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Ramsey Problems for Monotone Paths in Graphs and Hypergraphs Combinatorica (IF 1.1) Pub Date : 2024-02-28 Lior Gishboliner, Zhihan Jin, Benny Sudakov
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Fundamental exact sequence for the pro-étale fundamental group Algebra Number Theory (IF 1.3) Pub Date : 2024-02-26 Marcin Lara
The pro-étale fundamental group of a scheme, introduced by Bhatt and Scholze, generalizes formerly known fundamental groups — the usual étale fundamental group π1 ét defined in SGA1 and the more general π1SGA3 . It controls local systems in the pro-étale topology and leads to an interesting class of “geometric coverings” of schemes, generalizing finite étale coverings. We prove exactness of the fundamental
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Infinitesimal dilogarithm on curves over truncated polynomial rings Algebra Number Theory (IF 1.3) Pub Date : 2024-02-26 Sinan Ünver
We construct infinitesimal invariants of thickened one dimensional cycles in three dimensional space, which are the simplest cycles that are not in the Milnor range. This generalizes Park’s work on the regulators of additive cycles. The construction also allows us to prove the infinitesimal version of the strong reciprocity conjecture for thickenings of all orders. Classical analogs of our invariants
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Wide moments of L-functions I : Twists by class group characters of imaginary quadratic fields Algebra Number Theory (IF 1.3) Pub Date : 2024-02-26 Asbjørn Christian Nordentoft
We calculate certain “wide moments” of central values of Rankin–Selberg L-functions L(π ⊗Ω, 1 2) where π is a cuspidal automorphic representation of GL 2 over ℚ and Ω is a Hecke character (of conductor 1) of an imaginary quadratic field. This moment calculation is applied to obtain “weak simultaneous” nonvanishing results, which are nonvanishing results for different Rankin–Selberg L-functions where
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On Ozaki’s theorem realizing prescribed p-groups as p-class tower groups Algebra Number Theory (IF 1.3) Pub Date : 2024-02-26 Farshid Hajir, Christian Maire, Ravi Ramakrishna
We give a streamlined and effective proof of Ozaki’s theorem that any finite p-group Γ is the Galois group of the p-Hilbert class field tower of some number field F . Our work is inspired by Ozaki’s and applies in broader circumstances. While his theorem is in the totally complex setting, we obtain the result in any mixed signature setting for which there exists a number field k 0 with class number
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Supersolvable descent for rational points Algebra Number Theory (IF 1.3) Pub Date : 2024-02-26 Yonatan Harpaz, Olivier Wittenberg
We construct an analogue of the classical descent theory of Colliot-Thélène and Sansuc in which algebraic tori are replaced with finite supersolvable groups. As an application, we show that rational points are dense in the Brauer–Manin set for smooth compactifications of certain quotients of homogeneous spaces by finite supersolvable groups. For suitably chosen homogeneous spaces, this implies the
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On Kato and Kuzumaki’s properties for the Milnor K2 of function fields of p-adic curves Algebra Number Theory (IF 1.3) Pub Date : 2024-02-26 Diego Izquierdo, Giancarlo Lucchini Arteche
Let K be the function field of a curve C over a p-adic field k. We prove that, for each n,d ≥ 1 and for each hypersurface Z in ℙKn of degree d with d2 ≤ n, the second Milnor K-theory group of K is spanned by the images of the norms coming from finite extensions L of K over which Z has a rational point. When the curve C has a point in the maximal unramified extension of k, we generalize this result
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The Ungar Games Combinatorica (IF 1.1) Pub Date : 2024-02-21 Colin Defant, Noah Kravitz, Nathan Williams
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Finite size corrections for real eigenvalues of the elliptic Ginibre matrices Random Matrices Theory Appl. (IF 0.9) Pub Date : 2024-02-20 Sung-Soo Byun, Yong-Woo Lee
In this paper, we consider the elliptic Ginibre matrices in the orthogonal symmetry class that interpolates between the real Ginibre ensemble and the Gaussian orthogonal ensemble. We obtain the finite size corrections of the real eigenvalue densities in both the global and edge scaling regimes, as well as in both the strong and weak non-Hermiticity regimes. Our results extend and provide the rate of
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Quotients of admissible formal schemes and adic spaces by finite groups Algebra Number Theory (IF 1.3) Pub Date : 2024-02-16 Bogdan Zavyalov
We give a self-contained treatment of finite group quotients of admissible (formal) schemes and adic spaces that are locally topologically finite type over a locally strongly noetherian adic space.
