Volume 30, Issue 2

We study the variance of sums of the k -fold divisor function dk⁢(n){d_{k}(n)} over sparse arithmetic progressions, with averaging over both residue classes and moduli. In a restricted range, we confirm an averaged version of a recent conjecture about the asymptotics of this variance. This result is closely related to moments of Dirichlet L -functions, and our proof relies on the asymptotic large sieve.

We provide characterizations of Lie groups as compact-like groups in which all closed zero-dimensional metric (compact) subgroups are discrete. The “compact-like” properties we consider include (local) compactness, (local) ω-boundedness, (local) countable compactness, (local) precompactness, (local) minimality and sequential completeness. Below is A sample of our characterizations is as follows: (i) A topological group is a Lie group if and only if it is locally compact and has no infinite compact metric zero-dimensional subgroups. (ii) An abelian topological group G is a Lie group if and only if G is locally minimal, locally precompact and all closed metric zero-dimensional subgroups of G are discrete. (iii) An abelian topological group is a compact Lie group if and only if it is minimal and has no infinite closed metric zero-dimensional subgroups. (iv) An infinite topological group is a compact Lie group if and only if it is sequentially complete, precompact, locally minimal, contains a non-empty open connected subset and all its compact metric zero-dimensional subgroups are finite.

Given a modular form f{{f}} of even weight larger than two and an imaginary quadratic field K{{K}} satisfying a relaxed Heegner hypothesis, we construct a collection of CM cycles on a Kuga–Sato variety over a suitable Shimura curve which gives rise to a system of Galois cohomology classes attached to f{{f}} enjoying the compatibility properties of an Euler system. Then we use Kolyvagin’s method [21], as adapted by Nekovář [28] to higher weight modular forms, to bound the size of the relevant Selmer group associated to f{{f}} and K{{K}} and prove the finiteness of the (primary part) of the Shafarevich–Tate group, provided that a suitable cohomology class does not vanish.

Let G1{G^{1}} be an orthogonal, symplectic or unitary group over a local field and let P=M⁢N{P=MN} be a maximal parabolic subgroup. Then the Levi subgroup M is the product of a group of the same type as G1{G^{1}} and a general linear group, acting on vector spaces X and W , respectively. In this paper we decompose the unipotent radical N of P under the adjoint action of M , assuming dim⁡W≤dim⁡X{\dim W\leq\dim X}, excluding only the symplectic case with dim⁡W{\dim W} odd. The result is a Weyl-type integration formula for N with applications to the theory of intertwining operators for parabolically induced representations of G1{G^{1}}. Namely, one obtains a bilinear pairing on matrix coefficients, in the spirit of Goldberg–Shahidi, which detects the presence of poles of these operators at 0.

Let D be a division algebra over its center F of degree n . Consider the group μZ⁢(D)=μn⁢(F)/Z⁢(D′){\mu_{Z}(D)=\mu_{n}(F)/Z(D^{\prime})}, where μn⁢(F){\mu_{n}(F)} is the group of all the n -th roots of unity in F*{F^{*}}, and Z⁢(D′){Z(D^{\prime})} is the center of the commutator subgroup of the group of units D*{D^{*}} of D . It is shown that if μZ⁢(D⊗FL)≠1{\mu_{Z}(D\otimes_{F}L)\neq 1} for some L containing all the primitive nk{n^{k}}-th roots of unity for all positive integers k , then D is not split by any cyclic extension of F . This criterion is employed to prove that some special classes of division algebras are not cyclically split.

