Let Nn be an n×n complex random matrix, each of whose entries is an independent copy of a centered complex random variable z with finite nonzero variance σ2. The strong circular law, proved by Tao and Vu, states that almost surely, as n→∞, the empirical spectral distribution of Nn/(σn) converges to the uniform distribution on the unit disc in ℂ. A crucial ingredient in the proof of Tao and Vu, which

Let (𝕋f,𝔪f) denote the mod p local Hecke algebra attached to a normalized Hecke eigenform f, which is a commutative algebra over some finite field 𝔽q of characteristic p and with residue field 𝔽q. By a result of Carayol we know that, if the residual Galois representation ρ¯f:Gℚ→GL2(𝔽q) is absolutely irreducible, then one can attach to this algebra a Galois representation ρf:Gℚ→GL2(𝕋f) that is

We revisit Munshi’s proof of the t-aspect subconvex bound for GL(3)L-functions, and we are able to remove the “conductor lowering” trick. This simplification along with a more careful stationary phase analysis allows us to improve Munshi’s bound to L(1/2+it,π)≪π,𝜖(1+|t|)3/4−3/40+𝜖.

For x∈[0,1), let [d1(x),d2(x),…] be its Lüroth expansion and {pn(x)qn(x),n≥1} be the sequence of convergents of x. Define the sets 𝜀2(φ)={x∈[0,1):dn+1(x)dn(x)≥φ(n) for infinitely many n∈ℕ},U∗(τ)=x∈[0,1):x−pn(x)qn(x)<1qn(x)(τ+1) for n∈ℕ ultimately and F(τ)=x∈[0,1):limn→∞log(dn(x)dn+1(x))logqn(x)=τ, where φ:ℕ→[2,∞) is a positive function. In this paper, we calculate the Lebesgue measure of the set 𝜀2(φ)

Let 𝒮 be a finite cyclic semigroup written additively. An element e of 𝒮 is said to be idempotent if e+e=e. A sequence T over 𝒮 is called idempotent-sum-free provided that no idempotent of 𝒮 can be represented as a sum of one or more terms from T. We prove that an idempotent-sum-free sequence over 𝒮 of length over approximately a half of the size of 𝒮 is well structured. This result generalizes

S⊂{1,2,…,n} is called a Sidon set if a+b are all distinct for any a,b∈S. Let Sn be the largest cardinal number of such S. We are interested in the sum of elements in the Sidon set S. In this paper, we prove that for any 𝜖>0, ∑a∈Sa=12n3/2+O(n111/80+𝜖), where S⊂{1,2,…,n} is a Sidon set and |S|=Sn.

This paper provides a mean value theorem for arithmetic functions f defined by f(n)=∏d|ng(d), where g is an arithmetic function taking values in (0,1] and satisfying some generic conditions. As an application of our main result, we prove that the density μq(n) (respectively, ρq(n)) of normal (respectively, primitive) elements in the finite field extension 𝔽qn of 𝔽q are arithmetic functions of (nonzero)

Let C be a smooth plane curve of degree d≥4 defined over a global field k of characteristic p=0 or p>(d−1)(d−2)/2 (up to an extra condition on Jac(C)). Unless the curve is bielliptic of degree four, we observe that it always admits finitely many quadratic points. We further show that there are only finitely many quadratic extensions k(D) when k is a number field, in which we may have more points of

Let 𝔸[n] be the group of n-torsion points of a commutative algebraic group 𝔸 defined over a number field F. For a prime 𝔭 of F, we let N𝔭(𝔸[n]) be the number of 𝔽𝔭-solutions of the system of polynomial equations defining 𝔸[n] when reduced modulo 𝔭. Here, 𝔽𝔭 is the residue field at 𝔭. Let πF(x) denote the number of primes 𝔭 of F such that N(𝔭)≤x. We then, for algebraic groups of dimension

