Twisted Sums Research Articles

Analytic twists of GL2×GL2$\rm GL_2\times \rm GL_2$ automorphic forms

AbstractLet f and g be holomorphic or Maass cusp forms for with normalized Fourier coefficients and , respectively. In this paper, we prove nontrivial estimates for the sum where , , is a large parameter and is some nonlinear real‐valued smooth function. Applications of these estimates include a subconvex bound for the Rankin–Selberg L‐function in the t‐aspect, an improved estimate for a nonlinear exponential twisted sum and the following asymptotic formula for the sum of the Fourier coefficients of certain Eisenstein series for any .

Twisted sums of c 0(I)

We study in this paper a few remarkable properties of twisted sums Z(κ, X) of c 0(κ) and a Banach space X. We first prove a representation theorem for such twisted sums from which we will obtain, among others, the following: (a) twisted sums of c 0(κ) and c 0(I) are either subspaces of ℓ ∞(κ) or contain a complemented copy of c 0(κ +); (b) under the hypothesis [p = c], when K is either a suitable Corson compact, a separable Rosenthal compact or a scattered compact of finite height, there is a twisted sum of c 0 and C(K) that is not isomorphic to a space of continuous functions; (c) all twisted sums Z(κ, X) are isomorphically Lindenstrauss spaces when X is a Lindenstrauss space; (d) all twisted sums Z(κ, X) are isomorphically polyhedral when X is a polyhedral space with a σ-discrete boundary, which solves a problem of Castillo and Papini.

Brazil: A Biography

Brazil: A Biography

Multiple Dedekind Type Sums and Their Related Zeta Functions

The main purpose of this paper is to use the multiple twisted Bernoulli polynomials and their interpolation functions to construct multiple twisted Dedekind type sums. We investigate some properties of these sums. By use of the properties of multiple twisted zeta functions and the Bernoulli functions involving the Bernoulli polynomials, we derive reciprocity laws of these sums. Further developments and observations on these new Dedekind type sums are given.

Cancellation in algebraic twisted sums on GL_ m

Abstract Let π be an automorphic irreducible cuspidal representation of GL m {\operatorname{GL}_{m}} over ℚ {\mathbb{Q}} with unitary central character, and let λ π ⁢ ( n ) {\lambda_{\pi}(n)} be its n-th Dirichlet series coefficient. We study short sums of isotypic trace functions associated to some sheaves modulo primes q of bounded conductor, twisted by multiplicative functions λ π ⁢ ( n ) {\lambda_{\pi}(n)} and μ ⁢ ( n ) ⁢ λ π ⁢ ( n ) {\mu(n)\lambda_{\pi}(n)} . We are able to establish non-trivial bounds for these algebraic twisted sums with intervals of length of at least q 1 / 2 + ε {q^{1/2+\varepsilon}} for an arbitrary fixed ε > 0 {\varepsilon>0} .

Splitting chains, tunnels and twisted sums

We study splitting chains in $$\mathcal{P}(\omega)$$ , that is, families of subsets of ω which are linearly ordered by ⊆* and which are splitting. We prove that their existence is independent of axioms of ZFC. We show that they can be used to construct certain peculiar Banach spaces: twisted sums of C(ω*) = l∞/c0 and c0(c). Also, we consider splitting chains in a topological setting, where they give rise to the so called tunnels.

Twisted sums of c0 and C(K)-spaces: A solution to the CCKY problem

We consider the class of Banach spaces Y for which c0 admits a nontrivial twisted sum with Y. We present a characterization of such spaces Y in terms of properties of the weak⁎ topology on Y⁎. We prove that under the continuum hypothesis c0 has a nontrivial twisted sum with every space of the form Y=C(K), where K is compact and not metrizable. This gives a consistent affirmative solution to a problem posed by Cabello, Castillo, Kalton and Yost.

