On Rankin triple product L-functions over function fields: central critical values

The aim of this article is to study the special values of Rankin triple product $L$ -functions associated to Drinfeld type newforms of equal square-free levels. The functional equation of these $L$ -functions is deduced from a Garrett-type integral representation and the functional equation of Eisenstein series on the group of similitudes of a symplectic vector space of dimension $6$ . When the associated root number is positive, we present a function field analogue of Gross–Kudla formula for the central critical value. This formula is then applied to the non-vanishing of $L$ -functions coming from elliptic curves over function fields.     

Defect Functions of Holomorphic Contractive Operator Functions and the Scattering Suboperator Through the Internal Channels of a System: Part II

Let $\theta (\zeta )$ be a Schur operator function, i.e., it is defined on the unit disk ${\mathbb D}\,{:=}\,\{\zeta \in {\mathbb C}: |\zeta | < 1\}$ and its values are contractive operators acting from one Hilbert space into another one. In the first part of the paper the outer and $*$ -outer Schur operator functions $\varphi (\zeta )$ and $\psi (\zeta )$ which describe respectively the deviations of the function $\theta (\zeta )$ from inner and $*$ -inner operator functions are studied. If $\varphi (\zeta )\ne 0$ , then it means that in the scattering system for which $\theta (\zeta )$ is the transfer function a portion of “information” comes inward the system and does not go outward, i.e., it is left in the internal channels of the system ([11, Sect. 6]). The function $\psi (\zeta )$ has the analogous property. For this reason these functions are called defect ones of the function $\theta (\zeta )$ . The explicit form of the defect functions $\varphi (\zeta )$ and $\psi (\zeta )$ is obtained and the analytic connection of these functions with the function $\theta (\zeta )$ is described ([11, Sect. 3 and Sect. 5]). The operator functions $\left( \begin{matrix} \varphi (\zeta ) \\ \theta (\zeta ) \end{matrix}\right) $ and $(\psi (\zeta ), \theta (\zeta ))$ are Schur functions as well ([11, Sect. 3]). It is important that there exists the unique contractive operator function $\chi (t),t\in \partial {\mathbb D}$ , such that the operator function $\left( \begin{matrix} \chi (t) &{} \varphi (t) \\ \psi (t) &{} \theta (t) \end{matrix}\right) ,t\in \partial {\mathbb D},$ is also contractive (Sect. 6). The second part of the paper is devoted to introducing and studying the properties of the function $\chi (t)$ . Specifically, it is shown that the function $\chi (t)$ is the scattering suboperator through the internal channels of the scattering system for which $\theta (\zeta )$ is the transfer function (Sect. 6).     

On the pullback of an arithmetic theta function

In this paper, we describe a relationship between the simplest examples of arithmetic theta series. The first of these are the weight 1 theta series ${\widehat{\phi}_{\mathcal C}(\tau)}$ defined using arithmetic 0-cycles on the moduli space ${\mathcal C}$ of elliptic curves with CM by the ring of integers ${O_{\kappa}}$ of an imaginary quadratic field. The second such series ${\widehat{\phi}_{\mathcal M}(\tau)}$ has weight 3/2 and takes values in the arithmetic Chow group ${\widehat{{\rm CH}}^1(\mathcal{M})}$ of the arithmetic surface associated to an indefinite quaternion algebra ${B/\mathbb{Q}}$ . For an embedding ${O_\kappa \rightarrow O_B}$ , a maximal order in B, and a two sided O _B -ideal Λ, there is a morphism ${j_\Lambda:{\mathcal C} \rightarrow {\mathcal M}}$ and a pullback ${j_\Lambda^*: \widehat{{\rm CH}}^1(\mathcal{M}) \rightarrow \widehat{{\rm CH}}^1(\mathcal C)}$ . Our main result is an expression for the pullback ${j^*_\Lambda \widehat{\phi}_{\mathcal M}(\tau)}$ as a linear combination of products of ${\widehat{\phi}_{\mathcal C}(\tau)}$ ’s and classical weight ${\frac{1}{2}}$ theta series.     