Let F be the maximal totally real subfield of ℚ(ζ32), 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 𝒪 be a maximal order in D, and X0D(1) the Shimura curve attached to 𝒪. Let C = X0D(1)∕⟨wD⟩, where wD is the unique Atkin–Lehner involution on X0D(1). We show that the curve C has several

Let k be a perfect field of characteristic p > 2, and let K be a finite totally ramified extension over W(k)[1 p] of ramification degree e. Let R0 be a relative base ring over W(k)⟨t1±1,…,tm±1⟩ satisfying some mild conditions, and let R = R0 ⊗W(k)𝒪K. We show that if e < p − 1, then every crystalline representation of π1e ́ t(SpecR[1 p]) with Hodge–Tate weights in [0,1] arises from a p-divisible group

In this work, we use the Rankin–Selberg method to obtain results on the analytic properties of the standard L-function attached to vector-valued Siegel modular forms. In particular we provide a detailed description of its possible poles and obtain a non-vanishing result of the twisted L-function beyond the usual range of absolute convergence. Our results include also the case of metaplectic Siegel

In this paper, we investigate the Galois conjugates of a Pisot number q∈(m,m+1), m≥1. In particular, we conjecture that for q∈(1,2) we have |q′|≥5−12 for all conjugates q′ of q. Further, for m≥3, we conjecture that for all Pisot numbers q∈(m,m+1) we have |q′|≥m+1−m2+2m−32. A similar conjecture if made for m=2. We conjecture that all of these bounds are tight. We provide partial supporting evidence

“Quasi-elliptic” functions can be given a ring structure in two different ways, using either ordinary multiplication, or convolution. The map between the corresponding standard bases is calculated. A related structure has appeared recently in the computation of Feynman integrals. The two approaches are related by a sequence of polynomials closely tied to the Eulerian polynomials.

For each prime p≡1(mod4) consider the Legendre character χ=(⋅p). Let p±(n) be the number of partitions of n into parts λ>0 such that χ(λ)=±1. Petersson proved a beautiful limit formula for the ratio of p+(n) to p−(n) as n→∞ expressed in terms of important invariants of the real quadratic field K=ℚ(p). But his proof is not illuminating and Grosswald conjectured a more natural proof using a Tauberian

We prove that there exists positive even integer k such that ϕ(n)=ϕ(n+k) holds for infinitely many n. We also prove various estimates on number of solutions to ϕ(p−1)=ϕ(q−1) for distinct primes p and q.

We investigate the arithmetic of algebraic curves on coarse moduli spaces for special linear rank two local systems on surfaces with fixed boundary traces. We prove a structure theorem for morphisms from the affine line into the moduli space. We show that the set of integral points on any nondegenerate algebraic curve on the moduli space can be effectively determined.

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

We realize the moduli spaces of cubic fourfolds with specified group actions as arithmetic quotients of complex hyperbolic balls or type IV symmetric domains, and study their compactifications. We prove the geometric ( GIT) compactifications are naturally isomorphic to the Hodge theoretic (Looijenga, in many cases Baily–Borel) compactifications. The key ingredients of the proof are the global Torelli

We establish a short exact sequence about depth-graded motivic double zeta values of even weight relative to μ2. We find a basis for the depth-graded motivic double zeta values relative to μ2 of even weight and a basis for the depth-graded motivic triple zeta values relative to μ2 of odd weight. As an application of our main results, we prove Kaneko and Tasaka’s conjectures about the sum odd double

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

We consider two group actions on m-tuples of n × n matrices with entries in the field K. The first is simultaneous conjugation by GLn and the second is the left-right action of SLn × SLn. Let K¯ be the algebraic closure of the field K. Recently, a polynomial time algorithm was found to decide whether 0 lies in the Zariski closure of the SLn( K¯) × SLn( K¯)-orbit of a given m-tuple by Garg, Gurvits

We produce nontrivial asymptotic estimates for shifted sums of the form ∑a(h)b(m)c(2m−h), in which a(n),b(n),c(n) are un-normalized Fourier coefficients of holomorphic cusp forms. These results are unconditional, but we demonstrate how to strengthen them under the Riemann Hypothesis. As an application, we show that there are infinitely many three term arithmetic progressions n−h,n,n+h such that a(n−h)a(n)a(n+h)≠0

Let dk(n) denote the k-th divisor function. In this paper, we study the asymptotic formula of the sum∑1≤n1,n2,…,nl≤xdk(n12+n22+⋯+nl2), where n1,n2,…,nl∈Z+, k≥4 and l≥3 are integers. Previously only the cases of k=2,3 are studied.

