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

Let G be a bridgeless cubic graph. In 2023, the three authors solved a conjecture (also known as the \(S_4\)-Conjecture) made by Mazzuoccolo in 2013: there exist two perfect matchings of G such that the complement of their union is a bipartite subgraph of G. They actually show that given any \(1^+\)-factor F (a spanning subgraph of G such that its vertices have degree at least 1) and an arbitrary edge

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

Given a residually connected incidence geometry \(\Gamma \) that satisfies two conditions, denoted \((B_1)\) and \((B_2)\), we construct a new geometry \(H(\Gamma )\) with properties similar to those of \(\Gamma \). This new geometry \(H(\Gamma )\) is inspired by a construction of Lefèvre-Percsy, Percsy and Leemans (Bull Belg Math Soc Simon Stevin 7(4):583–610, 2000). We show how \(H(\Gamma )\) relates

For an odd integer \(n = 2d-1\), let \({\mathcal {B}}_d\) be the subgraph of the hypercube \(Q_n\) induced by the two largest layers. In this paper, we describe the typical structure of proper q-colorings of \(V({\mathcal {B}}_d)\) and give asymptotics on the number of such colorings when q is an even number. The proofs use various tools including information theory (entropy), Sapozhenko’s graph container

A spherical L-code, where \(L \subseteq [-1,\infty )\), consists of unit vectors in \(\mathbb {R}^d\) whose pairwise inner products are contained in L. Determining the maximum cardinality \(N_L(d)\) of an L-code in \(\mathbb {R}^d\) is a fundamental question in discrete geometry and has been extensively investigated for various choices of L. Our understanding in high dimensions is generally quite poor

A permutation \(\pi \in \mathbb {S}_n\) is k-balanced if every permutation of order k occurs in \(\pi \) equally often, through order-isomorphism. In this paper, we explicitly construct k-balanced permutations for \(k \le 3\), and every n that satisfies the necessary divisibility conditions. In contrast, we prove that for \(k \ge 4\), no such permutations exist. In fact, we show that in the case \(k

Abstract polytopes are combinatorial objects that generalise geometric objects such as convex polytopes, maps on surfaces and tilings of the space. Chiral polytopes are those abstract polytopes that admit full combinatorial rotational symmetry but do not admit reflections. In this paper we build chiral polytopes whose facets (maximal faces) are isomorphic to a prescribed regular cubic tessellation

We determine the excluded minors characterising the class of countable graphs that embed into some compact surface.

In this paper, we prove that the Fourier entropy of an n-dimensional boolean function f can be upper-bounded by \(O(I(f)+ \sum \limits _{k\in [n]}I_k(f)\log \frac{1}{I_k(f)})\), where I(f) is its total influence and \(I_k(f)\) is the influence of the k-th coordinate. There is no strict quantitative relationship between our bound with the known bounds for the Fourier-Min-Entropy-Influence conjecture

A subset \(\mathcal {C}\subseteq \{0,1,2\}^n\) is said to be a trifferent code (of block length n) if for every three distinct codewords \(x,y, z \in \mathcal {C}\), there is a coordinate \(i\in \{1,2,\ldots ,n\}\) where they all differ, that is, \(\{x(i),y(i),z(i)\}\) is same as \(\{0,1,2\}\). Let T(n) denote the size of the largest trifferent code of block length n. Understanding the asymptotic behavior

A majority edge-coloring of a graph without pendant edges is a coloring of its edges such that, for every vertex v and every color \(\alpha \), there are at most as many edges incident to v colored with \(\alpha \) as with all other colors. We extend some known results for finite graphs to infinite graphs, also in the list setting. In particular, we prove that every infinite graph without pendant edges

In this paper we give anticoncentration bounds for sums of independent random vectors in finite-dimensional vector spaces. In particular, we asymptotically establish a conjecture of Leader and Radcliffe (SIAM J Discrete Math 7:90–101, 1994) and a question of Jones (SIAM J Appl Math 34:1–6, 1978). The highlight of this work is an application of the strong perfect graph theorem by Chudnovsky et al. (Ann

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

For all integers \(n \ge k > d \ge 1\), let \(m_{d}(k,n)\) be the minimum integer \(D \ge 0\) such that every k-uniform n-vertex hypergraph \({\mathcal {H}}\) with minimum d-degree \(\delta _{d}({\mathcal {H}})\) at least D has an optimal matching. For every fixed integer \(k \ge 3\), we show that for \(n \in k \mathbb {N}\) and \(p = \Omega (n^{-k+1} \log n)\), if \({\mathcal {H}}\) is an n-vertex

A storage code on a graph G is a set of assignments of symbols to the vertices such that every vertex can recover its value by looking at its neighbors. We consider the question of constructing large-size storage codes on triangle-free graphs constructed as coset graphs of binary linear codes. Previously it was shown that there are infinite families of binary storage codes on coset graphs with rate

In the first part of this paper we study the coaction dual to the action of the cosmic Galois group on the motivic lift of the sunrise Feynman integral with generic masses and momenta, and we express its conjugates in terms of motivic lifts of Feynman integrals associated to related Feynman graphs. Only one of the conjugates of the motivic lift of the sunrise, other than itself, can be expressed in

We study vector spaces associated to a family of generalized Euler integrals. Their dimension is given by the Euler characteristic of a very affine variety. Motivated by Feynman integrals from particle physics, this has been investigated using tools from homological algebra and the theory of $D$-modules. We present an overview and uncover new relations between these approaches. We also provide new