Sturmian maximizing measures for the piecewise-linear cosine family

Let T be the angle-doubling map on the circle $\mathbb{T}$ , and consider the 1-parameter family of piecewise-linear cosine functions $f_\theta :\mathbb{T} \to \mathbb{R}$ , defined by $f_\theta (x) = 1 - 4d_\mathbb{T} (x,\theta )$ . We identify the maximizing T-invariant measures for this family: for each ?? the f _?? -maximizing measure is unique and Sturmian (i.e. with support contained in some closed semi-circle). For rational p/q, we give an explicit formula for the set of functions in the family whose maximizing measure is the Sturmian measure of rotation number p/q. This allows us to analyse the variation with ?? of the maximum ergodic average for f _?? .     

Orbit structure of a distinguished Stein invariant domain in the complexification of a Hermitian symmetric space

Recently, Bruinier and Ono proved that the coefficients of certain weight \(-1/2\) harmonic weak Maaß forms are given as “traces” of singular moduli for harmonic weak Maaß forms. Here, we prove that similar results hold for the coefficients of harmonic weak Maaß forms of weight \(3/2+k\) , \(k\) even, and weight \(1/2-k\) , \(k\) odd, by extending the theta lift of Bruinier–Funke and Bruinier–Ono. Moreover, we generalize these results to include twisted traces of singular moduli using earlier work of the author and Ehlen on the twisted Bruinier–Funke-lift. Employing a general duality result between weight \(k\) and \(2-k\) , we obtain formulas for all half-integral weights. We also show that the non-holomorphic part of the theta lift in weight \(1/2-k\) , \(k\) odd, is connected to the vanishing of the special value of the \(L\) -function of a certain derivative of the lifted function.     

Representation of Integers by Positive Quaternary Quadratic Forms

Let $f(x,y,x,w) = x^2 + y^2 + z^2 + Dw^2$ , where $D >1$ is an integer such that $D \ne d^2$ and ${{\sqrt n } \mathord{\left/ {\vphantom {{\sqrt n } {\sqrt D = n^\theta , 0 < \theta < {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}} \right. \kern-0em} {\sqrt D = n^\theta , 0 < \theta < {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}$ . Let $rf(n)$ be the number of representations of n by f. It is proved that $r_f (n) = \pi ^2 \frac{n}{{\sqrt D }}\sigma _f (n) + O\left( {\frac{{n^{1 + \varepsilon - c(\theta )} }}{{\sqrt D }}} \right),$ where $\sigma _f (n)$ is the singular series, $c(\theta ) >0$ , and ε is an arbitrarily small positive constant. Bibliography: 14 titles.     

Vector partition functions and generalized dahmen and micchelli spaces

This is the first of a series of papers on partition functions and the index theory of transversally elliptic operators. In this paper we only discuss algebraic and combinatorial issues related to partition functions. The applications to index theory are in [4], while in [5] and [6] we shall investigate the cohomological formulas generated by this theory. Here we introduce a space of functions on a lattice which generalizes the space of quasipolynomials satisfying the difference equations associated to cocircuits of a sequence of vectors X, introduced by Dahmen and Micchelli [8]. This space $ \mathcal{F}(X) $ contains the partition function $ {\mathcal{P}_{(X)}} $ . We prove a “localization formula” for any f in $ \mathcal{F}(X) $ , inspired by Paradan's decomposition formula [12]. In particular, this implies a simple proof that the partition function $ {\mathcal{P}_{(X)}} $ is a quasi-polynomial on the Minkowski differences $ \mathfrak{c} - B(X) $ , where c is a big cell and B(X) is the zonotope generated by the vectors in X, a result due essentially to Dahmen and Micchelli.     

New identities involving sums of the tails related to real quadratic fields

In previous work, the authors discovered new examples of q-hypergeometric series related to the arithmetic of $\mathbb {Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt{3})$ . Building on this work, we construct in this paper sum of the tails identities for which some which some of these functions occur as error terms. As an application, we obtain formulas for the generating function of a certain zeta functions for real quadratic fields at negative integers.     

Two Problems on Coinvariant Subspaces of the Shift Operator

Two problems are posed that involve the star-invariant subspace ${K^{p}_{\theta}}$ (in the Hardy space H ^p ) associated with an inner function ${\theta}$ . One of these asks for a characterization of the extreme points of the unit ball in ${K^{\infty}_{\theta}}$ , while the other concerns the Fermat equation f ⁿ + g ⁿ =  h ⁿ in ${K^{p}_{\theta}}$ .     

