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A bound for the exterior product of S-units Algebra Number Theory (IF 0.9) Pub Date : 2024-09-19 Shabnam Akhtari, Jeffrey D. Vaaler
We generalize an inequality for the determinant of a real matrix proved by A. Schinzel, to more general exterior products of vectors in Euclidean space. We apply this inequality to the logarithmic embedding of S-units contained in a number field k. This leads to a bound for the exterior product of S-units expressed as a product of heights. Using a volume formula of P. McMullen we show that our inequality
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Prime values of f(a,b2) and f(a,p2), f quadratic Algebra Number Theory (IF 0.9) Pub Date : 2024-09-19 Stanley Yao Xiao
We prove an asymptotic formula for primes of the shape f(a,b2) with a, b integers and of the shape f(a,p2) with p prime. Here f is a binary quadratic form with integer coefficients, irreducible over ℚ and has no local obstructions. This refines the seminal work of Friedlander and Iwaniec on primes of the form x2 + y4 and of Heath-Brown and Li on primes of the form a2 + p4, as well as earlier work of
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Affine Deligne–Lusztig varieties with finite Coxeter parts Algebra Number Theory (IF 0.9) Pub Date : 2024-09-19 Xuhua He, Sian Nie, Qingchao Yu
We study affine Deligne–Lusztig varieties Xw(b) when the finite part of the element w in the Iwahori–Weyl group is a partial σ-Coxeter element. We show that such w is a cordial element and Xw(b)≠∅ if and only if b satisfies a certain Hodge–Newton indecomposability condition. Our main result is that for such w and b, Xw(b) has a simple geometric structure: the σ-centralizer of b acts transitively on
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Semistable models for some unitary Shimura varieties over ramified primes Algebra Number Theory (IF 0.9) Pub Date : 2024-09-19 Ioannis Zachos
We consider Shimura varieties associated to a unitary group of signature (n − 2,2). We give regular p-adic integral models for these varieties over odd primes p which ramify in the imaginary quadratic field with level subgroup at p given by the stabilizer of a selfdual lattice in the hermitian space. Our construction is given by an explicit resolution of a corresponding local model.
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A unipotent realization of the chromatic quasisymmetric function Algebra Number Theory (IF 0.9) Pub Date : 2024-09-19 Lucas Gagnon
We realize two families of combinatorial symmetric functions via the complex character theory of the finite general linear group GL n(𝔽q): chromatic quasisymmetric functions and vertical strip LLT polynomials. The associated GL n(𝔽q) characters are elementary in nature and can be obtained by induction from certain well-behaved characters of the unipotent upper triangular groups UT n(𝔽q). The
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The strong maximal rank conjecture and moduli spaces of curves Algebra Number Theory (IF 0.9) Pub Date : 2024-09-18 Fu Liu, Brian Osserman, Montserrat Teixidor i Bigas, Naizhen Zhang
Building on recent work of the authors, we use degenerations to chains of elliptic curves to prove two cases of the Aprodu–Farkas strong maximal rank conjecture, in genus 22 and 23. This constitutes a major step forward in Farkas’ program to prove that the moduli spaces of curves of genus 22 and 23 are of general type. Our techniques involve a combination of the Eisenbud–Harris theory of limit linear
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Unramifiedness of weight 1 Hilbert Hecke algebras Algebra Number Theory (IF 0.9) Pub Date : 2024-09-18 Shaunak V. Deo, Mladen Dimitrov, Gabor Wiese
We prove that the Galois pseudorepresentation valued in the mod pn cuspidal Hecke algebra for GL (2) over a totally real number field F, of parallel weight 1 and level prime to p, is unramified at any place above p. The same is true for the noncuspidal Hecke algebra at places above p whose ramification index is not divisible by p−1. A novel geometric ingredient, which is also of independent interest
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Failure of the local-global principle for isotropy of quadratic forms over function fields Algebra Number Theory (IF 0.9) Pub Date : 2024-09-18 Asher Auel, V. Suresh
We prove the failure of the local-global principle, with respect to discrete valuations, for isotropy of quadratic forms in 2n variables over function fields of transcendence degree n ≥ 2 over an algebraically closed field of characteristic ≠2. Our construction involves the generalized Kummer varieties considered by Borcea and by Cynk and Hulek as well as new results on the nontriviality of unramified
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Application of a polynomial sieve: beyond separation of variables Algebra Number Theory (IF 0.9) Pub Date : 2024-09-18 Dante Bonolis, Lillian B. Pierce
Let a polynomial f ∈ ℤ[X1,… ,Xn] be given. The square sieve can provide an upper bound for the number of integral x ∈ [−B,B]n such that f(x) is a perfect square. Recently this has been generalized substantially: first to a power sieve, counting x ∈ [−B,B]n for which f(x) = yr is solvable for y ∈ ℤ; then to a polynomial sieve, counting x ∈ [−B,B]n for which f(x) = g(y) is solvable, for a given polynomial
