Montgomery and Soundararajan showed that the distribution of ψ(x + H) − ψ(x), for 0 ≤ x ≤ N, is approximately normal with mean ∼ H and variance ∼ Hlog ⁡ (N∕H), when Nδ ≤ H ≤ N1−δ . Their work depends on showing that sums Rk(h) of k-term singular series are μk(−hlog ⁡ h + Ah)k∕2 + Ok(hk∕2−1∕(7k)+𝜀), where A is a constant and μk are the Gaussian moment constants. We study lower-order terms in the size

A Thue–Mahler equation is a Diophantine equation of the form F(X,Y ) = a ⋅ p1z1 ⋯pvzv ,gcd ⁡ (X,Y ) = 1 where F is an irreducible binary form of degree at least 3 with integer coefficients, a is a nonzero integer and p1,… ⁡,pv are rational primes. Existing algorithms for resolving such equations require computations in the field L = ℚ(𝜃,𝜃′,𝜃′′), where 𝜃, 𝜃′, 𝜃′′ are distinct roots of F(X,1) =

We classify the automorphism groups of del Pezzo surfaces of degrees 1 and 2 over an algebraically closed field of characteristic 2. This finishes the classification of automorphism groups of del Pezzo surfaces in all characteristics.

Let Z be the germ of a complex hypersurface isolated singularity of equation f, with Z at least of dimension 2. We consider the family of analytic D-modules generated by the powers of 1∕f and describe it in terms of the pole order filtration on the de Rham cohomology of the complement of {f = 0} in the neighbourhood of the singularity.

We investigate a generalization of the classical notion of a Schur functor associated to a ribbon diagram. These functors are defined with respect to an arbitrary algebra, and in the case that the underlying algebra is the symmetric/exterior algebra, we recover the classical definition of Schur/Weyl functors, respectively. In general, we construct a family of 3-term complexes categorifying the classical

We develop obstruction theory for lifting characteristic-p local Galois representations valued in reductive groups of type Bl, Cl, Dl or G2. An application of the Emerton–Gee stack then reduces the existence of crystalline lifts to a purely combinatorial problem when p is not too small. As a toy example, we show for all local fields K∕ℚp, with p > 3, all representations ρ¯ : GK → G2(𝔽¯p) admit a crystalline

In this paper, we begin the study of the Fermat equation xn + yn = zn over real biquadratic fields. In particular, we prove that there are no nontrivial solutions to the Fermat equation over ℚ(2,3) for n ≥ 4.

We find lower bounds on higher moments of the error term in the Chebotarev density theorem. Inspired by the work of Bellaïche, we consider general class functions and prove bounds which depend on norms associated to these functions. Our bounds also involve the ramification and Galois theoretical information of the underlying extension L∕K. Under a natural condition on class functions (which appeared

Over a finite field 𝔽q, abelian varieties with commutative endomorphism rings can be described by using modules over orders in étale algebras. By exploiting this connection, we produce four theorems regarding groups of rational points and self-duality, along with explicit examples. First, when End ⁡ (A) is locally Gorenstein, we show that the group structure of A(𝔽q) is determined by End ⁡ (A). In

We study instances of Beilinson–Tate conjectures for automorphic representations of PGSp ⁡ 6 whose spin L-function has a pole at s = 1. We construct algebraic cycles of codimension 3 in the Siegel–Shimura variety of dimension 6 and we relate its regulator to the residue at s = 1 of the L-function of certain cuspidal forms of PGSp ⁡ 6. Using the exceptional theta correspondence between the split group

We formulate an analogue of the Breuil–Mézard conjecture for the group of units of a central division algebra over a p-adic local field, and we prove that it follows from the conjecture for GL ⁡ n. To do so we construct a transfer of inertial types and Serre weights between the maximal compact subgroups of these two groups, in terms of Deligne–Lusztig theory, and we prove its compatibility with mod

We prove the Pappas–Rapoport conjecture on the existence of canonical integral models of Shimura varieties with parahoric level structure in the case where the Shimura variety is defined by a torus. As an important ingredient, we show, using the Bhatt–Scholze theory of prismatic F-crystals, that there is a fully faithful functor from 𝒢-valued crystalline representations of Gal ⁡ (K¯∕K) to 𝒢-shtukas

We establish a geometrization of the Breuil–Mézard conjecture for potentially Barsotti–Tate representations, as well as of the weight part of Serre’s conjecture, for moduli stacks of two-dimensional mod p representations of the absolute Galois group of a p-adic local field. These results are first proved for the stacks of our earlier papers, and then transferred to the stacks of Emerton and Gee by

An arc space of an affine cone over a projective toric variety is known to be nonreduced in general. It was demonstrated recently that the reduced scheme structure of arc spaces is very meaningful from algebro-geometric, representation-theoretic and combinatorial points of view. In this paper we develop a general machinery for the description of the reduced arc spaces of affine cones over toric varieties

Let k be a positive integer. We show that, as n goes to infinity, almost every entry of the character table of Sn is divisible by k. This proves a conjecture of Miller.

We study the index of log Calabi–Yau pairs (X,B) of coregularity 0. We show that 2λ(KX + B) ∼ 0, where λ is the Weil index of (X,B). This is in contrast to the case of klt Calabi–Yau varieties, where the index can grow doubly exponentially with the dimension. Our sharp bound on the index extends to the context of generalized log Calabi–Yau pairs, semi-log canonical pairs, and isolated log canonical

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