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  • An intriguing hyperelliptic Shimura curve quotient of genus 16
    Algebra Number Theory (IF 0.92) Pub Date : 2020-11-19
    Lassina Dembélé

    Let F be the maximal totally real subfield of ℚ(ζ32), the cyclotomic field of 32-nd roots of unity. Let D be the quaternion algebra over F ramified exactly at the unique prime above 2 and 7 of the real places of F. Let 𝒪 be a maximal order in D, and X0D(1) the Shimura curve attached to 𝒪. Let C = X0D(1)∕⟨wD⟩, where wD is the unique Atkin–Lehner involution on X0D(1). We show that the curve C has several

    更新日期:2020-11-27
  • Relative crystalline representations and p-divisible groups in the small ramification case
    Algebra Number Theory (IF 0.92) Pub Date : 2020-11-19
    Tong Liu; Yong Suk Moon

    Let k be a perfect field of characteristic p > 2, and let K be a finite totally ramified extension over W(k)[1 p] of ramification degree e. Let R0 be a relative base ring over W(k)⟨t1±1,…,tm±1⟩ satisfying some mild conditions, and let R = R0 ⊗W(k)𝒪K. We show that if e < p − 1, then every crystalline representation of π1e ́ t(SpecR[1 p]) with Hodge–Tate weights in [0,1] arises from a p-divisible group

    更新日期:2020-11-27
  • Arithmetic of curves on moduli of local systems
    Algebra Number Theory (IF 0.92) Pub Date : 2020-11-19
    Junho Peter Whang

    We investigate the arithmetic of algebraic curves on coarse moduli spaces for special linear rank two local systems on surfaces with fixed boundary traces. We prove a structure theorem for morphisms from the affine line into the moduli space. We show that the set of integral points on any nondegenerate algebraic curve on the moduli space can be effectively determined.

    更新日期:2020-11-21
  • Curtis homomorphisms and the integral Bernstein center for GLn
    Algebra Number Theory (IF 0.92) Pub Date : 2020-11-19
    David Helm

    We describe two conjectures, one strictly stronger than the other, that give descriptions of the integral Bernstein center for GLn(F) (that is, the center of the category of smooth W(k)[GLn(F)]-modules, for F a p-adic field and k an algebraically closed field of characteristic ℓ different from p) in terms of Galois theory. Moreover, we show that the weak version of the conjecture (for m ≤ n), together

    更新日期:2020-11-21
  • Moduli spaces of symmetric cubic fourfolds and locally symmetric varieties
    Algebra Number Theory (IF 0.92) Pub Date : 2020-11-19
    Chenglong Yu; Zhiwei Zheng

    We realize the moduli spaces of cubic fourfolds with specified group actions as arithmetic quotients of complex hyperbolic balls or type IV symmetric domains, and study their compactifications. We prove the geometric ( GIT) compactifications are naturally isomorphic to the Hodge theoretic (Looijenga, in many cases Baily–Borel) compactifications. The key ingredients of the proof are the global Torelli

    更新日期:2020-11-21
  • Motivic multiple zeta values relative to μ2
    Algebra Number Theory (IF 0.92) Pub Date : 2020-11-19
    Zhongyu Jin; Jiangtao Li

    We establish a short exact sequence about depth-graded motivic double zeta values of even weight relative to μ2. We find a basis for the depth-graded motivic double zeta values relative to μ2 of even weight and a basis for the depth-graded motivic triple zeta values relative to μ2 of odd weight. As an application of our main results, we prove Kaneko and Tasaka’s conjectures about the sum odd double

    更新日期:2020-11-21
  • Generating series of a new class of orthogonal Shimura varieties
    Algebra Number Theory (IF 0.92) Pub Date : 2020-11-19
    Eugenia Rosu; Dylan Yott

    For a new class of Shimura varieties of orthogonal type over a totally real number field, we construct special cycles and show the modularity of Kudla’s generating series in the cohomology group.