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Functorial embedded resolution via weighted blowings up Algebra Number Theory (IF 0.9) Pub Date : 2024-09-18 Dan Abramovich, Michael Temkin, Jarosław Włodarczyk
We provide a simple procedure for resolving, in characteristic 0, singularities of a variety X embedded in a smooth variety Y by repeatedly blowing up the worst singularities, in the sense of stack-theoretic weighted blowings up. No history, no exceptional divisors, and no logarithmic structures are necessary to carry this out; the steps are explicit geometric operations requiring no choices; and the
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Serre weights for three-dimensional wildly ramified Galois representations Algebra Number Theory (IF 0.9) Pub Date : 2024-06-13 Daniel Le, Bao V. Le Hung, Brandon Levin, Stefano Morra
We formulate and prove the weight part of Serre’s conjecture for three-dimensional mod p Galois representations under a genericity condition when the field is unramified at p. This removes the assumption made previously that the representation be tamely ramified at p. We also prove a version of Breuil’s lattice conjecture and a mod p multiplicity one result for the cohomology of U(3)-arithmetic manifolds
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Combining Igusa’s conjectures on exponential sums and monodromy with semicontinuity of the minimal exponent Algebra Number Theory (IF 0.9) Pub Date : 2024-06-13 Raf Cluckers, Kien Huu Nguyen
We combine two of Igusa’s conjectures with recent semicontinuity results by Mustaţă and Popa to form a new, natural conjecture about bounds for exponential sums. These bounds have a deceivingly simple and general formulation in terms of degrees and dimensions only. We provide evidence consisting partly of adaptations of already known results about Igusa’s conjecture on exponential sums, but also some
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Exceptional characters and prime numbers in sparse sets Algebra Number Theory (IF 0.9) Pub Date : 2024-06-13 Jori Merikoski
We develop a lower bound sieve for primes under the (unlikely) assumption of infinitely many exceptional characters. Compared with the illusory sieve due to Friedlander and Iwaniec which produces asymptotic formulas, we show that less arithmetic information is required to prove nontrivial lower bounds. As an application of our method, assuming the existence of infinitely many exceptional characters
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Polyhedral and tropical geometry of flag positroids Algebra Number Theory (IF 0.9) Pub Date : 2024-06-13 Jonathan Boretsky, Christopher Eur, Lauren Williams
A flag positroid of ranks r := (r1 < ⋯ < rk) on [n] is a flag matroid that can be realized by a real rk × n matrix A such that the ri × ri minors of A involving rows 1,2,… ,ri are nonnegative for all 1 ≤ i ≤ k. In this paper we explore the polyhedral and tropical geometry of flag positroids, particularly when r := (a,a + 1,… ,b) is a sequence of consecutive numbers. In this case we show that the
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Maximal subgroups of exceptional groups and Quillen’s dimension Algebra Number Theory (IF 0.9) Pub Date : 2024-06-13 Kevin I. Piterman
Given a finite group G and a prime p, let 𝒜p(G) be the poset of nontrivial elementary abelian p-subgroups of G. The group G satisfies the Quillen dimension property at p if 𝒜p(G) has nonzero homology in the maximal possible degree, which is the p-rank of G minus 1. For example, D. Quillen showed that solvable groups with trivial p-core satisfy this property, and later, M. Aschbacher and S. D. Smith
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Refined height pairing Algebra Number Theory (IF 0.9) Pub Date : 2024-04-30 Bruno Kahn
For a d-dimensional regular proper variety X over the function field of a smooth variety B over a field k and for i ≥ 0, we define a subgroup CH i(X)(0) of CH i(X) and construct a “refined height pairing” CH i(X)(0) × CH d+1−i(X)(0) → CH 1(B) in the category of abelian groups up to isogeny. For i = 1,d, CH i(X)(0) is the group of cycles numerically equivalent to 0. This pairing relates
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Balmer spectra and Drinfeld centers Algebra Number Theory (IF 0.9) Pub Date : 2024-04-30 Kent B. Vashaw
The Balmer spectrum of a monoidal triangulated category is an important geometric construction which is closely related to the problem of classifying thick tensor ideals. We prove that the forgetful functor from the Drinfeld center of a finite tensor category C to C extends to a monoidal triangulated functor between their corresponding stable categories, and induces a continuous map between their Balmer
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On the p-adic interpolation of unitary Friedberg–Jacquet periods Algebra Number Theory (IF 0.9) Pub Date : 2024-04-30 Andrew Graham
We establish functoriality of higher Coleman theory for certain unitary Shimura varieties and use this to construct a p-adic analytic function interpolating unitary Friedberg–Jacquet periods.
