Almost perfect powers in consecutive integers (II)

Let k ≥ 4 be an integer. We find all integers of the form by^l where l ≥ 2 and the greatest prime factor of b is at most k (i.e. nearly a perfect power) such that they are also products of k consecutive integers with two terms omitted.     

Moduli for pairs of elliptic curves¶with isomorphic N-torsion

We study the moduli surface for pairs of elliptic curves together with an isomorphism between their N-torsion groups. The Weil pairing gives a “determinant” map from this moduli surface to (Z/N Z)^*; its fibers are the components of the surface. We define spaces of modular forms on these components and Hecke correspondences between them, and study how those spaces of modular forms behave as modules for the Hecke algebra. We discover that the component with determinant −1 is somehow the “dominant” one; we characterize the difference between its spaces of modular forms and the spaces of modular forms on the other components using forms with complex multiplication. In addition, we prove Atkin–Lehner-style results about these spaces of modular forms. Finally, we show some simplifications that arise when N is prime, including a complete determination of such CM-forms, and give numerical examples. Received: 20 September 2000 / Revised version: 7 February 2001     

A Hecke correspondence theorem for automorphic integrals with symmetric rational period functions on the Hecke groups

Marvin Knopp showed that entire automorphic integrals with rational period functions satisfy a Hecke correspondence theorem, provided the rational period functions have poles only at 0 or ∞. For other automorphic integrals the corresponding Dirichlet series has a functional equation with a remainder term that arises from the nonzero poles of the rational period function. In this paper we prove a Hecke correspondence theorem for a class of automorphic integrals with rational period functions on the Hecke groups. We restrict our attention to automorphic integrals of weight that is twice an odd integer and to rational period functions that satisfy a symmetry property we call “Hecke-symmetry.” Each remainder term satisfies two relations (the second of which is new in this paper) corresponding to the two relations for the rational period function.     

On Atkin and Swinnerton-Dyer congruence relations (3)

It is now well known that Hecke operators defined classically act trivially on genuine cuspforms for noncongruence subgroups of SL₂(Z). Atkin and Swinnerton-Dyer speculated the existence of p-adic Hecke operators so that the Fourier coefficients of their eigenfunctions satisfy three-term congruence recursions. In the previous two papers with the same title ([W.C. Li, L. Long, Z. Yang, On Atkin and Swinnerton-Dyer congruence relations, J. Number Theory 113 (1) (2005) 117-148] by W.C. Li, L. Long, Z. Yang and [A.O.L. Atkin, W.C. Li, L. Long, On Atkin and Swinnerton-Dyer congruence relations (2), Math. Ann. 340 (2) (2008) 335-358] by A.O.L. Atkin, W.C. Li, L. Long), the authors have studied two exceptional spaces of noncongruence cuspforms where almost all p-adic Hecke operators can be diagonalized simultaneously or semi-simultaneously. Moreover, it is shown that the l-adic Scholl representations attached to these spaces are modular in the sense that they are isomorphic, up to semisimplification, to the l-adic representations arising from classical automorphic forms.In this paper, we study an infinite family of spaces of noncongruence cuspforms (which includes the cases in [W.C. Li, L. Long, Z. Yang, On Atkin and Swinnerton-Dyer congruence relations, J. Number Theory 113 (1) (2005) 117-148; A.O.L. Atkin, W.C. Li, L. Long, On Atkin and Swinnerton-Dyer congruence relations (2), Math. Ann. 340 (2) (2008) 335-358]) under a general setting. It is shown that for each space in this family there exists a fixed basis so that the Fourier coefficients of each basis element satisfy certain weaker three-term congruence recursions. For a new case in this family, we will exhibit that the attached l-adic Scholl representations are modular and the p-adic Hecke operators can be diagonalized semi-simultaneously.     

Higher order spt-functions

Andrews? spt-function can be written as the difference between the second symmetrized crank and rank moment functions. Using the machinery of Bailey pairs a combinatorial interpretation is given for the difference between higher order symmetrized crank and rank moment functions. This implies an inequality between crank and rank moments that was only known previously for sufficiently large n and fixed order. This combinatorial interpretation is in terms of a weighted sum of partitions. A number of congruences for higher order spt-functions are derived.     

?-adic properties of smallest parts functions

We prove explicit congruences modulo powers of arbitrary primes for three smallest parts functions: one for partitions, one for overpartitions, and one for partitions without repeated odd parts. The proofs depend on ?-adic properties of certain modular forms and mock modular forms of weight 3/2 with respect to the Hecke operators T(?2m).     

On the first cohomology of arithmetic groups

We study the restriction to smaller subgroups, of cohomology classes on arithmetic groups (possibly after moving the class by Hecke correspondences), especially in the context of first cohomology of arithmetic groups. We obtain vanishing results for the first cohomology of cocompact arithmetic lattices in SU(n,1) which arise from hermitian forms over division algebras D of degree p ², p an odd prime, equipped with an involution of the second kind. We show that it is not possible for a ‘naive’ restriction of cohomology to be injective in general. We also establish that the restriction map is injective at the level of first cohomology for non co-compact lattices, extending a result of Raghunathan and Venkataramana for co-compact lattices. Received: 14 September 2000 / Accepted: 6 June 2001     

p-Adic liftings of the supersingular j-invariants and j-zeros of certain Eisenstein series

Let p>3 be a prime. We consider j-zeros of Eisenstein series Ek of weights k=p−1+Mpa(p²−1) with M,a?0 as elements of . If M=0, the j-zeros of Ep−1 belong to Qp(ζp²−1) by Hensel's lemma. Call these j-zeros p-adic liftings of supersingular j-invariants. We show that for every such lifting u there is a j-zero r of Ek such that ordp(r−u)>a. Applications of this result are considered. The proof is based on the techniques of formal groups.     