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Subconvexity bound for GL(3) × GL(2) L-functions : Hybrid level aspect Algebra Number Theory (IF 1.3) Pub Date : 2024-02-16 Sumit Kumar, Ritabrata Munshi, Saurabh Kumar Singh
Let F be a GL (3) Hecke–Maass cusp form of prime level P1 and let f be a GL (2) Hecke–Maass cuspform of prime level P2. We will prove a subconvex bound for the GL (3) × GL (2) Rankin–Selberg L-function L(s,F × f) in the level aspect for certain ranges of the parameters P1 and P2.
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A categorical Künneth formula for constructible Weil sheaves Algebra Number Theory (IF 1.3) Pub Date : 2024-02-16 Tamir Hemo, Timo Richarz, Jakob Scholbach
We prove a Künneth-type equivalence of derived categories of lisse and constructible Weil sheaves on schemes in characteristic p > 0 for various coefficients, including finite discrete rings, algebraic field extensions E ⊃ ℚℓ, ℓ≠p, and their rings of integers 𝒪E. We also consider a variant for ind-constructible sheaves which applies to the cohomology of moduli stacks of shtukas over global function
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Generalized Igusa functions and ideal growth in nilpotent Lie rings Algebra Number Theory (IF 1.3) Pub Date : 2024-02-16 Angela Carnevale, Michael M. Schein, Christopher Voll
We introduce a new class of combinatorially defined rational functions and apply them to deduce explicit formulae for local ideal zeta functions associated to the members of a large class of nilpotent Lie rings which contains the free class-2-nilpotent Lie rings and is stable under direct products. Our results unify and generalize a substantial number of previous computations. We show that the new
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On Tamagawa numbers of CM tori Algebra Number Theory (IF 1.3) Pub Date : 2024-02-16 Pei-Xin Liang, Yasuhiro Oki, Hsin-Yi Yang, Chia-Fu Yu
We investigate the problem of computing Tamagawa numbers of CM tori. This problem arises naturally from the problem of counting polarized abelian varieties with commutative endomorphism algebras over finite fields, and polarized CM abelian varieties and components of unitary Shimura varieties in the works of Achter, Altug, Garcia and Gordon and of Guo, Sheu and Yu, respectively. We make a systematic
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Decidability via the tilting correspondence Algebra Number Theory (IF 1.3) Pub Date : 2024-02-06 Konstantinos Kartas
We prove a relative decidability result for perfectoid fields. This applies to show that the fields ℚp(p1∕p∞ ) and ℚp(ζp∞) are (existentially) decidable relative to the perfect hull of 𝔽p((t)) and ℚpab is (existentially) decidable relative to the perfect hull of 𝔽¯p((t)). We also prove some unconditional decidability results in mixed characteristic via reduction to characteristic p.
