Abstract In this paper, we study the values of modular functions at real quadratic irrationalities which are defined in terms of cycle integrals of modular functions associated with closed geodesics. We refine a result of Bengoechea and Imamog̎lu to include values (and its limiting behaviour) of modular functions at a larger class of real quadratic irrationalities. As a consequence, we deduce the limiting behaviour of values of modular functions at powers of Pisot–Vijayaraghavan numbers. Further, we deduce distribution of class numbers of real quadratic fields having discriminants of a particular type.
Abstract In this paper, we study traces of Hecke operators on Drinfeld modular forms of level 1 in the case . We deduce closed‐form expressions for traces of Hecke operators corresponding to primes of degree at most 2 and provide algorithms for primes of higher degree. We improve the Ramanujan bound and deduce the decomposition of cusp forms of level into oldforms and newforms, as conjectured by Bandini–Valentino, under the hypothesis that each Hecke eigenvalue has multiplicity less than .
We provide a simple and new induction-based treatment of the problem of distinguishing cusp forms from the growth of the Fourier coefficients of modular forms. Our approach gives the best possible ranges of the weights for this problem, and has wide adaptability. We propose a conjecture which asks the same converse question based on information on the Fourier-Jacobi coefficients, and answer it partially. We also discuss how to recover cuspidality from the poles of the allied Rankin-Selberg -series.
This paper investigates a fourth-order weighted equation represented as where denotes the unit ball in and is a singular logarithmic weight. The nonlinearity in the equation comprises a reaction source takes the form of a radial function and plays a crucial role in the context of Adams' type exponential inequality and a polynomial function. The Kirchhoff function is positive and continuous. The proof of the existence result utilizes constrained minimization within the Nehari set, supplemented by the application of the quantitative deformation lemma and degree theory results.
We prove a bias towards the zero residue class in the distribution of the integers represented by binary quadratic forms. In most cases, we prove that the bias comes from a secondary term in an associated asymptotic expansion. This is unlike Chebyshev's bias, which exists somewhere at the level of . We make a conjecture on the general situation which includes all cases not proven. Results on the distribution of the integers represented by a quadratic form-some of which are of independent interest-are proven along the way. The paper concludes with some numerical data that is illustrative of the bias.
Let be a real quadratic number field, and let denote its cyclotomic -extension. For each integer , let be the unique intermediate field in such that . By studying the 2-adic divisibility of Dirichlet -series at negative integers, we derive an asymptotic formula that determines the order of the 2-primary part of even -groups of rings of integers of for sufficiently large . As a corollary, we determine their and invariants. We also establish a lower bound for beyond which this asymptotic formula holds. Our results have two main applications: (1) For , or with , we determine the structure of the 2-primary tame kernels . (2) We explicitly determine the three Iwasawa invariants for a family of real quadratic number fields, whose discriminants have arbitrarily many prime divisors.
We consider equations of the form where the variables are all taken to be primes. We define an analogue of the Hasse principle for solubility in the primes (which we call the prime Hasse principle), and prove that, whenever , , this holds for almost all such equations. This is based on work of Br & uuml;dern and Dietmann on the Hasse principle. We then prove some further results about prime solubility and the prime Hasse principle, including a partial converse, and some counterexamples. Of particular interest are counterexamples of degree 2, which show that the analogue of the Hasse-Minkowski theorem fails for prime solubility.
Let . Analogous to orthogonality in the Euclidean space , there exists a well-studied notion of ultrametric orthogonality in . In this paper, we extend the work of [4] on counting problems related to orthogonality in . For example, we resolve an open question posed in [4] by bounding the size of the largest "orthogonal sets" in . Furthermore, using similar ideas and techniques, we investigate analogues of Hadamard matrices over . Finally, we also use ultrametric orthogonality to compute the number of sublattices of with a certain geometric structure, and to determine the number of orthogonal bases of a sublattice in . The resulting formulas depend crucially on successive minima.