    更新日期:2020-11-21
  • Algorithms for orbit closure separation for invariants and semi-invariants of matrices
    Algebra Number Theory (IF 0.92) Pub Date : 2020-11-19
    Harm Derksen; Visu Makam

    We consider two group actions on m-tuples of n × n matrices with entries in the field K. The first is simultaneous conjugation by GLn and the second is the left-right action of SLn × SLn. Let K¯ be the algebraic closure of the field K. Recently, a polynomial time algorithm was found to decide whether 0 lies in the Zariski closure of the SLn( K¯) × SLn( K¯)-orbit of a given m-tuple by Garg, Gurvits

    更新日期:2020-11-21
  • Quadratic Chabauty for (bi)elliptic curves and Kim’s conjecture
    Algebra Number Theory (IF 0.92) Pub Date : 2020-10-13
    Francesca Bianchi

    We explore a number of problems related to the quadratic Chabauty method for determining integral points on hyperbolic curves. We remove the assumption of semistability in the description of the quadratic Chabauty sets 𝒳(ℤp)2 containing the integral points 𝒳(ℤ) of an elliptic curve of rank at most 1. Motivated by a conjecture of Kim, we then investigate theoretically and computationally the set-theoretic

    更新日期:2020-11-19
  • Invertible functions on nonarchimedean symmetric spaces
    Algebra Number Theory (IF 0.92) Pub Date : 2020-10-13
    Ernst-Ulrich Gekeler

    Let u be a nowhere vanishing holomorphic function on the Drinfeld space Ωr of dimension r − 1, where r ≥ 2. The logarithm logq|u| of its absolute value may be regarded as an affine function on the attached Bruhat–Tits building ℬ𝒯r. Generalizing a construction of van der Put in case r = 2, we relate the group 𝒪(Ωr)∗ of such u with the group H(ℬ𝒯r, ℤ) of integer-valued harmonic 1-cochains on ℬ𝒯r

    更新日期:2020-11-19
  • Iterated local cohomology groups and Lyubeznik numbers for determinantal rings
    Algebra Number Theory (IF 0.92) Pub Date : 2020-10-13
    András C. Lőrincz; Claudiu Raicu

    We give an explicit recipe for determining iterated local cohomology groups with support in ideals of minors of a generic matrix in characteristic zero, expressing them as direct sums of indecomposable 𝒟-modules. For nonsquare matrices these indecomposables are simple, but this is no longer true for square matrices where the relevant indecomposables arise from the pole order filtration associated

    更新日期:2020-11-19
  • The Brauer group of the moduli stack of elliptic curves
    Algebra Number Theory (IF 0.92) Pub Date : 2020-10-13
    Benjamin Antieau; Lennart Meier

    We compute the Brauer group of ℳ1,1, the moduli stack of elliptic curves, over Spec ℤ, its localizations, finite fields of odd characteristic, and algebraically closed fields of characteristic not 2. The methods involved include the use of the parameter space of Legendre curves and the moduli stack ℳ(2) of curves with full (naive) level 2 structure, the study of the Leray–Serre spectral sequence in

    更新日期:2020-10-16
  • Modular forms from Noether–Lefschetz theory
    Algebra Number Theory (IF 0.92) Pub Date : 2020-10-13
    François Greer

    We enumerate smooth rational curves on very general Weierstrass fibrations over hypersurfaces in projective space. The generating functions for these numbers lie in the ring of classical modular forms. The method of proof uses topological intersection products on a period stack and the cohomological theta correspondence of Kudla and Millson for special cycles on a locally symmetric space of orthogonal

    更新日期:2020-10-16
  • The Prasad conjectures for GSp4 and PGSp4
    Algebra Number Theory (IF 0.92) Pub Date : 2020-10-13
    Hengfei Lu

    We use the theta correspondence between GSp4(E) and GO(V ) to study the GSp4-distinction problems over a quadratic extension E∕F of nonarchimedean local fields of characteristic 0. With a similar strategy, we investigate the distinction problem for the pair (GSp4(E),GSp1,1(F)), where GSp1,1 is the unique inner form of GSp4 defined over F. Then we verify the Prasad conjecture for a discrete series representation