Sailing over three problems of Koszmider

We discuss three problems of Koszmider on the structure of the spaces of continuous functions on the Stone compact KA generated by an almost disjoint family A of infinite subsets of ω — we present a solution to two problems and develop previous results of Marciszewski and Pol answering the third one. We will show, in particular, that assuming Martin's axiom the space C(KA) is uniquely determined up to isomorphism by the cardinality of A whenever |A|<c, while there are 2c nonisomorphic spaces C(KA) with |A|=c. We also investigate Koszmider's problems in the context of the class of separable Rosenthal compacta and indicate the meaning of our results in the language of twisted sums of c0 and some C(K) spaces.

The behaviour of quasi-linear maps on C(K)-spaces

In this paper we combine topological and functional analysis methods to prove that a non-locally trivial quasi-linear map defined on a C(K) must be nontrivial on a subspace isomorphic to c0. We conclude the paper with a few examples showing that the result is optimal, and providing an application to the existence of nontrivial twisted sums of ℓ1 and c0.

Operator ideals and three-space properties of asymptotic ideal seminorms

We introduce asymptotic analogues of the Rademacher and martingale type and cotype of Banach spaces and operators acting on them. Some classical local theory results related, for example, to the `automatic-type' phenomenon, the type-cotype duality, or the Maurey-Pisier theorem, are extended to the asymptotic setting. We also investigate operator ideals corresponding to the asymptotic subtype/subcotype. As an application of this theory, we provide a sharp version of a result of Brooker and Lancien by showing that any twisted sum of Banach spaces with Szlenk power types $p$ and $q$ has Szlenk power type $\max\{p,q\}$.

ON A TWISTED VERSION OF LINNIK AND SELBERG'S CONJECTURE ON SUMS OF KLOOSTERMAN SUMS

We generalise the work of Sarnak-Tsimerman to twisted sums of Kloosterman sums and thus give evidence towards the twisted Linnik-Selberg Conjecture.

Nontrivial twisted sums for finite height spacesunder Martin’s Axiom

We show that if we assume Martin's Axiom, then there exists a nontrivial twisted sum of c_0 and C(K), for every compact space K with finite height and weight at least continuum. This result settles the problem of existence of nontrivial twisted sums of c_0 and C(K), for finite height spaces K, under Martin's Axiom.

On Twisted Sums of Sequence Spaces

We prove the existence of non trivial twisted sums involving the pth James space Jp(1 &lt; p &lt; 1), the Johnson-Lindenstrauss space JL; the James tree space JT , the Tsirelson’s space T and the Argyros and Deliyanni space AD . We also present non trivial twisted sums involving their duals and biduals. We show that there are strictly singular quasi-linear maps from the spaces T ;T* ; AD and JT into C[0; 1]. We discuss the Pelczynski’s property(u) for the twisted sums involving these spaces which extends a pthJames-Schreier spaces Vp(1 &lt; p &lt;1) or J2.

Sign changes of Kloosterman sums and exceptional characters

We prove that the existence of exceptional real zeroes of Dirichlet $L$-functions would lead to cancellations in the sum $\sum_{p\leq x} \Kl(1, p)$ of Kloosterman sums over primes, and also to sign changes of $\Kl(1, n)$, where $n$ runs over integers with exactly two prime factors. Our arguments involve a variant of Bombieri's sieve, bounds for twisted sums of Kloosterman sums, and work of Fouvry and Michel on sums of $\left| \Kl(1, n)\right|$.

Type, cotype and twisted sums induced by complex interpolation

This paper deals with extensions or twisted sums of Banach spaces that come induced by complex interpolation and the relation between the type and cotype of the spaces in the interpolation scale and the nontriviality and singularity of the induced extension. The results are presented in the context of interpolation of families of Banach spaces, and are applied to the study of submodules of Schatten classes. We also obtain nontrivial extensions of spaces without the CAP which also fail the CAP.