Modularity and the distinct rank function

If R(ω,q) denotes Dyson’s partition rank generating function, due to work of Bringmann and Ono, it is known that for roots of unity ω≠1, R(ω,q) is the “holomorphic part” of a harmonic weak Maass form. Dating back to Ramanujan, it is also known that $\widehat{R}(\omega,q):=R(\omega,q^{-1})$ is given by Eichler integrals and modular forms. In analogy to these results, more recently Monks and Ono have shown that modular forms arise in a natural way from G(ω,q), the generating function for ranks of partitions into distinct parts. Moreover, Monks and Ono pose the following problem: determine whether the function $\widehat{G}(\omega,q):=G(\omega,q^{-1})$ appears naturally in the theory of modular forms. Here we answer this question of Monks and Ono, and show that $\widehat{G}(\omega,q)$ , when combined with $\widehat{G}(\omega^{-1},q)$ and a twisted third-order mock theta of Ramanujan, form a weight 1 modular form. We provide a more general result on the modularity of certain expressions involving basic hypergeometric series and then show that our result on $\widehat {G}(\omega,q)$ may be deduced from this as a special case.     

Approximation of classes B_{p,\theta }^r of periodic functions of one and several variables

We obtain order-sharp estimates of best approximations to the classes $B_{p,\theta }^r$ of periodic functions of several variables in the space L _q , 1 ≤ p, q ≤ ∞ by trigonometric polynomials with “numbers” of harmonics from step hyperbolic crosses. In the one-dimensional case, we establish the order of deviation of Fourier partial sums of functions from the classes $ B_{1,\theta }^{r_1 } $ in the space L ₁.     

On the Density of the Winding Number of Planar Brownian Motion

We obtain a formula for the density \(f(\theta , t)\) of the winding number of a planar Brownian motion \(Z_t\) around the origin. From this formula, we deduce an expansion for \(f(\log (\sqrt{t})\,\theta ,\,t)\) in inverse powers of \(\log \sqrt{t}\) and \((1+\theta ^2)^{1/2}\) which in particular yields the corrections of any order to Spitzer’s asymptotic law (in Spitzer, Trans. Am. Math. Soc. 87:187–197, 1958). We also obtain an expansion for \(f(\theta ,t)\) in inverse powers of \(\log \sqrt{t}\) , which yields precise asymptotics as \(t \rightarrow \infty \) for a local limit theorem for the windings.     

Pointwise convergence of vector-valued Fourier series

We prove a vector-valued version of Carleson’s theorem: let $Y=[X,H]_\theta $ be a complex interpolation space between an unconditionality of martingale differences (UMD) space $X$ and a Hilbert space $H$ . For $p\in (1,\infty )$ and $f\in L^p(\mathbb T ;Y)$ , the partial sums of the Fourier series of $f$ converge to $f$ pointwise almost everywhere. Apparently, all known examples of UMD spaces are of this intermediate form $Y=[X,H]_\theta $ . In particular, we answer affirmatively a question of Rubio de Francia on the pointwise convergence of Fourier series of Schatten class valued functions.     

Harmonic vector fields on compact Lorentz surfaces

We study harmonic vector fields on a Lorentzian torus T ² i.e. critical points of the total bending functional ${\mathcal {B} : \mathcal {E} \to \mathbb {R}}$ were ${\mathcal {E}}$ consists of all unit timelike vector fields on T ². We derive the first variation formula for ${\mathcal {B}}$ in terms of the Lorentz angle function associated to each ${X \in \mathcal {E}}$ and give applications on flat Lorentzian tori.     

