We establish the Sato–Tate equidistribution of Hecke eigenvalues of the family of Hecke–Maass cusp forms on SL(n, ℤ)∖SL(n, ℝ)∕SO(n). As part of the proof, we establish a uniform upper-bound for spherical functions on semisimple Lie groups which is of independent interest. For each of the principal, symmetric square and exterior square L-functions, we deduce the level distribution with restricted support

Let C∕K be a smooth plane quartic over a discrete valuation field. We characterize the type of reduction (i.e., smooth plane quartic, hyperelliptic genus 3 curve or bad) over K in terms of the existence of a special plane quartic model and, over K¯, in terms of the valuations of certain algebraic invariants of C when the characteristic of the residue field is not 2,3,5 or 7. On the way, we gather several

We study certain mod p differential operators that act on automorphic forms over Shimura varieties of type A or C. We show that, over the ordinary locus, these operators agree with the mod p reduction of the p-adic theta operators previously studied by some of the authors. In the characteristic 0, p-adic case, there is an obstruction that makes it impossible to extend the theta operators to the whole

We prove that if X is a Grassmannian of type A, then the Schubert basis of the (small) quantum cohomology ring QH(X) is the only homogeneous deformation of the Schubert basis of the ordinary cohomology ring H∗(X) that multiplies with nonnegative structure constants. This implies that the (three point, genus zero) Gromov–Witten invariants of X are uniquely determined by Witten’s presentation of QH(X)

We prove a purely local form of a result of Saito and Yatagawa. They proved that the characteristic cycle of a constructible étale sheaf is determined by wild ramification of the sheaf along the boundary of a compactification. But they had to consider ramification at all the points of the compactification. We give a pointwise result, that is, we prove that the characteristic cycle of a constructible

We prove a certain transcendence property of the unipotent Albanese map of a smooth variety, conditional on the Ax–Schanuel conjecture for variations of mixed Hodge structure. We show that this property allows the Chabauty–Kim method to be generalized to higher-dimensional varieties. In particular, we conditionally generalize several of the main Diophantine finiteness results in Chabauty–Kim theory

We generalize to n-torsion a result of Kempf’s describing 2-torsion points lying on a theta divisor. This is accomplished by means of certain semihomogeneous vector bundles introduced and studied by Mukai and Oprea. As an application, we prove a sharp upper bound for the number of n-torsion points lying on a theta divisor and show that this is achieved only in the case of products of elliptic curves

Noncommutative near-group fusion categories were completely classified in the previous work of the first author by using an operator algebraic method (and hence under the assumption of unitarity), and they were shown to be group theoretical though the corresponding pointed categories were not identified. In this note we give a purely algebraic construction of the noncommutative near-group fusion categories

We show that the shape of a complex cubic field lies on the geodesic of the modular surface defined by the field’s trace-zero form. We also prove a general such statement for all orders in étale Q-algebras. Applying a method of Manjul Bhargava and Piper H to results of Bhargava and Ariel Shnidman, we prove that the shapes lying on a fixed geodesic become equidistributed with respect to the hyperbolic

We completely classify all subbundles of a given vector bundle on the Fargues–Fontaine curve, where by a subbundle we mean a locally free subsheaf. Our classification is given in terms of a simple and explicit condition on Harder–Narasimhan polygons. Our proof is inspired by the proof of the main theorem in our previous work (Hong 2019), but also involves a number of nontrivial adjustments.

We investigate cohomological support varieties for finite-dimensional Lie superalgebras defined over fields of odd characteristic. Verifying a conjecture from our previous work, we show the support variety of a finite-dimensional supermodule can be realized as an explicit subset of the odd nullcone of the underlying Lie superalgebra. We also show the support variety of a finite-dimensional supermodule

Let F be a finite extension of ℚp. Let W(k) denote the Witt vectors of an algebraically closed field k of characteristic ℓ different from p, and let 𝒵 be the spherical Hecke algebra for GLn(F) over W(k). Given a Hecke character λ : 𝒵→ R, where R is an arbitrary W(k)-algebra, we introduce the universal unramified module ℳλ,R. We show ℳλ,R embeds in its Whittaker space and is flat over R, resolving

