Recently Navarro proposed a strengthening of the unsolved McKay conjecture using Galois automorphisms. We prove that the Navarro conjecture and its blockwise version hold for the alternating groups.

Let A be an abelian variety over a global function field K of characteristic p. We study the μ-invariant appearing in the Iwasawa theory of A over the unramified ℤp-extension of K. Ulmer suggests that this invariant is equal to what he calls the dimension of the Tate–Shafarevich group of A and that it is indeed the dimension of some canonically defined group scheme. Our first result is to verify his

A Drinfeld module has a 𝔭-adic Tate module not only for every finite place 𝔭 of the coefficient ring but also for 𝔭 = ∞. This was discovered by J.-K. Yu in the form of a representation of the Weil group. Following an insight of Taelman we construct the ∞-adic Tate module by means of the theory of isocrystals. This applies more generally to pure A-motives and to pure F-isocrystals of p-adic cohomology

Let H be a weak Hopf algebra that is a finitely generated module over its affine center. We show that H has finite self-injective dimension and so the Brown–Goodearl conjecture holds in this special weak Hopf setting.

We study the inequities in the distribution of Frobenius elements in Galois extensions of the rational numbers with Galois groups that are either dihedral D2n or (generalized) quaternion ℍ2n of two-power order. In the spirit of recent work of Fiorilli and Jouve (2020), we study, under natural hypotheses, some families of such extensions, in a horizontal aspect, where the degree is fixed, and in a vertical

We prove that for any proper smooth formal scheme 𝔛 over 𝒪K, where 𝒪K is the ring of integers in a complete discretely valued nonarchimedean extension K of ℚp with perfect residue field k and ramification degree e, the i-th Breuil–Kisin cohomology group and its Hodge–Tate specialization admit nice decompositions when ie < p − 1. Thanks to the comparison theorems in the recent works of Bhatt, Morrow

We develop a general method for computing integral points on modular curves, based on Baker’s inequality. As an illustration, we show that for 11 ≤ p < 101, the only integral points on the curve Xns+(p) are the CM points.

A covering system is a finite collection of arithmetic progressions whose union is the set of integers. The study of covering systems with distinct moduli was initiated by Erdős in 1950, and over the following decades numerous problems were posed regarding their properties. One particularly notorious question, due to Erdős, asks whether there exist covering systems whose moduli are distinct and all

Let A be an abelian variety over a number field F and let p be a prime. Cohen–Lenstra–Delaunay-style heuristics predict that the Tate–Shafarevich group of (As) should contain an element of order p for a positive proportion of quadratic twists As of A. We give a general method to prove instances of this conjecture by exploiting independent isogenies of A. For each prime p, there is a large class of

We investigate in a statistical fashion the smallest height of a rational point on a Fano hypersurface defined over the field of rational numbers. Along the way, we establish an average version of Manin’s conjecture about the number of rational points of bounded height on Fano varieties for the complete family of Fano hypersurfaces of fixed degree and dimension.

We show, in the large q limit, that the average size of n-Selmer groups of elliptic curves of bounded height over 𝔽q(t) is the sum of the divisors of n. As a corollary, again in the large q limit, we deduce that 100% of elliptic curves of bounded height over 𝔽q(t) have rank 0 or 1.

We show that if a rational map is constant on each isomorphism class of unpolarized abelian varieties of a given dimension, then it is a constant map. Our results shed light on a question raised by Boneh et al. (J. Math. Cryptol. 14:1 (2020), 5–14) concerning a proposal for multiparty noninteractive key exchange.

Let k be a field of positive characteristic. We prove that the only linear relations between the Hodge numbers hi,j(X) = dimHj(X,ΩXi) that hold for every smooth proper variety X over k are the ones given by Serre duality. We also show that the only linear combinations of Hodge numbers that are birational invariants of X are given by the span of the hi,0(X) and the h0,j(X) (and their duals hi,n(X) and

It is known that if p > 37 is a prime number and E∕ℚ is an elliptic curve without complex multiplication, then the image of the mod p Galois representation ρ̄E,p : Gal(ℚ¯∕ℚ) → GL(E[p]) of E is either the whole of GL(E[p]), or is contained in the normaliser of a nonsplit Cartan subgroup of GL(E[p]). In this paper, we show that when p > 1.4 × 107, the image of ρ̄E,p is either GL(E[p]), or the full normaliser

Given a finite dimensional Lie algebra L, let I be the augmentation ideal in the universal enveloping algebra U(L). We study the conditions on L under which the Ext-groups Ext(k,k) for the trivial L-module k are the same when computed in the category of all U(L)-modules or in the category of I-torsion U(L)-modules. An application to cohomology of equivariant sheaves is given.

We show that the non-Archimedean skeleton of the Prym variety associated to an unramified double cover of an algebraic curve is naturally isomorphic (as a principally polarized tropical abelian variety) to the tropical Prym variety of the associated tropical double cover. This confirms a conjecture by Jensen and the first author. We prove a new upper bound on the dimension of the Prym–Brill–Noether

We use the Taylor–Wiles–Kisin patching method to prove some new cases of the Breuil–Schneider conjecture.

We produce a Grothendieck transformation from bivariant operational K-theory to Chow, with a Riemann–Roch formula that generalizes classical Grothendieck–Verdier–Riemann–Roch. We also produce Grothendieck transformations and Riemann–Roch formulas that generalize the classical Adams–Riemann–Roch and equivariant localization theorems. As applications, we exhibit a projective toric variety X whose equivariant

We use the Taylor–Wiles–Kisin patching method to investigate the multiplicities with which Galois representations occur in the mod ℓ cohomology of Shimura curves over totally real number fields. Our method relies on explicit computations of local deformation rings done by Shotton, which we use to compute the Weil class group of various deformation rings. Exploiting the natural self-duality of the cohomology

We investigate the birational section conjecture for curves over function fields of characteristic zero and prove that the conjecture holds over finitely generated fields over ℚ if it holds over number fields.

Using the universal torsor method due to Salberger, we study the approximation of a general fixed point by rational points on split toric varieties. We prove that under certain geometric hypothesis the best approximations (in the sense of McKinnon and Roth’s work) can be achieved on rational curves passing through the fixed point of minimal degree, confirming a conjecture of McKinnon. These curves

In this paper, we study the geometry of GIT configurations of n ordered points on ℙ1 both from the birational and the biregular viewpoint. In particular, we prove that any extremal ray of the Mori cone of effective curves of the quotient (ℙ1)n//  PGL(2), taken with the symmetric polarization, is generated by a one dimensional boundary stratum of the moduli space. Furthermore, we develop some technical

We show the boundedness of B-pluricanonical representa- tions of lc log Calabi–Yau pairs in dimension 2. As applica- tions, we prove the boundedness of indices of slc log Calabi–Yau pairs up to dimension 3 and that of nonklt lc log Calabi–Yau pairs in dimension 4.

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.