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Differentially large fields Algebra Number Theory (IF 1.3) Pub Date : 2024-02-06 Omar León Sánchez, Marcus Tressl
We introduce the notion of differential largeness for fields equipped with several commuting derivations (as an analogue to largeness of fields). We lay out the foundations of this new class of “tame” differential fields. We state several characterizations and exhibit plenty of examples and applications. Our results strongly indicate that differentially large fields will play a key role in differential
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p-groups, p-rank, and semistable reduction of coverings of curves Algebra Number Theory (IF 1.3) Pub Date : 2024-02-06 Yu Yang
We prove various explicit formulas concerning p-rank of p-coverings of pointed semistable curves over discrete valuation rings. In particular, we obtain a full generalization of Raynaud’s formula for p-rank of fibers over nonmarked smooth closed points in the case of arbitrary closed points. As an application, for abelian p-coverings, we give an affirmative answer to an open problem concerning boundedness
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A deterministic algorithm for Harder–Narasimhan filtrations for representations of acyclic quivers Algebra Number Theory (IF 1.3) Pub Date : 2024-02-06 Chi-Yu Cheng
Let M be a representation of an acyclic quiver Q over an infinite field k. We establish a deterministic algorithm for computing the Harder–Narasimhan filtration of M. The algorithm is polynomial in the dimensions of M, the weights that induce the Harder–Narasimhan filtration of M, and the number of paths in Q. As a direct application, we also show that when k is algebraically closed and when M is unstable
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Sur les espaces homogènes de Borovoi–Kunyavskii Algebra Number Theory (IF 1.3) Pub Date : 2024-02-06 Mạnh Linh Nguyễn
Nous établissons le principe de Hasse et l’approximation faible pour certains espaces homogènes de SL m à stabilisateur géométrique nilpotent de classe 2, construits par Borovoi et Kunyavskii. Ces espaces homogènes vérifient donc une conjecture de Colliot-Thélène concernant l’obstruction de Brauer–Manin pour les variétés géométriquement rationnellement connexes. We establish the Hasse principle and
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Partial sums of typical multiplicative functions over short moving intervals Algebra Number Theory (IF 1.3) Pub Date : 2024-02-06 Mayank Pandey, Victor Y. Wang, Max Wenqiang Xu
We prove that the k-th positive integer moment of partial sums of Steinhaus random multiplicative functions over the interval (x,x + H] matches the corresponding Gaussian moment, as long as H ≪ x∕(log x)2k2+2+o(1) and H tends to infinity with x. We show that properly normalized partial sums of typical multiplicative functions arising from realizations of random multiplicative functions have Gaussian
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An Upper Bound for the Height of a Tree with a Given Eigenvalue Combinatorica (IF 1.1) Pub Date : 2024-02-02 Artūras Dubickas
In this paper we prove that every totally real algebraic integer \(\lambda \) of degree \(d \ge 2\) occurs as an eigenvalue of some tree of height at most \(d(d+1)/2+3\). In order to prove this, for a given algebraic number \(\alpha \ne 0\), we investigate an additive semigroup that contains zero and is closed under the map \(x \mapsto \alpha /(1-x)\) for \(x \ne 1\). The problem of finding the smallest
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Asymptotic cyclic-conditional freeness of random matrices Random Matrices Theory Appl. (IF 0.9) Pub Date : 2024-01-31 Guillaume Cébron, Nicolas Gilliers
Voiculescu’s freeness emerges when computing the asymptotic spectra of polynomials on N×N random matrices with eigenspaces in generic positions: they are randomly rotated with a uniform unitary random matrix UN. In this paper, we elaborate on the previous result by proposing a random matrix model, which we name the Vortex model, where UN has the law of a uniform unitary random matrix conditioned to
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The distribution of sample mean-variance portfolio weights Random Matrices Theory Appl. (IF 0.9) Pub Date : 2024-01-31 Raymond Kan, Nathan Lassance, Xiaolu Wang
We present a simple stochastic representation for the joint distribution of sample estimates of three scalar parameters and two vectors of portfolio weights that characterize the minimum-variance frontier. This stochastic representation is useful for sampling observations efficiently, deriving moments in closed-form, and studying the distribution and performance of many portfolio strategies that are
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Asymptotics of the determinant of the modified Bessel functions and the second Painlevé equation Random Matrices Theory Appl. (IF 0.9) Pub Date : 2024-01-31 Yu Chen, Shuai-Xia Xu, Yu-Qiu Zhao