We consider a special theta lift 𝜃(f) from cuspidal Siegel modular forms f on Sp4 to “modular forms” 𝜃(f) on SO(4,4) in the sense of our prior work (Pollack 2020a). This lift can be considered an analogue of the Saito–Kurokawa lift, where now the image of the lift is representations of SO(4,4) that are quaternionic at infinity. We relate the Fourier coefficients of 𝜃(f) to those of f, and in particular

The aim of this paper is to study some modular contractions of the moduli space of stable pointed curves M¯g,n. These new moduli spaces, which are modular compactifications of Mg,n, are related to the minimal model program for M¯g,n and have been introduced by Codogni et al. (2018). We interpret them as log canonical models of adjoint divisors and we then describe the Shokurov decomposition of a region

We consider some additive properties of integers with restricted digit expansions. Let b ≥ 3 be an integer and Bb be the set of integers whose base b expansions have only digits {0,1}. Let a,b,c be three integers greater than 2. We give some estimates on the size of (Ba + Bb) ∩ Bc. In particular, under mild conditions, (Ba + Bb) ∩ Bc is a very thin set in the sense that for each 𝜖 > 0, as N →∞, #((Ba

Let p and q be multiplicatively independent natural numbers, and K the field ℂ(x1∕s∣s = 1,2,3…). Let p and q act on K as the Mahler operators x↦xp and x↦xq. Schäfke and Singer (2019) showed that a finite-dimensional vector space over K, carrying commuting structures of a p-Mahler module and a q-Mahler module, is obtained via base change from a similar object over ℂ. As a corollary, they gave a new

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 study the Diophantine geometry of algebraic curves on relative moduli of special linear rank two local systems over surfaces. We prove that the set of integral points on any nondegenerately embedded algebraic curve can be effectively determined. Under natural hypotheses on the embedding in relation to mapping class group dynamics of the moduli space, the set of all imaginary quadratic integral points

We describe two conjectures, one strictly stronger than the other, that give descriptions of the integral Bernstein center for GL_n(F) (that is, the center of the category of smooth W(k)[GL_n(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 at most n)

In this paper we realize the moduli spaces of cubic fourfolds with specified automorphism groups as arithmetic quotients of complex hyperbolic balls or type IV symmetric domains, and study their compactifications. Our results mainly depend on the well-known works about moduli space of cubic fourfolds, including the global Torelli theorem proved by Voisin ([Voi86]) and the characterization of the image

We establish a short exact sequence about depth-graded motivic double zeta values of even weight relative to $\mu_2$. And we find a basis for the depth-graded motivic double zeta values relative to $\mu_2$ of even weight and a basis for the depth-graded motivic triple zeta values relative to $\mu_2$ of odd weight. As an application of our main results, we prove part of Kaneko and Tasaka's conjectures

Let $F$ be the maximal totally real subfield of $\mathbf{Q}(\zeta_{32})$, 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 $\mathcal{O}$ be a maximal order in $D$, and $X_0^D(1)$ the Shimura curve attached to $\mathcal{O}$. Let $C = X_0^D(1)/\langle w_D \rangle$, where $w_D$

For a new class of Shimura varieties of orthogonal type over a totally real number field, we construct special cycles and show the the modularity of Kudla's generating series in the cohomology group.

Let $k$ be a perfect field of characteristic $p > 2$, and let $K$ be a finite totally ramified extension over $W(k)[\frac{1}{p}]$ of ramification degree $e$. Let $R_0$ be a relative base ring over $W(k)\langle t_1^{\pm 1}, \ldots, t_m^{\pm 1}\rangle$ satisfying some mild conditions, and let $R = R_0\otimes_{W(k)}\mathcal{O}_K$. We show that if $e < p-1$, then every crystalline representation of $\

We consider two group actions on $m$-tuples of $n \times n$ matrices. The first is simultaneous conjugation by $\operatorname{GL}_n$ and the second is the left-right action of $\operatorname{SL}_n \times \operatorname{SL}_n$. We give efficient algorithms to decide if the orbit closures of two points intersect. We also improve the known bounds for the degree of separating invariants in these cases.

We compute the Brauer group of the moduli stack of elliptic curves over the integers, localizations of the integers, 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 of curves with full (naive) level $2$ structure, the study of the descent spectral sequence

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 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 $\mathcal{X}(\mathbb{Z}_p)_2$ containing the integral points $\mathcal{X}(\mathbb{Z})$ of an elliptic curve of rank at most $1$. Motivated by a conjecture of Kim, we then investigate