The Gauss circle problem concerns the difference P2(n) between the area of a circle of radius n and the number of lattice points it contains. In this paper, we study the Dirichlet series with coefficients P2(n)2, and prove that this series has meromorphic continuation to ℂ. Using this series, we prove that the Laplace transform of P2(n)2 satisfies ∫ 0∞P2(t)2e−t∕Xdt = CX3∕2 − X + O(X1∕2+𝜖), which gives

In this article, we construct SLk-friezes using Plücker coordinates, making use of the cluster structure on the homogeneous coordinate ring of the Grassmannian of k-spaces in n-space via the Plücker embedding. When this cluster algebra is of finite type, the SLk-friezes are in bijection with the so-called mesh friezes of the corresponding Grassmannian cluster category. These are collections of positive

Motivated by questions from the study of relative trace formulae, we construct a generalization of Grothendieck’s simultaneous resolution over the regular locus of certain symmetric pairs. We use this space to prove a relative version of results of Donagi and Gaitsgory about the automorphism sheaf of regular stabilizers. We also obtain partial results toward applications in Springer theory for symmetric

We study the cancellation property of projective modules of rank 2 with a trivial determinant over Noetherian rings of dimension ≤ 4. If R is a smooth affine algebra of dimension 4 over an algebraically closed field k such that 6 ∈ k×, then we prove that stably free R-modules of rank 2 are free if and only if a Hermitian K-theory group V ˜SL(R) is trivial.

The purpose of this article is to prove a “Newton over Hodge” result for exponential sums on curves. Let X be a smooth proper curve over a finite field 𝔽q of characteristic p ≥ 3 and let V ⊂ X be an affine curve. For a regular function f̄ on V , we may form the L-function L(f̄,V,s) associated to the exponential sums of f̄. In this article, we prove a lower estimate on the Newton polygon of L(f̄,V

We relate the existence of linear models on central simple algebras to local root numbers.

An algebraic framework for noncommutative bundles with (quantum) homogeneous fibres is proposed. The framework relies on the use of principal coalgebra extensions which play the role of principal bundles in noncommutative geometry which might be additionally equipped with a Hopf algebra symmetry. The proposed framework is supported by two examples of noncommutative ℂPq1-bundles: the quantum flag manifold

We find new lower bounds on the torsion orders of very general Fano hypersurfaces over (uncountable) fields of arbitrary characteristic. Our results imply that unirational parametrizations of most Fano hypersurfaces need to have very large degree. Our results also hold in characteristic two, where they solve the rationality problem for hypersurfaces under a logarithmic degree bound, thereby extending

We construct a basis of the space S14(Sp12(ℤ)) of Siegel cusp forms of degree 6 and weight 14 consisting of harmonic theta series. One of these functions has vanishing order 2 at the boundary which implies that the Kodaira dimension of 𝒜6 is nonnegative.

We generalize the gcd results of Corvaja and Zannier and of Levin on 𝔾mn to more general settings. More specifically, we analyze the height of a closed subscheme of codimension at least 2 inside an n-dimensional Cohen–Macaulay projective variety, and show that this height is small when evaluated at integral points with respect to a divisor D when D is a sum of n + 1 effective divisors which are all

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.

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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.

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|≤

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

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

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

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

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

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

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

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.

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

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.

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

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

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

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.

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

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

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.

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