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Enumeration of conjugacy classes in affine groups Algebra Number Theory (IF 0.9) Pub Date : 2024-04-30 Jason Fulman, Robert M. Guralnick
We study the conjugacy classes of the classical affine groups. We derive generating functions for the number of classes analogous to formulas of Wall and the authors for the classical groups. We use these to get good upper bounds for the number of classes. These naturally come up as difficult cases in the study of the noncoprime k(GV ) problem of Brauer.
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On the ordinary Hecke orbit conjecture Algebra Number Theory (IF 0.9) Pub Date : 2024-04-16 Pol van Hoften
We prove the ordinary Hecke orbit conjecture for Shimura varieties of Hodge type at primes of good reduction. We make use of the global Serre–Tate coordinates of Chai as well as recent results of D’Addezio about the monodromy groups of isocrystals. The new ingredients in this paper are a general monodromy theorem for Hecke-stable subvarieties for Shimura varieties of Hodge type, and a rigidity result
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Locally analytic vector bundles on the Fargues–Fontaine curve Algebra Number Theory (IF 0.9) Pub Date : 2024-04-16 Gal Porat
We develop a version of Sen theory for equivariant vector bundles on the Fargues–Fontaine curve. We show that every equivariant vector bundle canonically descends to a locally analytic vector bundle. A comparison with the theory of (φ,Γ)-modules in the cyclotomic case then recovers the Cherbonnier–Colmez decompletion theorem. Next, we focus on the subcategory of de Rham locally analytic vector bundles
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Multiplicity structure of the arc space of a fat point Algebra Number Theory (IF 0.9) Pub Date : 2024-04-16 Rida Ait El Manssour, Gleb Pogudin
The equation xm = 0 defines a fat point on a line. The algebra of regular functions on the arc space of this scheme is the quotient of k[x,x′,x(2),… ] by all differential consequences of xm = 0. This infinite-dimensional algebra admits a natural filtration by finite-dimensional algebras corresponding to the truncations of arcs. We show that the generating series for their dimensions equals m∕(1 −
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Theta correspondence and simple factors in global Arthur parameters Algebra Number Theory (IF 0.9) Pub Date : 2024-04-16 Chenyan Wu
By using results on poles of L-functions and theta correspondence, we give a bound on b for (χ,b)-factors of the global Arthur parameter of a cuspidal automorphic representation π of a classical group or a metaplectic group where χ is a conjugate self-dual automorphic character and b is an integer which is the dimension of an irreducible representation of SL 2(ℂ). We derive a more precise relation
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Equidistribution theorems for holomorphic Siegel cusp forms of general degree: the level aspect Algebra Number Theory (IF 0.9) Pub Date : 2024-04-16 Henry H. Kim, Satoshi Wakatsuki, Takuya Yamauchi
This paper is an extension of Kim et al. (2020a), and we prove equidistribution theorems for families of holomorphic Siegel cusp forms of general degree in the level aspect. Our main contribution is to estimate unipotent contributions for general degree in the geometric side of Arthur’s invariant trace formula in terms of Shintani zeta functions in a uniform way. Several applications, including the
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Fundamental exact sequence for the pro-étale fundamental group Algebra Number Theory (IF 0.9) 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 0.9) 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 0.9) 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 0.9) 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 0.9) 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 0.9) 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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Quotients of admissible formal schemes and adic spaces by finite groups Algebra Number Theory (IF 0.9) 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 0.9) 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 0.9) 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 0.9) 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 0.9) 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 0.9) 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 0.9) 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 0.9) 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 0.9) 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 0.9) 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 0.9) 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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Degree growth for tame automorphisms of an affine quadric threefold Algebra Number Theory (IF 0.9) 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 0.9) 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 0.9) 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 0.9) 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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GKM-theory for torus actions on cyclic quiver Grassmannians Algebra Number Theory (IF 0.9) 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 0.9) 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 0.9) 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 0.9) 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 0.9) 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 0.9) 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
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Fitting ideals of class groups for CM abelian extensions Algebra Number Theory (IF 0.9) Pub Date : 2023-10-03 Mahiro Atsuta, Takenori Kataoka
Let K be a finite abelian CM-extension of a totally real field k and T a suitable finite set of finite primes of k. We determine the Fitting ideal of the minus component of the T-ray class group of K, except for the 2-component, assuming the validity of the equivariant Tamagawa number conjecture. As an application, we give a necessary and sufficient condition for the Stickelberger element to lie in
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The behavior of essential dimension under specialization, II Algebra Number Theory (IF 0.9) Pub Date : 2023-10-03 Zinovy Reichstein, Federico Scavia
Let G be a linear algebraic group over a field. We show that, under mild assumptions, in a family of primitive generically free G-varieties over a base variety B, the essential dimension of the geometric fibers may drop on a countable union of Zariski closed subsets of B and stays constant away from this countable union. We give several applications of this result.