    更新日期:2020-10-16
  • On a cohomological generalization of the Shafarevich conjecture for K3 surfaces
    Algebra Number Theory (IF 0.92) Pub Date : 2020-10-13
    Teppei Takamatsu

    The Shafarevich conjecture for K3 surfaces asserts the finiteness of isomorphism classes of K3 surfaces over a fixed number field admitting good reduction away from a fixed finite set of finite places. André proved this conjecture for polarized K3 surfaces of fixed degree, and recently She proved it for polarized K3 surfaces of unspecified degree. We prove a certain generalization of their results

    更新日期:2020-10-16
  • On asymptotic Fermat over ℤp-extensions of ℚ
    Algebra Number Theory (IF 0.92) Pub Date : 2020-10-13
    Nuno Freitas; Alain Kraus; Samir Siksek

    Let p be a prime and let ℚn,p denote the n-th layer of the cyclotomic ℤp-extension of ℚ. We prove the effective asymptotic FLT over ℚn,p for all n ≥ 1 and all primes p ≥ 5 that are non-Wieferich, i.e., 2p−1≢1(modp2). The effectivity in our result builds on recent work of Thorne proving modularity of elliptic curves over ℚn,p.

    更新日期:2020-10-16
  • On iterated product sets with shifts, II
    Algebra Number Theory (IF 0.92) Pub Date : 2020-09-18
    Brandon Hanson; Oliver Roche-Newton; Dmitrii Zhelezov

    The main result of this paper is the following: for all b ∈ ℤ there exists k = k(b) such that max{|A(k)|,|(A + u)(k)|}≥|A|b, for any finite A ⊂ ℚ and any nonzero u ∈ ℚ. Here, |A(k)| denotes the k-fold product set {a1⋯ak : a1,…,ak ∈ A}. Furthermore, our method of proof also gives the following l∞ sum-product estimate. For all γ > 0 there exists a constant C = C(γ) such that for any A ⊂ ℚ with |AA|≤

    更新日期:2020-10-15
  • The dimension growth conjecture, polynomial in the degree and without logarithmic factors
    Algebra Number Theory (IF 0.92) Pub Date : 2020-09-18
    Wouter Castryck; Raf Cluckers; Philip Dittmann; Kien Huu Nguyen

    We study Heath-Brown’s and Serre’s dimension growth conjecture (proved by Salberger) when the degree d grows. Recall that Salberger’s dimension growth results give bounds of the form OX,𝜀(Bdim X+𝜀) for the number of rational points of height at most B on any integral subvariety X of ℙℚn of degree d ≥ 2, where one can write Od,n,𝜀 instead of OX,𝜀 as soon as d ≥ 4. We give the following simplified

    更新日期:2020-10-15
  • Toroidal orbifolds, destackification, and Kummer blowings up
    Algebra Number Theory (IF 0.92) Pub Date : 2020-09-18
    Dan Abramovich; Michael Temkin; Jarosław Włodarczyk

    We show that any toroidal DM stack X with finite diagonalizable inertia possesses a maximal toroidal coarsening Xtcs such that the morphism X → Xtcs is logarithmically smooth. Further, we use torification results of Abramovich and Temkin (2017) to construct a destackification functor, a variant of the main result of Bergh (2017), on the category of such toroidal stacks X. Namely, we associate to X

    更新日期:2020-09-20
  • Auslander correspondence for triangulated categories
    Algebra Number Theory (IF 0.92) Pub Date : 2020-09-18
    Norihiro Hanihara

    We give analogues of the Auslander correspondence for two classes of triangulated categories satisfying certain finiteness conditions. The first class is triangulated categories with additive generators and we consider their endomorphism algebras as the Auslander algebras. For the second one, we introduce the notion of [1]-additive generators and consider their graded endomorphism algebras as the Auslander

    更新日期:2020-09-20
  • Supersingular locus of Hilbert modular varieties, arithmetic level raising and Selmer groups
    Algebra Number Theory (IF 0.92) Pub Date : 2020-09-18
    Yifeng Liu; Yichao Tian

