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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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.

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.

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

The ring R of real-exponent polynomials in n variables over any field has global dimension n + 1 and flat dimension n. In particular, the residue field k ⁡ = R∕𝔪 of R modulo its maximal graded ideal 𝔪 has flat dimension n via a Koszul-like resolution. Projective and flat resolutions of all R-modules are constructed from this resolution of k ⁡ . The same results hold when R is replaced by the monoid

We develop machinery to explicitly determine, in many instances, when the difference x2 − yn is divisible only by powers of a given fixed prime. This combines a wide variety of techniques from Diophantine approximation (bounds for linear forms in logarithms, both archimedean and nonarchimedean, lattice basis reduction, methods for solving Thue–Mahler and S-unit equations, and the primitive divisor

We introduce a new class of Hodge cycles with nonreduced associated Hodge loci; we call them fake linear cycles. We characterize them for all Fermat varieties and show that they exist only for degrees d = 3,4,6, where there are infinitely many in the space of Hodge cycles. These cycles are pathological in the sense that the Zariski tangent space of their associated Hodge locus is of maximal dimension

Let F be a nonarchimedean local field and G the F-points of a connected simply connected reductive group over F. We study the unipotent ℓ-blocks of G, for ℓ≠p. To that end, we introduce the notion of (d,1)-series for finite reductive groups. These series form a partition of the irreducible representations and are defined using Harish-Chandra theory and d-Harish-Chandra theory. The ℓ-blocks are then

We prove a characteristic p version of a theorem of Silverman on integral points in orbits over number fields and establish a primitive prime divisor theorem for polynomials in this setting. In characteristic p, the Thue–Siegel–Dyson–Roth theorem is false, so the proof requires new techniques from those used by Silverman. The problem is largely that isotriviality can arise in subtle ways, and we define

We classify additive operations in connective K-theory with various torsion-free coefficients. We discover that the answer for the integral case requires understanding of the ℤ^ case. Moreover, although integral additive operations are topologically generated by Adams operations, these are not reduced to infinite linear combinations of the latter ones. We describe a topological basis for stable operations

We establish structure results for Frobenius kernels of automorphism group schemes for surfaces of general type in positive characteristic. It turns out that there are surprisingly few possibilities. This relies on properties of the famous Witt algebra, which is a simple Lie algebra without finite-dimensional counterpart over the complex numbers, together with its twisted forms. The result actually

Let X ⊂ ℙn+e be any n-dimensional closed subscheme. We are mainly interested in two notions related to syzygies: one is the property Nd,p(d ≥ 2,p ≥ 1), which means that X is d-regular up to p-th step in the minimal free resolution and the other is a new notion ND ⁡ (ℓ) which generalizes the classical “being nondegenerate” to the condition that requires a general finite linear section not to be contained

We prove a spectral reciprocity formula for automorphic forms on GL ⁡ (2) over a number field that is reminiscent of one found by Blomer and Khan. Our approach uses period representations of L-functions and the language of automorphic representations.

We prove that a smooth complete intersection of two quadrics of dimension at least 2 over a number field has index dividing 2, i.e., that it possesses a rational 0-cycle of degree 2.