In the paper, we consider the extended Gross–Witten–Wadia unitary matrix model by introducing a logarithmic term in the potential. The partition function of the model can be expressed equivalently in terms of the Toeplitz determinant with the (i,j)-entry being the modified Bessel functions of order i−j−ν, ν∈ℂ. When the degree n is finite, we show that the Toeplitz determinant is described by the isomonodromy
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The Boolean quadratic forms and tangent law Random Matrices Theory Appl. (IF 0.9) Pub Date : 2024-01-29 Wiktor Ejsmont, Patrycja Hęćka
In [W. Ejsmont and F. Lehner, The free tangent law, Adv. Appl. Math. 121 (2020) 102093], we study the limit sums of free commutators and anticommutators and show that the generalized tangent function tanz1−xtanz describes the limit distribution. This is the generating function of the higher order tangent numbers of Carlitz and Scoville (see (1.6) in [L. Carlitz and R. Scoville, Tangent numbers and
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Laplace transform of the $x-y$ symplectic transformation formula in Topological Recursion Commun. Number Theory Phys. (IF 1.9) Pub Date : 2024-01-24 Alexander Hock
The functional relation coming from the $x-y$ symplectic transformation of Topological Recursion has a lot of applications; for instance it is the higher order moment-cumulant relation in free probability or can be used to compute intersection numbers on the moduli space of complex curves. We derive the Laplace transform of this functional relation, which has a very nice and compact form as a formal
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Cohomological Hall algebras and perverse coherent sheaves on toric Calabi–Yau $3$-folds Commun. Number Theory Phys. (IF 1.9) Pub Date : 2024-01-24 Miroslav Rapčák, Yan Soibelman, Yaping Yang, Gufang Zhao
We study the Drinfeld double of the (equivariant spherical) Cohomological Hall algebra in the sense of Kontsevich and Soibelman, associated to a smooth toric Calabi–Yau $3$-fold $X$. By general reasons, the COHA acts on the cohomology of the moduli spaces of certain perverse coherent systems on $X$ via “raising operators”. Conjecturally the COHA action extends to an action of the Drinfeld double by
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Numerical experiments on coefficients of instanton partition functions Commun. Number Theory Phys. (IF 1.9) Pub Date : 2024-01-24 Aradhita Chattopadhyaya, Jan Manschot
We analyze the coefficients of partition functions of Vafa–Witten (VW) theory on a four-manifold. These partition functions factorize into a product of a function enumerating pointlike instantons and a function enumerating smooth instantons. For gauge groups $SU(2)$ and $SU(3)$ and four-manifold the complex projective plane $\mathbb{CP}^2$, we experimentally study the latter functions, which are examples
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On the Generating Rank and Embedding Rank of the Hexagonic Lie Incidence Geometries Combinatorica (IF 1.1) Pub Date : 2024-01-05
Abstract Given a (thick) irreducible spherical building \(\Omega \) , we establish a bound on the difference between the generating rank and the embedding rank of its long root geometry and the dimension of the corresponding Weyl module, by showing that this difference does not grow when taking certain residues of \(\Omega \) (in particular the residue of a vertex corresponding to a point of the long
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Euler’s Theorem for Regular CW-Complexes Combinatorica (IF 1.1) Pub Date : 2024-01-05 Richard H. Hammack, Paul C. Kainen
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On special solutions to the Ermakov–Painlevé XXV equation Random Matrices Theory Appl. (IF 0.9) Pub Date : 2024-01-03 Alexander Chichurin, Galina Filipuk
In this paper, we study a nonlinear second-order ordinary differential equation which we call the Ermakov–Painlevé XXV equation since under certain restrictions on its coefficients it can be reduced either to the Ermakov or the Painlevé XXV equation. The Ermakov–Painlevé XXV equation arises from a generalized Riccati equation and the related third-order linear differential equation via the Schwarzian
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A Topological Version of Hedetniemi’s Conjecture for Equivariant Spaces Combinatorica (IF 1.1) Pub Date : 2023-12-19 Vuong Bui, Hamid Reza Daneshpajouh
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Tight Bound on Treedepth in Terms of Pathwidth and Longest Path Combinatorica (IF 1.1) Pub Date : 2023-12-19
Abstract We show that every graph with pathwidth strictly less than a that contains no path on \(2^b\) vertices as a subgraph has treedepth at most 10ab. The bound is best possible up to a constant factor.
