Discrete Universality of L-Functions for New Forms

A Voronin-type discrete universality theorem for the L-functions of new forms is proved.     

L-Functions of Jacobi forms with Shimura type

For every Jacobi form of Shimura type over H × C, a system of L-functions associated to it is given. These L-functions can be analytically continued to the whole complex plane and satisfy a kind of functional equation. As a consequence, Hecke's inverse theorem on modular forms is extended to the context of Jacobi forms with Shimura type.     

Subconvexity for Rankin-Selberg L-Functions of Maass Forms

In this paper we prove a subconvexity bound for Rankin–Selberg L-functions associated with a Maass cusp form f and a fixed cusp form g in the aspect of the Laplace eigenvalue 1/4 + k² of f, on the critical line Re s = 1/2. Using this subconvexity bound, we prove the equidistribution conjecture of Rudnick and Sarnak [RS] on quantum unique ergodicity for dihedral Maass forms, following the work of Sarnak [S2] and Watson [W]. Also proved here is that the generalized Lindelöf hypothesis for the central value of our L-function is true on average.     

Algebraic Boundaries Among Typical Ranks for Real Binary Forms of Arbitrary Degree

Foundations of Computational Mathematics - We show that the algebraic boundaries of the regions of real binary forms with fixed typical rank are always unions of dual varieties to suitable...     

Generating Jacobi Forms

In this paper we explore the relationship between vector-valued modular forms and Jacobi forms and give explicit relations over various congruence subgroups. The main result is that a Jacobi form of square-free index on the full Jacobi group is uniquely determined by any of the associated vector components. In addition, an explicit construction is given to determine the other vector components from this single component. In other words, we give an explicit construction of a Jacobi form from a subset of its Fourier coefficients. This leads to results about how the transformation properties are affected by congruence restrictions on the Fourier expansion. 2000 Mathematics Subject Classification: Primary—11F50; Secondary—11F30     

The L-Functions and Modular Forms Database Project

The Langlands Programme, formulated by Robert Langlands in the 1960s and since much developed and refined, is a web of interrelated theory and conjectures concerning many objects in number theory, their interconnections, and connections to other fields. At the heart of the Langlands Programme is the concept of an L-function. The most famous L-function is the Riemann zeta function, and as well as being ubiquitous in number theory itself, L-functions have applications in mathematical physics and cryptography. Two of the seven Clay Mathematics Million Dollar Millennium Problems, the Riemann Hypothesis and the Birch and Swinnerton-Dyer Conjecture, deal with their properties. Many different mathematical objects are connected in various ways to L-functions, but the study of those objects is highly specialized, and most mathematicians have only a vague idea of the objects outside their specialty and how everything is related. Helping mathematicians to understand these connections was the motivation for the L-functions and Modular Forms Database (LMFDB) project. Its mission is to chart the landscape of L-functions and modular forms in a systematic, comprehensive, and concrete fashion. This involves developing their theory, creating and improving algorithms for computing and classifying them, and hence discovering new properties of these functions, and testing fundamental conjectures. In the lecture I gave a very brief introduction to L-functions for non-experts and explained and demonstrated how the large collection of data in the LMFDB is organized and displayed, showing the interrelations between linked objects, through our website www.lmfdb.org. I also showed how this has been created by a worldwide open-source collaboration, which we hope may become a model for others.     

Congruences of Modular Forms and Tensor Product L-Functions

Two normalised Hecke eigenforms f and g are considered in thispaper, of level one and weights k' > k, lying in a p-adicfamily, such that f g(mod p^r as q-expansions. Interpretingthe congruence in terms of the associated Galois representationsleads to the existence of non-trivial global p^r-torsion forthe motive associated with the tensor product L-function ats = k' – 1. (It must be assumed that the (mod p) Galoisrepresentation attached to g is irreducible.) This contributesa factor of p^r to the denominator of the Bloch–Kato conjecturalformula for the L-value. The p-part of the numerator is considered,using recent work of Diamond, Flach and Guo. Using Shimura'sRankin–Selberg integral formula and the Clausen–vonStaudt theorem, the ratio of L-values is examined at s = k'– 1, for the tensor product and a quadratic twist; confirmationis given (under certain conditions) that, at p, this is as predictedby Bloch and Kato. 2000 Mathematics Subject Classification 11F33,11F67, 11G40.     