Eisenstein series and theta functions to the septic base

We develop a theory for Eisenstein series to the septic base, which was started by S. Ramanujan in his “Lost Notebook.” We show that two types of septic Eisenstein series may be parameterized in terms of the septic theta function and the eta quotient η⁴(7τ)/η⁴(τ). This is accomplished by constructing elliptic functions which have the septic Eisenstein series as Taylor coefficients. The elliptic functions are shown to be solutions of a differential equation, and this leads to a recurrence relation for the septic Eisenstein series.     

The algebraic structure of relative twisted vertex operators

Twisted vertex operators based on rational lattices have had many applications in vertex operator algebra theory and conformal field theory. In this paper, “relativized” twisted vertex operators are constructed in a general context based on isometries of rational lattices, and a generalized twisted Jacobi identity is established for them. This result generalizes many previous results. Relatived untwisted vertex operators had been studied in a monograph by the authors. The present paper includes as a special case the proof of the main relations among twisted vertex operators based on even lattices announced some time ago by the second author.     

Slopes of integral lattices

We use the dyadic trace to define the concept of slope for integral lattices. We present an introduction to the theory of the slope invariant. The main theorem states that a Siegel modular cusp form f of slope strictly less than the slope of an integral lattice with Gram matrix s satisfies f(sτ)=0 for all τ in the upper half plane. We compute the dyadic trace and the slope of each root lattice and we give applications to Siegel modular cusp forms.     

Special values of L-functions by a Siegel-Weil-Kudla-Rallis formula

We study the arithmeticity of special values of L-functions attached to cuspforms which are Hecke eigenfunctions on hermitian quaternion groups Sp^∗(m,0) which form a reductive dual pair with G=O^∗(4n). For f₁ and f₂ two cuspforms on H, consider their theta liftings θf₁ and θf₂ on G. Then we compute a Rankin-Selberg type integral and obtain an integral representation of the standard L-function:

G〈θf₁⋅Es,θf₂〉=H〈f₁,f₂〉⋅Lstd(f₁,s).     

Iwasawa invariants for the False-Tate extension and congruences between modular forms

For an ordinary prime p?3, we consider the Hida family associated to modular forms of a fixed tame level, and their Selmer groups defined over certain Galois extensions of Q(μp) whose Galois group is G≅Zp?Zp. For Selmer groups defined over the cyclotomic Zp-extension of Q(μp), we show that if the μ-invariant of one member of the Hida family is zero, then so are the μ-invariants of the other members, while the λ-invariants remain the same only in a branch of the Hida family. We use these results to study the behavior of some invariants from non-commutative Iwasawa theory in the Hida family.     

Newforms of squarefree level and theta series

We show that the cuspidal part of the span of the theta series associated to maximal integral lattices on a definite quaternion algebra over is precisely the space of newforms. Furthermore, using the Eichler commutation relation and an idea coming out of the Yoshida lifting, we show how to express each newform as an explicit linear combination of such theta series. The coefficients of these linear combinations come from cuspidal eigenvectors of Brandt matrices.     

The sixth and eighth moments of Fourier coefficients of cusp forms

Let λ(n) be the nth normalized Fourier coefficient of a holomorphic Hecke eigencuspform f(z) of even integral weight k for the full modular group. In this paper we are able to prove the following results.

(i)

For any ε>0, we have

     

A note on linear permutation polynomials

Kai Zhou (2008) [8] gave an explicit representation of the class of linear permutation polynomials and computed the number of them. In this paper, we give a simple proof of the above results.     

Second order average estimates on local data of cusp forms

We specify sufficient conditions for the square modulus of the local parameters of a family of GL_n cusp forms to be bounded on average. These conditions are global in nature and are satisfied for n ≦ 4. As an application, we show that Rankin-Selberg L-functions on , for n_i ≦ 4, satisfy the standard convexity bound. Received: 21 June 2005     

On some entire modular forms of weights 5 and 6 for the congruence group Г(in0)(4N)

Two entire modular forms of weight 5 and two of weight 6 for the congruence subgroup Γ₀ (4N)are constructed, which will be useful for revealing the arithmetical sense of additional terms in formulas for the number of representations of positive integers by quadratic forms in 10 and 12 variables.     

Exceptional ergodic directions in Eaton lenses

We construct examples of ergodic vortical flows in periodic configurations of Eaton lenses of fixed radius. We achieve this by studying a family of infinite translation surfaces that are ?²-covers of slit tori. We show that the Hausdorff dimension of lattices for which the vertical flow is ergodic is bigger than 3/2. Moreover, the lattices are explicitly constructed.     

On the representation of integers as sums of triangular numbers

Summary In this survey article we discuss the problem of determining the number of representations of an integer as sums of triangular numbers. This study yields several interesting results. Ifn 0 is a non-negative integer, then thenth triangular number isT _n =n(n + 1)/2. Letk be a positive integer. We denote by _k (n) the number of representations ofn as a sum ofk triangular numbers. Here we use the theory of modular forms to calculate _k (n). The case wherek = 24 is particularly interesting. It turns out that, ifn 3 is odd, then the number of points on the 24 dimensional Leech lattice of norm 2n is 2¹²(2¹² – 1) ₂₄(n – 3). Furthermore the formula for ₂₄(n) involves the Ramanujan(n)-function. As a consequence, we get elementary congruences for(n). In a similar vein, whenp is a prime, we demonstrate ₂₄(p ^k – 3) as a Dirichlet convolution of ₁₁(n) and(n). It is also of interest to know that this study produces formulas for the number of lattice points insidek-dimensional spheres.