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Edge-Connectivity and Pairwise Disjoint Perfect Matchings in Regular Graphs Combinatorica (IF 1.1) Pub Date : 2023-12-19 Yulai Ma, Davide Mattiolo, Eckhard Steffen, Isaak H. Wolf
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A Characterization of Graphs Whose Small Powers of Their Edge Ideals Have a Linear Free Resolution Combinatorica (IF 1.1) Pub Date : 2023-11-27 Nguyen Cong Minh, Thanh Vu
Let I(G) be the edge ideal of a simple graph G. We prove that \(I(G)^2\) has a linear free resolution if and only if G is gap-free and \({{\,\textrm{reg}\,}}I(G) \le 3\). Similarly, we show that \(I(G)^3\) has a linear free resolution if and only if G is gap-free and \({{\,\textrm{reg}\,}}I(G) \le 4\). We deduce these characterizations by establishing a general formula for the regularity of powers
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A Group Ring Approach to Fuglede’s Conjecture in Cyclic Groups Combinatorica (IF 1.1) Pub Date : 2023-11-27 Tao Zhang
Fuglede’s conjecture states that a subset \(\Omega \subseteq \mathbb {R}^{n}\) with positive and finite Lebesgue measure is a spectral set if and only if it tiles \(\mathbb {R}^{n}\) by translation. However, this conjecture does not hold in both directions for \(\mathbb {R}^n\), \(n\ge 3\). While the conjecture remains unsolved in \(\mathbb {R}\) and \(\mathbb {R}^2\), cyclic groups are instrumental
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Tail bounds on the spectral norm of sub-exponential random matrices Random Matrices Theory Appl. (IF 0.9) Pub Date : 2023-11-24 Guozheng Dai, Zhonggen Su, Hanchao Wang
Let X be an n×n symmetric random matrix with independent but non-identically distributed entries. The deviation inequalities of the spectral norm of X with Gaussian entries have been obtained by using the standard concentration of Gaussian measure results. This paper establishes an upper tail bound of the spectral norm of X with sub-exponential entries. Our method relies upon a crucial ingredient of
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Universal Planar Graphs for the Topological Minor Relation Combinatorica (IF 1.1) Pub Date : 2023-11-21 Florian Lehner
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Tiling Edge-Coloured Graphs with Few Monochromatic Bounded-Degree Graphs Combinatorica (IF 1.1) Pub Date : 2023-11-21 Jan Corsten, Walner Mendonça
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Degree growth for tame automorphisms of an affine quadric threefold Algebra Number Theory (IF 1.3) Pub Date : 2023-11-22 Nguyen-Bac Dang
We consider the degree sequences of the tame automorphisms preserving an affine quadric threefold. Using some valuative estimates derived from the work of Shestakov and Umirbaev and the action of this group on a CAT (0), Gromov-hyperbolic square complex constructed by Bisi, Furter and Lamy, we prove that the dynamical degrees of tame elements avoid any value strictly between 1 and 4 3. As an application
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A weighted one-level density of families of L-functions Algebra Number Theory (IF 1.3) Pub Date : 2023-11-22 Alessandro Fazzari
This paper is devoted to a weighted version of the one-level density of the nontrivial zeros of L-functions, tilted by a power of the L-function evaluated at the central point. Assuming the Riemann hypothesis and the ratio conjecture, for some specific families of L-functions, we prove that the same structure suggested by the density conjecture also holds in this weighted investigation, if the exponent
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Semisimple algebras and PI-invariants of finite dimensional algebras Algebra Number Theory (IF 1.3) Pub Date : 2023-11-22 Eli Aljadeff, Yakov Karasik
Let Γ be the T-ideal of identities of an affine PI-algebra over an algebraically closed field F of characteristic zero. Consider the family ℳΓ of finite dimensional algebras Σ with Id (Σ) = Γ. By Kemer’s theory ℳΓ is not empty. We show there exists A ∈ℳΓ with Wedderburn–Malcev decomposition A≅ Ass ⊕ JA, where JA is the Jacobson’s radical and Ass is a semisimple supplement with the property that