Extension operators and twisted sums of c0 and C(K) spaces

We investigate the following problem posed by Cabello Sanchéz, Castillo, Kalton, and Yost:Let K be a nonmetrizable compact space. Does there exist a nontrivial twisted sum of c0 and C(K), i.e., does there exist a Banach space X containing a non-complemented copy Y of c0 such that the quotient space X/Y is isomorphic to C(K)?Using additional set-theoretic assumptions we give the first examples of compact spaces K providing a negative answer to this question. We show that under Martin's axiom and the negation of the continuum hypothesis, if either K is the Cantor cube 2ω1 or K is a separable scattered compact space of height 3 and weight ω1, then every twisted sum of c0 and C(K) is trivial.We also construct nontrivial twisted sums of c0 and C(K) for K belonging to several classes of compacta. Our main tool is an investigation of pairs of compact spaces K⊆L which do not admit an extension operator C(K)→C(L).

Nonvanishing of Central Values of L-functions of Newforms in S2(Γ0(dp2)) Twisted by Quadratic Characters

AbstractWe prove that for d ∊ {2, 3, 5, 7, 13} and K a quadratic (or rational) field of discriminant D and Dirichlet character 𝜒, if a prime p is large enough compared to D, there is a new form f ∊ S2Γ0(dp2)) with sign (+1) with respect to the Atkin–Lehner involution wp2 such that L( f ⊗ 𝜒, •)≠ 0. This result is obtained through an estimate of a weighted sum of twists of L-functions that generalises a result of Ellenberg. It relies on the approximate functional equation for the L-functions L( f ⊗ 𝜒, · ) and a Petersson trace formula restricted to Atkin–Lehner eigenspaces. An application of this nonvanishing theorem will be given in terms of existence of rank zero quotients of some twisted jacobians, which generalises a result of Darmon and Merel.

Singular twisted sums generated by complex interpolation

We present new methods to obtain singular twisted sums X ⊕ Ω X X\oplus _\Omega X (i.e., exact sequences 0 → X → X ⊕ Ω X → X → 0 0\to X\to X\oplus _\Omega X \to X\to 0 in which the quotient map is strictly singular) when X X is an interpolation space arising from a complex interpolation scheme and Ω \Omega is the induced centralizer. Although our methods are quite general, we are mainly concerned with the choice of X X as either a Hilbert space or Ferenczi’s uniformly convex Hereditarily Indecomposable space. In the first case, we construct new singular twisted Hilbert spaces (which includes the only known example so far: the Kalton-Peck space Z 2 Z_2 ). In the second case we obtain the first example of an H.I. twisted sum of an H.I. space. During our study of singularity we introduce the notion of a disjointly singular twisted sum of Köthe function spaces and construct several examples involving reflexive p p -convex Köthe function spaces (which includes the function space version of the Kalton-Peck space Z 2 Z_2 ). We then use Rochberg’s description of iterated twisted sums to show that there is a sequence F n \mathcal F_n of H.I. spaces so that F m + n \mathcal F_{m+n} is a singular twisted sum of F m \mathcal F_m and F n \mathcal F_n , while for l &gt; n l&gt;n the direct sum F n ⊕ F l + m \mathcal F_n \oplus \mathcal F_{l+m} is a nontrivial twisted sum of F l \mathcal F_l and F m + n \mathcal F_{m+n} .

Galois representations attached to tensor products of arithmetic cohomology

We compute the action of Hecke operators on tensor products of cohomology classes of lower congruence subgroups of SL(n,Z) in trivial weight. We use this computation to prove that if each representation in a collection of Galois representations is attached to a cohomology class of a lower congruence subgroup in trivial weight, then a sum of certain twists of the representations consistent with the main conjectures of [5,10] is also attached to such a cohomology class.

Small Valdivia compacta and trees

We present a characterization of Valdivia compact spaces of small weight in terms of path spaces of trees and we use it to obtain (under $\diamondsuit$) a counterexample to a conjecture related to an open problem concerning twisted sums of $C(K)$ spaces.