The s -th higher topological complexity TCs⁡(X){\operatorname{TC}_{s}(X)} of a space X can be estimated from above by homotopical methods, and from below by homological methods. We give a thorough analysis of the gap between such estimates when X=ℝ⁢Pm{X=\operatorname{\mathbb{R}P}^{m}}, the real projective space of dimension m . In particular, we describe a number r⁢(m){r(m)}, which depends on the structure of zeros and ones in the binary expansion of m , and with the property that 0≤s⁢m-TCs⁡(ℝ⁢Pm)≤δs⁢(m){0\leq sm-\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})\leq\delta_{s}(% m)} for s≥r⁢(m){s\geq r(m)}, where δs⁢(m)=(0,1,0){\delta_{s}(m)=(0,1,0)} for m≡(0,1,2)mod4{m\equiv(0,1,2)\bmod 4}. Such an estimation for TCs⁡(ℝ⁢Pm){\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})} appears to be closely related to the determination of the Euclidean immersion dimension of ℝ⁢Pm{\operatorname{\mathbb{R}P}^{m}}. We illustrate the phenomenon in the case m=3⋅2a{m=3\cdot 2^{a}}. In addition, we show that, for large enough s and even m , TCs⁡(ℝ⁢Pm){\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})} is characterized as the smallest positive integer t=t⁢(m,s){t=t(m,s)} for which there is a suitable equivariant map from Davis’ projective product space P𝐦s{\mathrm{P}_{\mathbf{m}_{s}}} to the (t+1){(t+1)}-st join-power ((ℤ2)s-1)∗(t+1){((\mathbb{Z}_{2})^{s-1})^{\ast(t+1)}}. This is a (partial, but conjecturally complete) generalization of the work of Farber, Tabachnikov and Yuzvinsky relating TC2{\operatorname{TC}_{2}} to the immersion dimension of real projective spaces.

The purpose of this note is to show that 2⁢K{2K} of any smooth compact complex 2-ball quotient is very ample, except possibly for four pairs of fake projective planes of minimal type, where K is the canonical line bundle. For the four pairs of fake projective planes, the sections of 2⁢KM{2K_{M}} give an embedding of M except possibly for at most two points on M .

We study the irreducible algebraic equation xn+a1⁢xn-1+⋯+an=0,with n≥4,x^{n}+a_{1}x^{n-1}+\cdots+a_{n}=0,\quad\text{with ${n\geq 4}$,} on the differential field (𝔽=ℂ⁢(t),δ=dd⁢t){(\mathbb{F}=\mathbb{C}(t),\delta=\frac{d}{dt})}. We assume that a root of the equation is a solution to the Riccati differential equation u′+B0+B1⁢u+B2⁢u2=0{u^{\prime}+B_{0}+B_{1}u+B_{2}u^{2}=0}, where B0{B_{0}}, B1{B_{1}}, B2{B_{2}} are in 𝔽{\mathbb{F}}. We show how to construct a large class of polynomials as in the above algebraic equation, i.e., we prove that there exists a polynomial Fn⁢(x,y)∈ℂ⁢(x)⁢[y]{F_{n}(x,y)\in\mathbb{C}(x)[y]} such that for almost T∈𝔽∖ℂ{T\in\mathbb{F}\setminus\mathbb{C}}, the algebraic equation Fn⁢(x,T)=0{F_{n}(x,T)=0} is of the same type as the above stated algebraic equation. In other words, all its roots are solutions to the same Riccati equation. On the other hand, we give an example of a degree 3 irreducible polynomial equation satisfied by certain weight 2 modular forms for the subgroup Γ⁢(2){\Gamma(2)}, whose roots satisfy a common Riccati equation on the differential field (ℂ⁢(E2,E4,E6),dd⁢τ){(\mathbb{C}(E_{2},E_{4},E_{6}),\frac{d}{d\tau})}, with Ei⁢(τ){E_{i}(\tau)} being the Eisenstein series of weight i . These solutions are related to a Darboux–Halphen system. Finally, we deal with the following problem: For which “potential” q∈ℂ⁢(℘,℘′){q\in\mathbb{C}(\wp,\wp^{\prime})} does the Riccati equation d⁢Yd⁢z+Y2=q{\frac{dY}{dz}+Y^{2}=q} admit algebraic solutions over the differential field ℂ⁢(℘,℘′){\mathbb{C}(\wp,\wp^{\prime})}, with ℘{\wp} being the classical Weierstrass function? We study this problem via Darboux polynomials and invariant theory and show that the minimal polynomial Φ⁢(x){\Phi(x)} of an algebraic solution u must have a vanishing fourth transvectant τ4⁢(Φ)⁢(x){\tau_{4}(\Phi)(x)}.