We use the classical circle method to give a relatively simple proof that
Several problems are treated about minimizing the absolute value of a real ternary quadratic form and of a real ternary cubic form, when restricted to the integer points on a quadric surface. Among the results are an estimate for the minimum that holds for all ternary quadratic forms and, for certain decomposable ternary cubics, a best possible result that supplements a theorem of Davenport.
We prove a conjecture of R. Oberlin and H & eacute;ra on the dimension of unions of -planes. Let be integers, and . If , with , then . The proof combines a recent idea of Zahl and the Brascamp-Lieb inequality.
Let and be the algebra of all bounded linear operators on a complex Hilbert space and the Jordan algebra of all self-adjoint operators in , respectively. In this paper, we give characterizations of rank one operators by the pseudospectrum on -Lie product of bounded linear operators and discuss some properties about the pseudospectrum. As applications, we obtain the structures of all surjective maps preserving the pseudospectrum of -Lie product on and , respectively.
For all sufficiently large , in any arithmetic progression in which and are relatively prime there exists a positive integer with at most two prime factors (counted with multiplicity) which is asymptotically less than q(1.8165). The proof uses the weighted sieve of Greaves-Halberstam-Richert with bilinear remainder terms and Selberg's sieve.
In this paper, we study traces of Hecke operators on Drinfeld modular forms of level 1 in the case A = F-q[T]. We deduce closed-form expressions for traces of Hecke operators corresponding to primes of degree at most 2 and provide algorithms for primes of higher degree. We improve the Ramanujan bound and deduce the decomposition of cusp forms of level Gamma(0)(& pfr;) into oldforms and newforms, as conjectured by Bandini-Valentino, under the hypothesis that each Hecke eigenvalue has multiplicity less than p.
In 2021, Ordentlich, Regev, and Weiss made a breakthrough that the lattice covering density of any -dimensional convex body is upper bounded by , improving on the best previous bound established by Rogers in 1959. However, for the Euclidean ball, Rogers obtained the better upper bound , and this result was extended to certain symmetric convex bodies by Gritzmann. The constant above is independent on . In this paper, we show that such a bound can be achieved for more general classes of convex bodies without symmetry, including anti-blocking bodies, locally anti-blocking bodies and -dimensional polytopes with vertices.
We establish new geometric inequalities comparing the volumes of sections and projections of a convex body, whose barycenter or Santal & oacute; point is at the origin, with those of its inner and outer regularizations. We also provide functional extensions of these inequalities to the setting of log-concave functions. Our approach relies on the recent optimal -estimate of Bizeul and Klartag for isotropic convex bodies.
In 2019 Kleinbock and Wadleigh proved a "zero-one law" for uniform inhomogeneous Diophantine approximations. We generalize this statement to arbitrary weight functions and establish a new and simple proof of this statement, based on the transference principle. We also give a complete description of the sets of -Dirichlet pairs with a fixed matrix in this setthe up from Lebesgue measure point of view. As an application, we consider the set of badly approximable matrices and give a characterization of bad approximability in terms of inhomogeneous approximations. All the aforementioned metrical descriptions work (and sometimes can be strengthened) for weighted Diophantine approximations.
In this paper, we establish estimates for the expectation and variance of the mixed (2,2)-moment of two Hecke eigenforms of distinct weights. Our results yield applications to triple product -functions. The proofs are based on moments of -functions.
The characterization of commutators in associative algebras is a classical problem in ring theory. In this paper, we address this problem for the natural class of generalized block-triangular algebras. To this end, we introduce a new invariant: the multitrace of an arbitrary element in an associative unital algebra, and prove that in a generalized block-triangular algebra, an element is a commutator if and only if its multitrace vanishes. As a consequence, we show that the set of commutators is closed under addition in these algebras. Our main result extends the classical Albert-Muckenhoupt-Shoda theorem for full matrix algebras to the broader setting of generalized block-triangular algebras.