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Projective orbifolds of Nikulin type Algebra Number Theory (IF 1.3) Pub Date : 2023-11-22 Chiara Camere, Alice Garbagnati, Grzegorz Kapustka, Michał Kapustka
We study projective irreducible symplectic orbifolds of dimension four that are deformations of partial resolutions of quotients of hyperkähler manifolds of K3[2]-type by symplectic involutions; we call them orbifolds of Nikulin type. We first classify those projective orbifolds that are really quotients, by describing all families of projective fourfolds of K3[2]-type with a symplectic involution
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Kempe Equivalent List Colorings Combinatorica (IF 1.1) Pub Date : 2023-11-16 Daniel W. Cranston, Reem Mahmoud
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Weyl invariant $E_8$ Jacobi forms and $E$-strings Commun. Number Theory Phys. (IF 1.9) Pub Date : 2023-11-07 Kaiwen Sun, Haowu Wang
In 1992 Wirthmüller showed that for any irreducible root system not of type $E_8$ the ring of weak Jacobi forms invariant under Weyl group is a polynomial algebra. However, it has recently been proved that for $E_8$ the ring is not a polynomial algebra. Weyl invariant $E_8$ Jacobi forms have many applications in string theory and it is an open problem to describe such forms. The scaled refined free
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Whittaker Fourier type solutions to differential equations arising from string theory Commun. Number Theory Phys. (IF 1.9) Pub Date : 2023-11-07 Ksenia Fedosova, Kim Klinger-Logan
In this article, we find the full Fourier expansion for solutions of $(\Delta-\lambda)f(z) = -E_k (z) E_\ell (z)$ for $z = x + i y \in \mathfrak{H}$ for certain values of parameters $k$, $\ell$ and $\lambda$. When such an $f$ is fully automorphic these functions are referred to as generalized non-holomorphic Eisenstein series. We give a connection of the boundary condition on such Fourier series with
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Enumeration of hypermaps and Hirota equations for extended rationally constrained KP Commun. Number Theory Phys. (IF 1.9) Pub Date : 2023-11-07 G. Carlet, J. van de Leur, H. Posthuma, S. Shadrin
We consider the Hurwitz Dubrovin–Frobenius manifold structure on the space of meromorphic functions on the Riemann sphere with exactly two poles, one simple and one of arbitrary order. We prove that the all genera partition function (also known as the total descendant potential) associated with this Dubrovin–Frobenius manifold is a tau function of a rational reduction of the Kadomtsev–Petviashvili
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Resurgence, Stokes constants, and arithmetic functions in topological string theory Commun. Number Theory Phys. (IF 1.9) Pub Date : 2023-11-07 Claudia Rella
The quantization of the mirror curve to a toric Calabi–Yau threefold gives rise to quantum-mechanical operators, whose fermionic spectral traces produce factorially divergent power series in the Planck constant. These asymptotic expansions can be promoted to resurgent trans-series. They show infinite towers of periodic singularities in their Borel plane and infinitely many rational Stokes constants
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A Book Proof of the Middle Levels Theorem Combinatorica (IF 1.1) Pub Date : 2023-11-06 Torsten Mütze
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On Unique Sums in Abelian Groups Combinatorica (IF 1.1) Pub Date : 2023-11-01 Benjamin Bedert
Let A be a subset of the cyclic group \({\textbf{Z}}/p{\textbf{Z}}\) with p prime. It is a well-studied problem to determine how small |A| can be if there is no unique sum in \(A+A\), meaning that for every two elements \(a_1,a_2\in A\), there exist \(a_1',a_2'\in A\) such that \(a_1+a_2=a_1'+a_2'\) and \(\{a_1,a_2\}\ne \{a_1',a_2'\}\). Let m(p) be the size of a smallest subset of \({\textbf{Z}}/p{\textbf{Z}}\)
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Sweeps, Polytopes, Oriented Matroids, and Allowable Graphs of Permutations Combinatorica (IF 1.1) Pub Date : 2023-10-23 Arnau Padrol, Eva Philippe
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A Structural Theorem for Sets with Few Triangles Combinatorica (IF 1.1) Pub Date : 2023-10-12 Sam Mansfield, Jonathan Passant