Recent works [2], [5], [13], [14] studied how Hecke operators map between subspaces of modular forms of the type Ar,s={η(δz)rF(δz):F(z)∈Ms} for suitable δ. For primes p≥5, we give results on how Up maps between subspaces of these spaces with p-integral coefficients modulo powers of p. As a corollary, we prove congruences between Shimura images of a natural family of half-integral weight eigenforms

We prove a lower bound for the number of conductors q≤Q with q≡1(modr) having t prime divisors of Dirichlet characters χ of fixed order r and with given values χ(p) for primes p≤B. We also give an elliptic curve variant of this result.

In this paper, we study the solutions to the titular Diophantine equation in integers n≥m≥0, y≥2 and a≥2. We show that there are only finitely many of them for a fixed y, and we provide a bound on the largest such solution. As an application, we find all the solutions when y∈[2,1000]. We also show that the abc-conjecture implies that there are only finitely many integer solutions (n,m,y,a) with min⁡{y

We establish arithmetic duality theorems for short complexes of tori associated to reductive groups over p-adic function fields. Using arithmetic dualities, we deduce obstructions to weak approximation for certain reductive groups (especially quasi-split ones) and relate this obstruction to an unramified Galois cohomology group.

The Oesterlé bound shows that a curve of genus 8 over the finite field F4 can have at most 24 rational points, and Niederreiter and Xing used class field theory to show that there exists such a curve with 21 points. We improve both of these results: We show that a genus-8 curve over F4 can have at most 23 rational points, and we provide an example of such a curve with 22 points, namely the curve defined

We classify the possible torsion structures of rational elliptic curves over sextic number fields. Among these possible torsion group structures, all groups except C3⊕C18 are known to appear as subgroups of E(K)tors for some elliptic curve E/Q and for some sextic number field K. We prove that if the image of mod 2 Galois representation of E is not equal to the Borel subgroup of GL2(Z/2Z), then E(K)tors

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 𝒳(ℤp)2 containing the integral points 𝒳(ℤ) of an elliptic curve of rank at most 1. Motivated by a conjecture of Kim, we then investigate theoretically and computationally the set-theoretic

Let u be a nowhere vanishing holomorphic function on the Drinfeld space Ωr of dimension r − 1, where r ≥ 2. The logarithm logq|u| of its absolute value may be regarded as an affine function on the attached Bruhat–Tits building ℬ𝒯r. Generalizing a construction of van der Put in case r = 2, we relate the group 𝒪(Ωr)∗ of such u with the group H(ℬ𝒯r, ℤ) of integer-valued harmonic 1-cochains on ℬ𝒯r

We give an explicit recipe for determining iterated local cohomology groups with support in ideals of minors of a generic matrix in characteristic zero, expressing them as direct sums of indecomposable 𝒟-modules. For nonsquare matrices these indecomposables are simple, but this is no longer true for square matrices where the relevant indecomposables arise from the pole order filtration associated

Using a modular equation of level 3 and degree 23 due to Chan and Liaw, we prove the fastest known (conjectured to be the fastest one) convergent rational Ramanujan-type series for 1/π of level 3.

Based on the general strategy described by Borel and Serre and the Voronoi algorithm for computing unit groups of orders we present an algorithm for finding presentations of S-unit groups of orders. The algorithm is then used for some investigations concerning the congruence subgroup property.

