Volume 29, Issue 1

We prove that over a field of characteristic zero the maximal dimension of a proper unital subalgebra in the n×n${n\times n}$ matrix algebra is n2-n+1${n^{2}-n+1}$ and furthermore this upper bound is attained for the so-called parabolic subalgebras. We also investigate the corresponding notion of parabolic coideals for matrix coalgebras and prove that the minimal dimension of a non-zero coideal of the matrix coalgebra ℳn⁢(k)${{\mathcal{M}}^{n}(k)}$ is n-1${n-1}$.

We study the coefficients of a natural basis for the space of harmonic weak Maass forms of weight 5/2${5/2}$ on the full modular group. The non-holomorphic part of the first element of this infinite basis encodes the values of the partition function p⁢(n)${p(n)}$. We show that the coefficients of these harmonic Maass forms are given by traces of singular invariants. These are values of non-holomorphic modular functions at CM points or their real quadratic analogues: cycle integrals of such functions along geodesics on the modular curve. The real quadratic case relates to recent work of Duke, Imamoḡlu, and Tóth on cycle integrals of the j -function, while the imaginary quadratic case recovers the algebraic formula of Bruinier and Ono for the partition function.

A new characterization of the singular packing subspaces of general bounded self-adjoint operators is presented, which is used to show that the set of operators whose spectral measures have upper packing dimension equal to one is a Gδ${G_{\delta}}$ (in suitable metric spaces). As an application, it is proven that, generically (in space of continuous sampling functions), spectral measures of the limit-periodic Schrödinger operators have upper packing dimensions equal to one. Consequently, in a generic set, these operators are quasiballistic.

Let X be a complex projective variety of dimension n with only isolated normal singularities. In this paper, we prove, using mixed Hodge theory, that if the link of each singular point of X is (n-2)${(n-2)}$-connected, then X is a formal topological space. This result applies to a large class of examples, such as normal surface singularities, varieties with ordinary multiple points, hypersurfaces with isolated singularities and, more generally, complete intersections with isolated singularities. We obtain analogous results for contractions of subvarieties.

We prove a precise formula relating the Bessel period of certain automorphic forms on GSp4⁢(𝔸F)${\mathrm{GSp}_{4}(\mathbb{A}_{F})}$ to a central L -value. This is a special case of the refined Gan–Gross–Prasad conjecture for the groups (SO5,SO2)${(\mathrm{SO}_{5},\mathrm{SO}_{2})}$ as set out by Ichino–Ikeda [12] and Liu [14]. This conjecture is deep and hard to prove in full generality; in this paper we succeed in proving the conjecture for forms lifted, via automorphic induction, from GL2⁢(𝔸E)${\mathrm{GL}_{2}(\mathbb{A}_{E})}$ where E is a quadratic extension of F . The case where E=F×F${E=F\times F}$ has been previously dealt with by Liu [14].

In this paper we apply a group action approach to the study of Erdős–Falconer-type problems in vector spaces over finite fields and use it to obtain non-trivial exponents for the distribution of simplices. We prove that there exists s0⁢(d)<d${s_{0}(d)<d}$ such that if E⊂𝔽qd${E\subset{\mathbb{F}}_{q}^{d}}$, d≥2${d\geq 2}$, with |E|≥C⁢qs0${|E|\geq Cq^{s_{0}}}$, then |Tdd⁢(E)|≥C′⁢q(d+12)${|T^{d}_{d}(E)|\geq C^{\prime}q^{d+1\choose 2}}$, where Tkd⁢(E)${T^{d}_{k}(E)}$ denotes the set of congruence classes of k -dimensional simplices determined by k+1${k+1}$-tuples of points from E . Non-trivial exponents were previously obtained by Chapman, Erdogan, Hart, Iosevich and Koh [4] for Tkd⁢(E)${T^{d}_{k}(E)}$ with 2≤k≤d-1${2\leq k\leq d-1}$. A non-trivial result for T22⁢(E)${T^{2}_{2}(E)}$ in the plane was obtained by Bennett, Iosevich and Pakianathan [2]. These results are significantly generalized and improved in this paper. In particular, we establish the Wolff exponent 43${\frac{4}{3}}$, previously established in [4] for the q≡3⁢ mod ⁢4${q\equiv 3\textup{ mod }4}$ case to the case q≡1⁢ mod ⁢4${q\equiv 1\textup{ mod }4}$, and this results in a new sum-product type inequality. We also obtain non-trivial results for subsets of the sphere in 𝔽qd${{\mathbb{F}}_{q}^{d}}$, where previous methods have yielded nothing. The key to our approach is a group action perspective which quickly leads to natural and effective formulae in the style of the classical Mattila integral from geometric measure theory.

Suppose F is a p -adic field, and H1${\mathrm{H}_{1}}$ is (a z -extension of) a group that is twisted endoscopic to a connected reductive quasi-split group G over F . Suppose G satisfies the strong form of the generic packet conjecture (also called tempered packet conjecture in literature). Under certain assumptions, we show that the twisted endoscopic character identities associated to this situation imply the strong form of the generic packet conjecture for H1${\mathrm{H}_{1}}$. This generalizes a result of T. Konno, and lets us deduce, under the assumption that appropriate character identities are satisfied, the generic packet conjecture for general spin (GSpin) groups.

We prove that products of at most two vector valued Eisenstein series that originate in level 1 span all spaces of cusp forms for congruence subgroups. This can be viewed as an analogue in the level aspect to a result that goes back to Rankin, and Kohnen and Zagier, which focuses on the weight aspect. The main feature of the proof are vector valued Hecke operators. We recover several classical constructions from them, including classical Hecke operators, Atkin–Lehner involutions, and oldforms. As a corollary to our main theorem, we obtain a vanishing condition for modular forms reminiscent of period relations deduced by Kohnen and Zagier in the context their previously mentioned result.

In this paper we obtain regularity estimates in weighted Orlicz spaces for the Calderón–Zygmund singular integral operators under certain optimal conditions.

We calculate the homology of the free loop space of (n-1)${(n-1)}$-connected closed manifolds of dimension at most 3⁢n-2${3n-2}$ (n≥2${n\geq 2}$), with the Chas–Sullivan loop product and loop bracket. Over a field of characteristic zero, we obtain an expression for the BV-operator. We also give explicit formulas for the Betti numbers, showing they grow exponentially. Our main tool is the connection between formality, coformality and Koszul algebras that was elucidated by the first author [6].

In this article we prove a Riesz potential estimate and a Sobolev inequality for general generalized Orlicz spaces. Our assumptions are natural generalizations of the log${\log}$-Hölder continuity that is commonly used in the variable exponent case. We also provide a number of useful auxiliary results including a normalization of the Φ-function and its behavior under duality

The purpose of this paper is to study products of Fourier coefficients of an elliptic cusp form, a⁢(n)⁢a⁢(n+r)${a(n)a(n+r)}$ (n≥1${n\geq 1}$) for a fixed positive integer r , concerning both non-vanishing and non-negativity.