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On the Maximum of the Sum of the Sizes of Non-trivial Cross-Intersecting Families Combinatorica (IF 1.1) Pub Date : 2023-10-12 P. Frankl
Let \(n \ge 2k \ge 4\) be integers, \({[n]\atopwithdelims ()k}\) the collection of k-subsets of \([n] = \{1, \ldots , n\}\). Two families \({\mathcal {F}}, {\mathcal {G}} \subset {[n]\atopwithdelims ()k}\) are said to be cross-intersecting if \(F \cap G \ne \emptyset \) for all \(F \in {\mathcal {F}}\) and \(G \in {\mathcal {G}}\). A family is called non-trivial if the intersection of all its members
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GKM-theory for torus actions on cyclic quiver Grassmannians Algebra Number Theory (IF 1.3) Pub Date : 2023-10-08 Martina Lanini, Alexander Pütz
We define and investigate algebraic torus actions on quiver Grassmannians for nilpotent representations of the equioriented cycle. Examples of such varieties are type A flag varieties, their linear degenerations and finite-dimensional approximations of both the affine flag variety and affine Grassmannian for GL n. We show that these quiver Grassmannians equipped with our specific torus action are
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The de Rham–Fargues–Fontaine cohomology Algebra Number Theory (IF 1.3) Pub Date : 2023-10-08 Arthur-César Le Bras, Alberto Vezzani
We show how to attach to any rigid analytic variety V over a perfectoid space P a rigid analytic motive over the Fargues–Fontaine curve 𝒳(P) functorially in V and P. We combine this construction with the overconvergent relative de Rham cohomology to produce a complex of solid quasicoherent sheaves over 𝒳(P), and we show that its cohomology groups are vector bundles if V is smooth and proper over
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On the variation of Frobenius eigenvalues in a skew-abelian Iwasawa tower Algebra Number Theory (IF 1.3) Pub Date : 2023-10-08 Asvin G.
We study towers of varieties over a finite field such as y2 = f(xℓn ) and prove that the characteristic polynomials of the Frobenius on the étale cohomology show a surprising ℓ-adic convergence. We prove this by proving a more general statement about the convergence of certain invariants related to a skew-abelian cohomology group. The key ingredient is a generalization of Fermat’s little theorem to
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Limit multiplicity for unitary groups and the stable trace formula Algebra Number Theory (IF 1.3) Pub Date : 2023-10-08 Mathilde Gerbelli-Gauthier
We give upper bounds on limit multiplicities of certain nontempered representations of unitary groups U(a,b), conditionally on the endoscopic classification of representations. Our result applies to some cohomological representations, and we give applications to the growth of cohomology of cocompact arithmetic subgroups of unitary groups. The representations considered are transfers of products of
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A number theoretic characterization of E-smooth and (FRS) morphisms : estimates on the number of ℤ∕pkℤ-points Algebra Number Theory (IF 1.3) Pub Date : 2023-10-08 Raf Cluckers, Itay Glazer, Yotam I. Hendel
We provide uniform estimates on the number of ℤ∕pkℤ-points lying on fibers of flat morphisms between smooth varieties whose fibers have rational singularities, termed (FRS) morphisms. For each individual fiber, the estimates were known by work of Avni and Aizenbud, but we render them uniform over all fibers. The proof technique for individual fibers is based on Hironaka’s resolution of singularities
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On self-correspondences on curves Algebra Number Theory (IF 1.3) Pub Date : 2023-10-03 Joël Bellaïche
We study the algebraic dynamics of self-correspondences on a curve. A self-correspondence on a (proper and smooth) curve C over an algebraically closed field is the data of another curve D and two nonconstant separable morphisms π1 and π2 from D to C. A subset S of C is complete if π1−1(S) = π2−1(S). We show that self-correspondences are divided into two classes: those that have only finitely many