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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. Dark Energy Survey Year 1 results: constraints on intrinsic alignments and their colour dependence from galaxy clustering and weak lensing

   https://doi.org/10.1093/mnras/stz2197  

   Samuroff, S; Blazek, J; Troxel, M A; MacCrann, N; Krause, E; Leonard, C D; Prat, J; Gruen, D; Dodelson, S; Eifler, T F; et al (November 2019, Monthly Notices of the Royal Astronomical Society)

   Abstract We perform a joint analysis of intrinsic alignments and cosmology using tomographic weak lensing, galaxy clustering, and galaxy–galaxy lensing measurements from Year 1 (Y1) of the Dark Energy Survey. We define early- and late-type subsamples, which are found to pass a series of systematics tests, including for spurious photometric redshift error and point spread function correlations. We analyse these split data alongside the fiducial mixed Y1 sample using a range of intrinsic alignment models. In a fiducial non-linear alignment model analysis, assuming a flat Λ cold dark matter cosmology, we find a significant difference in intrinsic alignment amplitude, with early-type galaxies favouring $$A_\mathrm{IA} = 2.38^{+0.32}_{-0.31}$$ and late-type galaxies consistent with no intrinsic alignments at $$0.05^{+0.10}_{-0.09}$$. The analysis is repeated using a number of extended model spaces, including a physically motivated model that includes both tidal torquing and tidal alignment mechanisms. In multiprobe likelihood chains in which cosmology, intrinsic alignments in both galaxy samples and all other relevant systematics are varied simultaneously, we find the tidal alignment and tidal torquing parts of the intrinsic alignment signal have amplitudes $$A_1 = 2.66 ^{+0.67}_{-0.66}$$, $$A_2=-2.94^{+1.94}_{-1.83}$$, respectively, for early-type galaxies and $$A_1 = 0.62 ^{+0.41}_{-0.41}$$, $$A_2 = -2.26^{+1.30}_{-1.16}$$ for late-type galaxies. In the full (mixed) Y1 sample the best constraints are $$A_1 = 0.70 ^{+0.41}_{-0.38}$$, $$A_2 = -1.36 ^{+1.08}_{-1.41}$$. For all galaxy splits and IA models considered, we report cosmological parameter constraints consistent with the results of the main DES Y1 cosmic shear and multiprobe cosmology papers. 

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