Linear Transformations and Reduction Formulas for the Gelfand Hypergeometric Functions Associated with the Grassmannians $$G_{2,4} $$ and $$G_{3,6} $$

We show that the Gelfand hypergeometric functions associated with the Grassmannians $G_{2,4} $ and $G_{3,6} $ with some special relations imposed on the parameters can be represented in terms of hypergeometric series of a simpler form. In particular, a function associated with the Grassmannian $G_{2,4} $ (the case of three variables) can be represented (depending on the form of the additional conditions on the parameters of the series) in terms of the Horn series $H_2 ,G_2 $ , of the Appell functions $F_1 ,F_2 ,F_3 $ and of the Gauss functions $F_1^2 $ , while the functions associated with the Grassmannian $G_{3,6} $ (the case of four variables) can be represented in terms of the series $G_2 ,F_1 ,F_2 ,F_3 $ and $F_1^2 $ . The relation between certain formulas and the Gelfand--Graev--Retakh reduction formula is discussed. Combined linear transformations and universal elementary reduction rules underlying the method were implemented by a computer program developed by the authors on the basis of the computer algebra system Maple V-4.     

On the dependency for asymptotically independent estimates

Suppose ${\widehat{\theta}_1}$ and ${\widehat{\theta}_2}$ are asymptotically independent non-lattice with a joint second order Edgeworth expansion in n ^?1/2. Then the ?? dependency coefficient is $$\alpha \left(\widehat{\theta}_1, \widehat{\theta}_2 \right) = n^{-1/2} C + O \left(n^{-1} \right),$$ where ${C = (4 \pi)^{-1}\exp (-1/2) (\tau^2_1 + \tau^2_2) ^{1/2}}$ for ${\tau_1, \tau_2}$ their joint skewness coefficients.     

A proof of a theorem by Fried and MacRae and applications to the composition of polynomial functions

Fried and MacRae (Math. Ann. 180, 220?C226 (1969)) proved that for univariate polynomials ${p,q, f, g \in \mathbb{K}[t]}$ ( ${\mathbb{K}}$ a field) with p, q nonconstant, p(x) ? q(y) divides f(x) ? g(y) in ${\mathbb{K}[x,y]}$ if and only if there is ${h \in \mathbb{K}[t]}$ such that f?=?h(p(t)) and g?=?h(q(t)). Schicho (Arch. Math. 65, 239?C243 (1995)) proved this theorem from the viewpoint of category theory, thereby providing several generalizations to multivariate polynomials. In the present note, we give a new proof of one of these generalizations. The theorem by Fried and MacRae yields a way to prove the following fact for nonconstant functions f, g from ${\mathbb{C}}$ to ${\mathbb{C}}$ : if both the composition ${f \circ g}$ and g are polynomial functions, then f has to be a polynomial function as well. We give an algebraic proof of this fact and present a generalization to multivariate polynomials over algebraically closed fields. This provides a way to prove a generalization of a result by Carlitz (Acta Sci. Math. (Szeged) 24, 196?C203 (1963)) that describes those univariate polynomials over finite fields that induce bijective functions on all of their finite extensions.     

Uniform existential interpretation of arithmetic in rings of functions of positive characteristic

We show that first order integer arithmetic is uniformly positive-existentially interpretable in large classes of (subrings of) function fields of positive characteristic over some languages that contain the language of rings. One of the main intermediate results is a positive existential definition (in these classes), uniform among all characteristics p, of the binary relation “ $y=x^{p^{s}}$ or $x=y^{p^{s}}$ for some integer s≥0”. A natural consequence of our work is that there is no algorithm to decide whether or not a system of polynomial equations over $\mathbb {Z}[z]$ has solutions in all but finitely many polynomial rings $\mathbb {F}_{p}[z]$ . Analogous consequences are deduced for the rational function fields $\mathbb {F}_{p}(z)$ , over languages with a predicate for the valuation ring at zero.     

Burgess-like Subconvex Bounds for GL2 × GL1

Let F be a number field, π an irreducible cuspidal representation of \({{\rm GL}_2(\mathbb{A}_F)}\) with unitary central character, and χ a Hecke character of analytic conductor Q. Then \({L(1/2, \pi \otimes \chi) \ll Q^{\frac{1}{2} - \frac{1}{8}(1-2\theta)+\epsilon}}\) , where \({0 \leq \theta \leq 1/2}\) is any exponent towards the Ramanujan–Petersson conjecture. The proof is based on an idea of unipotent translation originated from P. Sarnak then developed by Ph. Michel and A. Venkatesh, combined with a method of amplification.