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1. Additive Energies on Discrete Cubes

   de Dios Pont, Jaume; Greenfeld, Rachel; Ivanisvili, Paata; Madrid, Jose (September 2023, Discrete analysis)

   Gowers, Tim (Ed.)

   We prove that for $$d\geq 0$$ and $$k\geq 2$$, for any subset $$A$$ of a discrete cube $$\{0,1\}^d$$, the $k-$higher energy of $$A$$ (i.e., the number of $2k-$tuples $$(a_1,a_2,\dots,a_{2k})$$ in $$A^{2k}$$ with $$a_1-a_2=a_3-a_4=\dots=a_{2k-1}-a_{2k}$$) is at most $$|A|^{\log_{2}(2^k+2)}$$, and $$\log_{2}(2^k+2)$$ is the best possible exponent. We also show that if $$d\geq 0$$ and $$2\leq k\leq 10$$, for any subset $$A$$ of a discrete cube $$\{0,1\}^d$$, the $k-$additive energy of $$A$$ (i.e., the number of $2k-$tuples $$(a_1,a_2,\dots,a_{2k})$$ in $$A^{2k}$$ with $$a_1+a_2+\dots+a_k=a_{k+1}+a_{k+2}+\dots+a_{2k}$$) is at most $$|A|^{\log_2{ \binom{2k}{k}}}$$, and $$\log_2{ \binom{2k}{k}}$$ is the best possible exponent. We discuss the analogous problems for the sets $$\{0,1,\dots,n\}^d$$ for $$n\geq2$$. 

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2. Multiplicative series, modular forms, and Mandelbrot polynomials

   https://doi.org/10.1090/mcom/3564  

   Larsen, Michael (January 2021, Mathematics of Computation)

   null (Ed.)

   We say a power series ∑ n = 0 ∞ a n q n \sum _{n=0}^\infty a_n q^n is multiplicative if the sequence 1 , a 2 / a 1 , … , a n / a 1 , … 1,a_2/a_1,\ldots ,a_n/a_1,\ldots is so. In this paper, we consider multiplicative power series f f such that f 2 f^2 is also multiplicative. We find a number of examples for which f f is a rational function or a theta series and prove that the complete set of solutions is the locus of a (probably reducible) affine variety over C \mathbb {C} . The precise determination of this variety turns out to be a finite computational problem, but it seems to be beyond the reach of current computer algebra systems. The proof of the theorem depends on a bound on the logarithmic capacity of the Mandelbrot set. 

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3. Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields

   https://doi.org/10.1017/fmp.2022.14  

   Shankar, Ananth N.; Shankar, Arul; Tang, Yunqing; Tayou, Salim (January 2022, Forum of Mathematics, Pi)

   Abstract Given a K3 surface X over a number field K with potentially good reduction everywhere, we prove that the set of primes of K where the geometric Picard rank jumps is infinite. As a corollary, we prove that either $$X_{\overline {K}}$$ has infinitely many rational curves or X has infinitely many unirational specialisations. Our result on Picard ranks is a special case of more general results on exceptional classes for K3 type motives associated to GSpin Shimura varieties. These general results have several other applications. For instance, we prove that an abelian surface over a number field K with potentially good reduction everywhere is isogenous to a product of elliptic curves modulo infinitely many primes of K . 

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4. The Voronoi summation formula for $${\textrm{GL}}_n$$ and the Godement–Jacquet kernels

   https://doi.org/10.1007/s00209-025-03695-w  

   Jiang, Dihua; Li, Zhaolin (April 2025, Mathematische Zeitschrift)

   Let $$\BA$$ be the ring of adeles of a number field $$k$$ and $$\pi$$ be an irreducible cuspidal automorphic representation of $$\GL_n(\BA)$$. In Jiang and Luo (Pac J Math 318:339–374. https://doi.org/10.2140/pjm.2022.318.339, 2022, Pac J Math 326: 301–372. https://doi.org/10.2140/pjm.2023.326.301, 2023), the authors introduced $$\pi$$-Schwartz space $$\CS_\pi(\BA^\times)$$ and $$\pi$$-Fourier transform $$\CF_{\pi,\psi}$$ with a non-trivial additive character $$\psi$$ of $$k\bs\BA$$, proved the associated Poisson summation formula over $$\BA^\times$$, based on the Godement-Jacquet theory for the standard $$L$$-functions $$L(s,\pi)$$, and provided interesting applications. In this paper, in addition to the further development of the local theory, we found two global applications. First, we find a Poisson summation formula proof of the Voronoi summation formula for $$\GL_n$$ over a number field, which was first proved by A. Ichino and N. Templier (Am J Math 135:65–101. https://doi.org/10.1353/ajm.2013.0005, 2013, Theorem 1). Then we introduce the notion of the Godement-Jacquet kernels $$H_{\pi,s}$$ and their dual kernels $$K_{\pi,s}$$ for any irreducible cuspidal automorphic representation $$\pi$$ of $$\GL_n(\BA)$$ and show in Theorems \ref{thm:H=FK} and \ref{thm:CTh-pi} that $$H_{\pi,s}$$ and $$K_{\pi,1-s}$$ are related by the nonlinear $$\pi_\infty$$-Fourier transform if and only if $$s\in\BC$$ is a zero of $$L_f(s,\pi_f)=0$$, the finite part of the standard automorphic $$L$$-function $$L(s,\pi)$$, which are the $$(\GL_n,\pi)$$-versions of Clozel (J Number Theory 261: 252–298 https://doi.org/10.1016/j.jnt.2024.02.018, 2024, Theorem 1.1), where the Tate kernel with $n=1$ and $$\pi$$ the trivial character are considered. 

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5. Heights on stacks and a generalized Batyrev–Manin–Malle conjecture

   https://doi.org/10.1017/fms.2023.5  

   Ellenberg, Jordan S.; Satriano, Matthew; Zureick-Brown, David (January 2023, Forum of Mathematics, Sigma)

   Abstract We define a notion of height for rational points with respect to a vector bundle on a proper algebraic stack with finite diagonal over a global field, which generalizes the usual notion for rational points on projective varieties. We explain how to compute this height for various stacks of interest (for instance: classifying stacks of finite groups, symmetric products of varieties, moduli stacks of abelian varieties, weighted projective spaces). In many cases, our uniform definition reproduces ways already in use for measuring the complexity of rational points, while in others it is something new. Finally, we formulate a conjecture about the number of rational points of bounded height (in our sense) on a stack $$\mathcal {X}$$ , which specializes to the Batyrev–Manin conjecture when $$\mathcal {X}$$ is a scheme and to Malle’s conjecture when $$\mathcal {X}$$ is the classifying stack of a finite group. 

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