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Differential operators, retracts, and toric face rings Algebra Number Theory (IF 0.9) Pub Date : 2023-10-03 Christine Berkesch, C-Y. Jean Chan, Patricia Klein, Laura Felicia Matusevich, Janet Page, Janet Vassilev
We give explicit descriptions of rings of differential operators of toric face rings in characteristic 0. For quotients of normal affine semigroup rings by radical monomial ideals, we also identify which of their differential operators are induced by differential operators on the ambient ring. Lastly, we provide a criterion for the Gorenstein property of a normal affine semigroup ring in terms of its
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Bézoutians and the 𝔸1-degree Algebra Number Theory (IF 0.9) Pub Date : 2023-10-03 Thomas Brazelton, Stephen McKean, Sabrina Pauli
We prove that both the local and global 𝔸1-degree of an endomorphism of affine space can be computed in terms of the multivariate Bézoutian. In particular, we show that the Bézoutian bilinear form, the Scheja–Storch form, and the 𝔸1-degree for complete intersections are isomorphic. Our global theorem generalizes Cazanave’s theorem in the univariate case, and our local theorem generalizes Kass–Wickelgren’s
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Axiomatizing the existential theory of 𝔽q((t)) Algebra Number Theory (IF 0.9) Pub Date : 2023-10-03 Sylvy Anscombe, Philip Dittmann, Arno Fehm
We study the existential theory of equicharacteristic henselian valued fields with a distinguished uniformizer. In particular, assuming a weak consequence of resolution of singularities, we obtain an axiomatization of — and therefore an algorithm to decide — the existential theory relative to the existential theory of the residue field. This is both more general and works under weaker resolution hypotheses
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The diagonal coinvariant ring of a complex reflection group Algebra Number Theory (IF 0.9) Pub Date : 2023-10-03 Stephen Griffeth
For an irreducible complex reflection group W of rank n containing N reflections, we put g = 2N∕n and construct a (g + 1)n-dimensional irreducible representation of the Cherednik algebra which is (as a vector space) a quotient of the diagonal coinvariant ring of W. We propose that this representation of the Cherednik algebra is the single largest representation bearing this relationship to the diagonal
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Special cycles on the basic locus of unitary Shimura varieties at ramified primes Algebra Number Theory (IF 0.9) Pub Date : 2023-09-19 Yousheng Shi
We study special cycles on the basic locus of certain unitary Shimura varieties over the ramified primes and their local analogs on the corresponding Rapoport–Zink spaces. We study the support and compute the dimension of these cycles.
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Hybrid subconvexity bounds for twists of GL(3) × GL(2) L-functions Algebra Number Theory (IF 0.9) Pub Date : 2023-09-19 Bingrong Huang, Zhao Xu
We prove hybrid subconvexity bounds for GL (3) × GL (2) L-functions twisted by a primitive Dirichlet character modulo M (prime) in the M- and t-aspects. We also improve hybrid subconvexity bounds for twists of GL (3) L-functions in the M- and t-aspects.
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Separation of periods of quartic surfaces Algebra Number Theory (IF 0.9) Pub Date : 2023-09-19 Pierre Lairez, Emre Can Sertöz
We give a computable lower bound for the distance between two distinct periods of a given quartic surface defined over the algebraic numbers. The main ingredient is the determination of height bounds on components of the Noether–Lefschetz loci. This makes it possible to study the Diophantine properties of periods of quartic surfaces and to certify a part of the numerical computation of their Picard groups