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 between Modular Forms, Cyclic Isogenies of Modular Elliptic Curves, and Integrality of -Functions

Let be a congruence subgroup of type and of level . We study congruences between weight 2 normalized newforms and Eisenstein series on modulo a prime above a rational prime . Assume that , is a common eigenfunction for all Hecke operators and is ordinary at . We show that the abelian variety associated to and the cuspidal subgroup associated to intersect non-trivially in their -torsion points. Let be a modular elliptic curve over with good ordinary reduction at . We apply the above result to show that an isogeny of degree divisible by from the optimal curve in the -isogeny class of elliptic curves containing to extends to an étale morphism of Néron models over if . We use this to show that -adic distributions associated to the -adic -functions of are -valued.

     

Modular Form Congruences and Selmer Groups

Motivated by Cremona and Mazur's notion of visibility of elementsin Shafarevich–Tate groups [6, 27], there have been anumber of recent works which test its compatibility with theBirch and Swinnerton–Dyer conjecture and the Bloch–Katoconjecture. These conjectures provide formulas for the ordersof Shafarevich–Tate groups in terms of values of L-functions.For example, one may see recent work of Agashe, Dummigan, Steinand Watkins [1, 2, 10, 11]. In their examples, they find thatthe presence of visible elements agrees with the expected divisibilityproperties of the relevant L-values.     

L-Functions for Jacobi Forms of Arbitrary Degree

A finite number ofL-functions are associated to every Jacobi cusp form of degreen. TheseL-functions are infinite series constructed with the Fourier coefficients of the form and a variables in ℂⁿ. It is proved that eachL-function has an integral representation, admits a holomorphic continuation to the whole space ℂⁿ, and the row vector formed with them satisfies a particular matrix functional equation.     

Discrete Universality of L-Functions for New Forms

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

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.     

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

半模的张量积

本文引入了半模的张量积的泛性定义，给出了半模范畴中的有向正向系的上极限以及任意一个半模同态的上核，讨论了半模张量积的若干性质，主要结果是：证明了张量函子保持半模的右正合性和上积，并且上积保持半模的平坦性，半模范畴中的张量函子保持上极限和上核．     

Congruence Restricted Modular Forms

Given f(z), a modular form on a congruence subgroup (of the full modular group), we construct the function f(z;r,t) by summing over the terms of the Fourier expansion of f(z) with index congruent to r modulo t. In this paper, we determine a condition on the multiplier system of f(z) which guarantees that f(z;r,t) is itself a modular form on a (smaller) congruence subgroup.2000 Mathematics Subject Classification: Primary—11F11; Secondary—11F30     

On Picard Modular Forms

We study moduli spaces of principally polarized abelian varieties with an automorphism of finite order. After some examples (e. g. hermitian modular forms) we compute the ring of Picard modular forms in the case considered by Picard.     

Holomorphic Almost Modular Forms

Holomorphic almost modular forms are holomorphic functions ofthe complex upper half plane that can be approximated arbitrarilywell (in a suitable sense) by modular forms of congruence subgroupsof large index in SL(2,Z). It is proved that such functionshave a rotation-invariant limit distribution when the argumentapproaches the real axis. An example of a holomorphic almostmodular form is the logarithm of . The paper is motivated by the author's previous studies [Int.Math. Res. Not. 39 (2003) 2131–2151] on the connectionbetween almost modular functions and the distribution of thesequence n²x modulo one. 2000 Mathematics Subject Classification11F11 (primary), 11F06, 11J71 (secondary).     

R-n模张量积与张量函子

文[1]引进了左R-n模范畴_RM_n~L,本文是在_RM_n中,建立相应的张量积,证明了它的存在性与唯一性,并讨论了张量函子与Hom函子的伴随性。 文中沿用[1]的记号。     

Chaotic Tensor Product Semigroups

Suppose {G₁(t)}_{t ≥ 0} and {G₂(t)_{t ≥ 0} be two semigroups on an infinite dimensional separable reflexive Banach space X. In this paper we give sufficient conditions for tensor product semigroup G(t): X → G₂(t)X G₁(t) to become chaotic in L with the strong operator topology and chaotic in the ideal of compact operators on X with the norm operator topology.     

L—fuzzy模范畴的张量积与张量函子

本文在Ｌ—ｆｕｚｚｙ模范畴中，建立了相应的张量积，给出了它的结构性、存在性与唯一性定理，并讨论了张量函子与Ｈｏｍ函子的伴随性。所得结果为通常张量积的“良好推广”（ｇｏｏｄｅｘｔｅｎｓｉｏｎ）。     

Heegner Divisors and Nonholomorphic Modular Forms

We consider an embedded modular curve in a locally symmetric space M attached to an orthogonal group of signature (p, 2) and associate to it a nonholomorphic elliptic modular form by integrating a certain theta function over the modular curve. We compute the Fourier expansion and identify the generating series of the (suitably defined) intersection numbers of the Heegner divisors in M with the modular curve as the holomorphic part of the modular form. This recovers and generalizes parts of work of Hirzebruch and Zagier.     

Tensor Product of Massey Products

In this paper, we interpret Massey products in terms of realizations （twitsting cochains） of certain differential graded coalgebras with values in differential graded algebras. In the case where the target algebra is the cobar construction of a differential graded commutative Hopf algebra, we construct the tensor product of realizations and show that the tensor product is strictly associative, and commutative up to homotopy.     

Derivations of Tensor Product Algebras

It is known that under certain finite dimensionality condition the derivation algebra of tensor product of two algebras can be obtained in terms of the derivation algebras and the centroids of the involved algebras. We extend this theorem to infinite dimensional case and as an application, we determine the derivation algebra of the fixed point algebra of the tensor product of two algebras, with respect to the tensor product of two finite order automorphisms. These provide the framework for calculating the derivations of some infinite dimensional Lie algebras.