In this paper, we present a local Tb theorem for the non-homogeneous Littlewood–Paley gλ*{g_{\lambda}^{*}}-function with non-convolution type kernels and upper power bound measure μ. Actually, we show that, under the assumptions supp⁡bQ⊂Q{\operatorname{supp}b_{Q}\subset Q}, |∫QbQdμ|≳μ(Q){\lvert\int_{Q}b_{Q}\,d\mu|\gtrsim\mu(Q)} and ∥bQ∥Lp⁢(μ)p≲μ⁢(Q){\|b_{Q}\|^{p}_{L^{p}(\mu)}\lesssim\mu(Q)}, the norm inequality ∥gλ*⁢(f)∥Lp⁢(μ)≲∥f∥Lp⁢(μ){\|g_{\lambda}^{*}(f\/)\|_{L^{p}(\mu)}\lesssim\|f\/\|_{L^{p}(\mu)}} holds if and only if the following testing condition holds: supQ:cubes in ⁢ℝn1μ⁢(Q)∫Q(∫0ℓ⁢(Q)∫ℝn(tt+|x-y|)m⁢λ|θt(bQ)(y,t)|2d⁢μ⁢(y)⁢d⁢ttm+1)p/2dμ(x)<∞.\sup_{Q:\text{cubes in }\mathbb{R}^{n}}\frac{1}{\mu(Q)}\int_{Q}\Biggr{(}\int_{% 0}^{\ell(Q)}\int_{\mathbb{R}^{n}}\Bigl{(}\frac{t}{t+|x-y|}\Big{)}^{m\lambda}|% \theta_{t}(b_{Q})(y,t)|^{2}\frac{d\mu(y)\,dt}{t^{m+1}}\Bigg{)}^{p/2}d\mu(x)<\infty. This is the first time to investigate the gλ*{g_{\lambda}^{*}}-function in the simultaneous presence of three attributes: local, non-homogeneous and Lp{L^{p}}-testing condition. It is important to note that the testing condition here is an Lp{L^{p}} type with p∈(1,2]{p\in(1,2]}.

We investigate the saturation rank of a finite group scheme defined over an algebraically closed field 𝕜{\Bbbk} of positive characteristic p . We begin by exploring the saturation rank for finite groups and infinitesimal group schemes. Special attention is given to reductive Lie algebras and the second Frobenius kernel of the algebraic group SLn{\operatorname{SL}_{n}}.

We prove that there exist exactly three non-equivalent symplectic semifield spreads of PG⁡(5,q2){\operatorname{PG}(5,q^{2})}, for q2>2⋅38{q^{2}>2\cdot 3^{8}} odd, whose associated semifield has center containing 𝔽q{\mathbb{F}_{q}}. Equivalently, we classify, up to isotopy, commutative semifields of order q6{q^{6}}, for q2>2⋅38{q^{2}>2\cdot 3^{8}} odd, with middle nucleus containing 𝔽q2{\mathbb{F}_{q^{2}}} and center containing 𝔽q{\mathbb{F}_{q}}.

We use a landmark result in the theory of Riesz spaces – Freudenthal’s 1936 spectral theorem – to canonically represent any Archimedean lattice-ordered group G with a strong unit as a (non-separating) lattice-group of real-valued continuous functions on an appropriate G -indexed zero-dimensional compactification wG⁢ZG{w_{G}Z_{G}} of its space ZG{Z_{G}} of minimal prime ideals. The two further ingredients needed to establish this representation are the Yosida representation of G on its space XG{X_{G}} of maximal ideals, and the well-known continuous surjection of ZG{Z_{G}} onto XG{X_{G}}. We then establish our main result by showing that the inclusion-minimal extension of this representation of G that separates the points of ZG{Z_{G}} – namely, the sublattice subgroup of C⁡(ZG){\operatorname{C}(Z_{G})} generated by the image of G along with all characteristic functions of clopen (closed and open) subsets of ZG{Z_{G}} which are determined by elements of G – is precisely the classical projectable hull of G . Our main result thus reveals a fundamental relationship between projectable hulls and minimal spectra, and provides the most direct and explicit construction of projectable hulls to date. Our techniques do require the presence of a strong unit.

Golodness of two-dimensional simplicial complexes is studied through polyhedral products, and combinatorial and topological characterizations of Golodness of surface triangulations are given. An answer to the question of Berglund is also given so that there is a two-dimensional simplicial complex which is rationally Golod but not Golod over ℤ/p{\mathbb{Z}/p}.