Constructing Modular Forms from Harmonic Maass Jacobi Forms

Czechoslovak Mathematical Journal - We construct a family of modular forms from harmonic Maass Jacobi forms by considering their Taylor expansion and using the method of holomorphic projection. As...     

Jacobi Condition for Elliptic Forms in Hilbert Spaces

A definition given in Ref. 1 is corrected together with its consequences.     

Symmetric Squares of Hecke L-Functions and Fourier Coefficients of Cusp Forms

Let S _k (₀(N)) be the space of cusp forms of even weight k for ₀ (N), let be the set of all newforms in S _k ( ₀ (N)), and let be the symmetric square of the Hecke L-function of a form . It is proved that for N=p we have

Round Forms of Higher Degree

We study forms ${\varphi}$ of higher degree over a field k which are round; i.e., where each element in k which is represented by ${\varphi}$ is a similarity factor of ${\varphi}$ .     

Bounds for Arcs of Arbitrary Degree in Finite Desarguesian Planes

This paper examines subsets with at most n points on a line in the projective plane . A lower bound for the size of complete ‐arcs is established and shown to be a generalisation of a classical result by Barlotti. A sufficient condition ensuring that the trisecants to a complete (k, 3)‐arc form a blocking set in the dual plane is provided. Finally, combinatorial arguments are used to show that, for , plane (k, 3)‐arcs satisfying a prescribed incidence condition do not attain the best known upper bound.     

Norm Principles for Forms of Higher Degree Permitting Composition

Let F be a field of characteristic 0 or greater than d. Scharlau's norm principle holds for finite field extensions K over F, for certain forms ? of degree d over F which permit composition.     

A Limit Theorem for Dirichlet L-Functions

We prove a limit theorem on the weak convergence of probability measures in the space of continuous functions for Dirichlet L-functions. The result generalizes a similar theorem for the Riemann zeta-function.     

On Improved Bounds for Bounded Degree Spanning Trees for Points in Arbitrary Dimension

Given points in Euclidean space of arbitrary dimension, we prove that there exists a spanning tree having no vertices of degree greater than 3 with weight at most 1.559 times the weight of the minimum spanning tree. We also prove that there is a set of points such that no spanning tree of maximal degree 3 exists that has this ratio be less than 1.447. Our central result is based on the proof of the following claim: Given n points in Euclidean space with one special point v, there exists a Hamiltonian path with an endpoint at v that is at most 1.559 times longer than the sum of the distances of the points to v. These proofs also lead to a way to find the tree in linear time given the minimal spanning tree.     

L-函数的唯一性

本文利用Nevanlinna理论,研究了在广义Selberg类中的L-函数的唯一性.证明了存在两个集合S_1(含有一个或两个元素),S2(含有三个元素),使得当E(S_i,f)=E(S_i,L),i=1,2时,有f≡L.     

Inverses of Hardy L-Functions

Hardy conjectured that there exists an L-function whose inverseis not asymptotic to any L-function. We show that this is trueof the L-function log(logx).log(log(logx)).     

Integral Moments of L-Functions

We give a new heuristic for all of the main terms in the integralmoments of various families of primitive L-functions. The resultsagree with previous conjectures for the leading order terms.Our conjectures also have an almost identical form to exactexpressions for the corresponding moments of the characteristicpolynomials of either unitary, orthogonal, or symplectic matrices,where the moments are defined by the appropriate group averages.This lends support to the idea that arithmetical L-functionshave a spectral interpretation, and that their value distributionscan be modelled using Random Matrix Theory. Numerical examplesshow good agreement with our conjectures. 2000 Mathematics SubjectClassification 11M26, 15A52.