We show that affine subspaces of Euclidean space are of Khintchine type for divergence under certain multiplicative Diophantine conditions on the parameterizing matrix. This provides evidence towards the conjecture that all affine subspaces of Euclidean space are of Khintchine type for divergence, or that Khintchine’s theorem still holds when restricted to the subspace. This result is proved as a special

We introduce combinatorial interpretations of the coefficients of two second-order mock theta functions. Then, we provide a bijection that relates the two combinatorial interpretations for each function. By studying other special cases of the multivariate identity proved by the bijection, we obtain new combinatorial interpretations for the coefficients of Watson’s third-order mock theta function ω(q)

For β>1, a sequence (cn)n≥1∈ℤℕ+ with 0≤cn<β is the beta expansion of x with respect to β if x=∑n=1∞cnβ−n. Defining dβ(x) to be the greedy beta expansion of x with respect to β, it is known that dβ(1) is eventually periodic as long as β is a Pisot number. It is conjectured that the same is true for Salem numbers, but is only currently known to be true for Salem numbers of degree 4. Heuristic arguments

We provide a generalization and new proofs of the formulas of Collins et al. for the spectrum of polynomials in cyclic monotone elements. This is applied to Random Matrices with discrete spectrum.

We discuss the process convergence of the time dependent fluctuations of linear eigenvalue statistics of random circulant matrices with independent Brownian motion entries, as the dimension of the matrix tends to ∞. Our derivation is based on the trace formula of circulant matrix, method of moments and some combinatorial techniques.

Erdős proved that F(A):=∑a∈A1alog⁡a converges for any primitive set of integers A and later conjectured this sum is maximized when A is the set of primes. Banks and Martin further conjectured that F(P1)>⋯>F(Pk)>F(Pk+1)>⋯, where Pj is the set of integers with j prime factors counting multiplicity, though this was recently disproven by Lichtman. We consider the corresponding problems over the function

For an algebraic number field K and a prime number p, let K̃/K be the maximal multiple ℤp-extension. Greenberg’s generalized conjecture (GGC) predicts that the Galois group of the maximal unramified abelian pro-p extension of K̃ is pseudo-null over the completed group ring ℤp[[Gal(K̃/K)]]. We show that GGC holds for some imaginary quartic fields containing imaginary quadratic fields and some prime

Long and Ramakrishna [Some supercongruences occurring in truncated hyper- geometric series, Adv. Math.290 (2016) 773–808] generalized the (H.2) supercongruence of Van Hamme to the modulus p3 case. In this paper, we give a q-analogue of Long and Ramakrishna’s result for p≡3(mod4). A q-congruence modulo the fourth power of a cyclotomic polynomial, which is a deeper q-analogue of the (A.2) supercongruence

We construct a normal projective rigid analytic compactification of an arbitrary Drinfeld modular variety whose boundary is stratified by modular varieties of smaller dimensions. This generalizes work of Kapranov. Using an algebraic modular compactification that generalizes Pink and Schieder's, we show that the analytic compactification is naturally isomorphic to the analytification of Pink's normal

The inequality between rank and crank moments was conjectured and later proved by Garvan himself in 2011. Recently, Dixit and the authors introduced finite analogues of rank and crank moments for vector partitions while deriving a finite analogue of Andrews’ famous identity for smallest parts function. In the same paper, they also conjectured an inequality between finite analogues of rank and crank

We prove a new mock theta function identity related to the partition rank modulo 3 and 9. As a consequence, we obtain the 3-dissection of the rank generating function modulo 9. We also evaluate all of the components of the rank–crank differences modulo 9. These are analogous to conjectures of Lewis [The generating functions of the rank and crank modulo 8, Ramanujan J.18 (2009) 121–146] on rank–crank

We prove that the equation (x−y)4+x4+(x+y)4=zn has no integer solutions x,y,z with gcd(x,y)=1 for all integers n>1. We mainly use a modular approach with two Frey ℚ-curves defined over the field ℚ(30).