    This article has three goals: First, we generalize the result of Deuring and Serre on the characterization of supersingular locus to all Shimura varieties given by totally indefinite quaternion algebras over totally real number fields. Second, we generalize the result of Ribet on arithmetic level raising to such Shimura varieties in the inert case. Third, as an application to number theory, we use

    更新日期:2020-09-20
  • Burch ideals and Burch rings
    Algebra Number Theory (IF 0.92) Pub Date : 2020-09-18
    Hailong Dao; Toshinori Kobayashi; Ryo Takahashi

    We introduce the notion of Burch ideals and Burch rings. They are easy to define, and can be viewed as generalization of many well-known concepts, for example integrally closed ideals of finite colength and Cohen–Macaulay rings of minimal multiplicity. We give several characterizations of these objects. We show that they satisfy many interesting and desirable properties: ideal-theoretic, homological

    更新日期:2020-09-20
  • Sous-groupe de Brauer invariant et obstruction de descente itérée
    Algebra Number Theory (IF 0.92) Pub Date : 2020-09-18
    Yang Cao

    Pour une variété quasi-projective, lisse, géométriquement intègre sur un corps de nombres k, on montre que l’obstruction de descente itérée est équivalente à l’obstruction de descente. Ceci généralise un résultat de Skorobogatov, et ceci répond à une question ouverte de Poonen. Les outils principaux sont la notion de sous-groupe de Brauer invariant et la notion d’obstruction de Brauer–Manin étale invariante

    更新日期:2020-09-20
  • Most words are geometrically almost uniform
    Algebra Number Theory (IF 0.92) Pub Date : 2020-09-18
    Michael Jeffrey Larsen

    If w is a word in d > 1 letters and G is a finite group, evaluation of w on a uniformly randomly chosen d-tuple in G gives a random variable with values in G, which may or may not be uniform. It is known that if G ranges over finite simple groups of given root system and characteristic, a positive proportion of words w give a distribution which approaches uniformity in the limit as |G|→∞. In this paper

    更新日期:2020-09-20
  • On a conjecture of Yui and Zagier
    Algebra Number Theory (IF 0.92) Pub Date : 2020-09-18
    Yingkun Li; Tonghai Yang

    We prove the conjecture of Yui and Zagier concerning the factorization of the resultants of minimal polynomials of Weber class invariants. The novelty of our approach is to systematically express differences of certain Weber functions as products of Borcherds products.

    更新日期:2020-09-20
  • Moments of quadratic twists of elliptic curve L-functions over function fields
    Algebra Number Theory (IF 0.92) Pub Date : 2020-08-18
    Hung M. Bui; Alexandra Florea; Jonathan P. Keating; Edva Roditty-Gershon

    We calculate the first and second moments of L-functions in the family of quadratic twists of a fixed elliptic curve E over 𝔽q[x], asymptotically in the limit as the degree of the twists tends to infinity. We also compute moments involving derivatives of L-functions over quadratic twists, enabling us to deduce lower bounds on the correlations between the analytic ranks of the twists of two distinct

    更新日期:2020-09-18
  • Nonvanishing of hyperelliptic zeta functions over finite fields
    Algebra Number Theory (IF 0.92) Pub Date : 2020-08-18
    Jordan S. Ellenberg; Wanlin Li; Mark Shusterman

    Fixing t ∈ ℝ and a finite field 𝔽q of odd characteristic, we give an explicit upper bound on the proportion of genus g hyperelliptic curves over 𝔽q whose zeta function vanishes at 1 2 + it. Our upper bound is independent of g and tends to 0 as q grows.

