)\in A^k$$ $(x_1,\cdots ,x_k)\in A^k$to be distinct. If we relax this condition and only demand that$$x_1,\dots ,x_k$$ $x_1,\cdots ,x_k$are not all equal, then the statement would follow easily from Tao’s slice rank polynomial method. However, handling the distinctness condition is much harder, and requires a new approach. While all previous combinatorial applications of the slice rank polynomial method have relied on the slice rank of diagonal tensors, we use a slice rank argument for a non-diagonal tensor in combination with combinatorial and probabilistic arguments. 

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5. Remarks on the existence of minimal models of log canonical generalized pairs

   https://doi.org/10.1007/s00209-024-03489-6  

   Tsakanikas, Nikolaos; Xie, Lingyao (April 2024, Mathematische Zeitschrift)

   Abstract Given an NQC log canonical generalized pair$$(X,B+M)$$ $(X,B+M)$whose underlying varietyXis not necessarily$$\mathbb {Q}$$ $Q$-factorial, we show that one may run a$$(K_X+B+M)$$ $(K_X+B+M)$-MMP with scaling of an ample divisor which terminates, provided that$$(X,B+M)$$ $(X,B+M)$has a minimal model in a weaker sense or that$$K_X+B+M$$ $K_X+B+M$is not pseudo-effective. We also prove the existence of minimal models of pseudo-effective NQC log canonical generalized pairs under various additional assumptions, for instance when the boundary contains an ample divisor. 

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