Let F be a non-Archimedean local field of characteristic zero. Let X be the p-adic symmetric space X=H\G, where G=GL2n(F) and H=GLn(F)×GLn(F). We verify a conjecture of Sakellaridis and Venkatesh on the Langlands parameters of certain representations in the discrete spectrum of X.

Let K be a number field, and let α1,…,αr be elements of K× which generate a subgroup of K× of rank r. Consider the cyclotomic-Kummer extensions of K given by K(ζn,α1n1,…,αrnr), where ni divides n for all i. There is an integer x such that these extensions have maximal degree over K(ζg,α1g1,…,αrgr), where g=gcd(n,x) and gi=gcd(ni,x). We prove that the constant x is computable. This result reduces to

In this work, two space-time fractional nonlinear systems named Drinfeld–Sokolov–Satsuma–Hirota system and Broer Kaup system are considered with fractional derivatives in Riemann–Liouville type. The symmetry approach and power series expansion technique are applied to derive the explicit solutions of both the systems. In addition, the nontrivial conserved vectors are reported using the nonlinear self-adjointness

Topological structure of translation-invariant noncommutative Yang–Mills theories are studied by means of a cohomology theory, the so-called ⋆-cohomology, which plays an intermediate role between de Rham and cyclic (co)homology theory for noncommutative algebras and gives rise to a cohomological formulation comparable to Seiberg–Witten map.

We study hypergraphs which represent finite quantum event structures. We contribute to results of graph theory, regarding bounds on the number of edges, given the number of vertices. We develop a missing one for 3-graphs of girth 4. As an application of the graph-theoretical approach to quantum structures, we show that the smallest orthoalgebra with an empty state space has 10 atoms. Optimized constructions

The geometric phase that appears in the effects of Aharonov–Bohm type is interpreted in the frame of Deligne's version of the Riemann–Hilbert correspondence. We extend also the concept of flat gauge field to B-branes on a complex manifold X, so that such a field on a B-brane turns it into an object of the category of constructible sheaves on X.

In this paper we start the investigation of a metric-related weakening of positive expansiveness. We will show some of its fundamental properties and relation to notions such as weak positive expansiveness and cover expansiveness. Finally, we will see how to use the Bing–Hanner modification to produce examples of positively subexpansive dynamical systems with nonmetrizable phase space.

We prove a local in time existence theorem to a Cauchy problem for the coupled Yang–Mills–Boltzmann system in Bianchi type 1 space-time background.

In this work, we define a spinor covariant derivative for degenerate manifolds with 4-dimensions. To perform this, we have found the principal bundle by using a degenerate spin group. Then, we benefit from a covering map to establish a relationship between the local connection forms of principal bundles. After that, we define a covariant derivative on a degenerate spinor bundle which is an associated

In this study, nonlocal phenomena in quantum mechanics are investigated by making use of fractional calculus. in this context, fractional creation and annihilation operators are introduced and quantum mechanical harmonic oscillator has been generalized as an important tool in quantum field theory. Therefore wave functions and energy eigenvalues of harmonic oscillator are obtained with respect to the

Given a prime power q and an integer n≥2, we establish a sufficient condition for the existence of a primitive pair (α,f(α)) where α∈Fq and f(x)∈Fq(x) is a rational function of degree sum n. (Here f=f1/f2, where f1,f2 are coprime polynomials of degree n1,n2, respectively, and the sum of their degrees n1+n2=n.) For any n, such a pair is guaranteed to exist for sufficiently large q. Indeed, when n=2

We give a precise determination of the algebraicity of the critical values of L-functions associated to Siegel modular forms of half-integral weight and arbitrary degree. We generalise and improve on similar results for the integral-weight case by adapting the Rankin-Selberg method to this setting, with the aid of Shimura's theory of half-integral weight modular forms and recent work on precise algebraicity

In the first part of this article, we review a formalism of local zeta integrals attached to spherical reductive prehomogeneous vector spaces, which partially extends M. Sato's theory by incorporating the generalized matrix coefficients of admissible representations. We summarize the basic properties of these integrals such as the convergence, meromorphic continuation and an abstract functional equation