    更新日期:2020-09-18
  • p-adic Asai L-functions of Bianchi modular forms
    Algebra Number Theory (IF 0.92) Pub Date : 2020-08-18
    David Loeffler; Chris Williams

    The Asai (or twisted tensor) L-function of a Bianchi modular form Ψ is the L-function attached to the tensor induction to ℚ of its associated Galois representation. When Ψ is ordinary at p we construct a p-adic analogue of this L-function: that is, a p-adic measure on ℤp× that interpolates the critical values of the Asai L-function twisted by Dirichlet characters of p-power conductor. The construction

    更新日期:2020-08-20
  • Pro-unipotent harmonic actions and dynamical properties of p-adic cyclotomic multiple zeta values
    Algebra Number Theory (IF 0.92) Pub Date : 2020-08-18
    David Jarossay

    p-adic cyclotomic multiple zeta values depend on the choice of a number of iterations of the crystalline Frobenius of the pro-unipotent fundamental groupoid of ℙ1 ∖{0,μN,∞}. In this paper we study how the iterated Frobenius depends on the number of iterations, in relation with the computation of p-adic cyclotomic multiple zeta values in terms of cyclotomic multiple harmonic sums. This provides new

    更新日期:2020-08-20
  • Nouvelles cohomologies de Weil en caractéristique positive
    Algebra Number Theory (IF 0.92) Pub Date : 2020-08-18
    Joseph Ayoub

    Soit K un corps valué de hauteur 1 et d’inégales caractéristiques (0,p), et soit k son corps résiduel. Dans cet article, nous construisons une nouvelle cohomologie de Weil pour les k-schémas de type fini à valeurs dans les AK-modules, avec AK une K-algèbre de « périodes abstraites p-adiques » qui admet une description explicite par générateurs et relations. Nous démontrons des théorèmes de comparaison

    更新日期:2020-08-20
  • Elliptic curves over totally real cubic fields are modular
    Algebra Number Theory (IF 0.92) Pub Date : 2020-08-18
    Maarten Derickx; Filip Najman; Samir Siksek

    We prove that all elliptic curves defined over totally real cubic fields are modular. This builds on previous work of Freitas, Le Hung and Siksek, who proved modularity of elliptic curves over real quadratic fields, as well as recent breakthroughs due to Thorne and to Kalyanswamy.

    更新日期:2020-08-20
  • Motivic Gauss–Bonnet formulas
    Algebra Number Theory (IF 0.92) Pub Date : 2020-08-18
    Marc Levine; Arpon Raksit

    The apparatus of motivic stable homotopy theory provides a notion of Euler characteristic for smooth projective varieties, valued in the Grothendieck–Witt ring of the base field. Previous work of the first author and recent work of Déglise, Jin and Khan established a motivic Gauss–Bonnet formula relating this Euler characteristic to pushforwards of Euler classes in motivic cohomology theories. We apply

    更新日期:2020-08-20
  • Burgess bounds for short character sums evaluated at forms
    Algebra Number Theory (IF 0.92) Pub Date : 2020-08-18
    Lillian B. Pierce; Junyan Xu

    We establish a Burgess bound for short multiplicative character sums in arbitrary dimensions, in which the character is evaluated at a homogeneous form that belongs to a very general class of “admissible” forms. This n-dimensional Burgess bound is nontrivial for sums over boxes of sidelength at least qβ, with β > 1∕2 − 1∕(2(n + 1)). This is the first Burgess bound that applies in all dimensions to

    更新日期:2020-08-20
  • Galois action on the principal block and cyclic Sylow subgroups
    Algebra Number Theory (IF 0.92) Pub Date : 2020-08-18
    Noelia Rizo; A. A. Schaeffer Fry; Carolina Vallejo

    We characterize finite groups G having a cyclic Sylow p-subgroup in terms of the action of a specific Galois automorphism on the principal p-block of G, for p = 2,3. We show that the analog statement for blocks with arbitrary defect group would follow from the blockwise McKay–Navarro conjecture.

    更新日期:2020-08-20
  • Abelian extensions in dynamical Galois theory
    Algebra Number Theory (IF 0.92) Pub Date : 2020-08-18
    Jesse Andrews; Clayton Petsche

    We propose a conjectural characterization of when the dynamical Galois group associated to a polynomial is abelian, and we prove our conjecture in several cases, including the stable quadratic case over ℚ. In the postcritically infinite case, the proof uses algebraic techniques, including a result concerning ramification in towers of cyclic p-extensions. In the postcritically finite case, the proof

    更新日期:2020-08-20
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