We introduce a modification of the linear sieve whose weights satisfy strong factorization properties, and consequently equidistribute primes up to size x in arithmetic progressions to moduli up to x10∕17. This surpasses the level of distribution x4∕7 with the linear sieve weights from well-known work of Bombieri, Friedlander, and Iwaniec, and which was recently extended to x7∕12 by Maynard. As an

Let A be an abelian variety over a number field F, and suppose that ℤ[ζn] embeds in End ⁡ F¯A, for some root of unity ζn of order n = 3m. Assuming that the Galois action on the finite group A[1 − ζn] is sufficiently reducible, we bound the average rank of the Mordell–Weil groups Ad(F), as Ad varies through the family of μ2n-twists of A. Combining this result with the recently proved uniform Mordell–Lang

Let X be a K3 surface over a number field. We prove that X has infinitely many specializations where its Picard rank jumps, hence extending our previous work with Shankar, Shankar and Tang to the case where X has bad reduction. We prove a similar result for generically ordinary nonisotrivial families of K3 surfaces over curves over 𝔽¯p which extends previous work of Maulik, Shankar and Tang. As a

Let K be a number field, and S a finite set of nonarchimedean places of K, and write 𝒪× for the group of S-units of K. A famous theorem of Siegel asserts that the S-unit equation 𝜀 + δ = 1, with 𝜀, δ ∈𝒪×, has only finitely many solutions. A famous theorem of Shafarevich asserts that there are only finitely many isomorphism classes of elliptic curves over K with good reduction outside S. Now instead

We prove vanishing results for the coherent cohomology of the good reduction modulo p of the Siegel modular variety with coefficients in some automorphic bundles. We show that for an automorphic bundle with highest weight λ near the walls of the antidominant Weyl chamber, there is an integer e ≥ 0 such that the cohomology is concentrated in degrees [0,e]. The accessible weights with our method are

Let E be a field of characteristic p. In a previous paper of ours, we defined and studied super-Hölder vectors in certain E-linear representations of ℤp. In the present paper, we define and study super-Hölder vectors in certain E-linear representations of a general p-adic Lie group. We then consider certain p-adic Lie extensions K∞∕K of a p-adic field K, and compute the super-Hölder vectors in the

We study the variation of linear sections of hypersurfaces in ℙn. We completely classify all plane curves, necessarily singular, whose line sections do not vary maximally in moduli. In higher dimensions, we prove that the family of hyperplane sections of any smooth degree d hypersurface in ℙn varies maximally for d ≥ n + 3. In the process, we generalize the classical Grauert–Mülich theorem about lines

We describe separating G2-invariants of several copies of the algebra of octonions over an algebraically closed field of characteristic two. We also obtain a minimal separating and a minimal generating set for G2-invariants of several copies of the algebra of octonions in case of a field of odd characteristic.

We construct scattering diagrams for Chekhov–Shapiro generalized cluster algebras where exchange polynomials are factorized into binomials, generalizing the cluster scattering diagrams of Gross, Hacking, Keel and Kontsevich. They turn out to be natural objects arising in Fock and Goncharov’s cluster duality. Analogous features and structures (such as positivity and the cluster complex structure) in

We study a family of exponential sums that arises in the study of expanding horospheres on GL ⁡ n. We prove an explicit version of general purity and find optimal bounds for these sums.

We prove that the Galois equidistribution of torsion points of the algebraic torus 𝔾md extends to the singular test functions of the form log ⁡ |P|, where P is a Laurent polynomial having algebraic coefficients that vanishes on the unit real d-torus in a set whose Zariski closure in 𝔾md has codimension at least 2. Our result includes a power-saving quantitative estimate of the decay rate of the equidistribution

We show the image of rooted tree maps forms a subspace of the kernel of the evaluation map of multiple L-values. To prove this, we define the diamond product as a modified harmonic product and describe its properties. We also show that τ-conjugate rooted tree maps are their antipodes.

The local structure of terminal Brauer classes on arithmetic surfaces was classified (2021), generalising the classification on geometric surfaces (2005). Part of the interest in these classifications is that it enables the minimal model program to be applied to the noncommutative setting of orders on surfaces. We give étale local structure theorems for terminal orders on arithmetic surfaces, at least

Fix a finite field K of order q and a word w in a free group F on r generators. A w-random element in GL ⁡ N(K) is obtained by sampling r independent uniformly random elements g1,… ⁡,gr ∈ GL ⁡ N(K) and evaluating w(g1,… ⁡,gr). Consider 𝔼w[fix ⁡ ], the average number of vectors in KN fixed by a w-random element. We show that 𝔼w[fix ⁡ ] is a rational function in qN. If w = ud with u a nonpower, then

We investigate the distribution of the maximum of character sums over the family of primitive quadratic characters attached to fundamental discriminants |d|≤ x. In particular, our work improves results of Montgomery and Vaughan, and gives strong evidence that the Omega result of Bateman and Chowla for quadratic character sums is optimal. We also obtain similar results for real characters with prime

We explain how logarithmic structures select principal components in an intersection of schemes. These manifest in Chow homology and can be understood using strict transforms under logarithmic blowups. Our motivation comes from Gromov–Witten theory. The toric contact cycles in the moduli space of curves parameterize curves that admit a map to a fixed toric variety with prescribed contact orders. We

Spectral moment formulae of various shapes have proven very successful in studying the statistics of central L-values. We establish, in a completely explicit fashion, such formulae for the family of GL ⁡ (3) × GL ⁡ (2) Rankin–Selberg L-functions using the period integral method. Our argument does not rely on either the Kuznetsov or Voronoi formulae. We also prove the essential analytic properties and

We prove that the double (or canonical unramified) wavefront set of an irreducible depth-0 supercuspidal representation of a reductive p-adic group is a singleton provided p > 3(h − 1), where h is the Coxeter number. We deduce that the geometric wavefront set is also a singleton in this case, proving a conjecture of Mœglin and Waldspurger. When the group is inner to split and the representation belongs

We associate with a generalised deep hole of the Leech lattice vertex operator algebra a generalised hole diagram. We show that this Dynkin diagram determines the generalised deep hole up to conjugacy and that there are exactly 70 such diagrams. In an earlier work we proved a bijection between the generalised deep holes and the strongly rational, holomorphic vertex operator algebras of central charge

Given a smooth projective toric variety X of Picard rank 2, we resolve the diagonal sheaf on X × X by a linear complex of length dim ⁡ X consisting of finite direct sums of line bundles. As applications, we prove a new case of a conjecture of Berkesch, Ermana and Smith that predicts a version of Hilbert’s syzygy theorem for virtual resolutions, and we obtain a Horrocks-type splitting criterion for

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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.

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

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

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 −

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

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

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

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

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

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

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

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

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.

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.

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

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

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

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.

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

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

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

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