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.     

Period relations and p-adic measures

Let (π,σ) be a pair of cuspidal automorphic representations of GL ⁿ × GL ^{n −1} over the adele ring of ℚ having non-vanishing cohomology with constant coefficients. The p-adic distribution interpolating the critical values of the twisted corresponding Rankin–Selberg convolution is shown to be p-adically bounded thus leading to an associated p-adic L-function. Received: 18 October 2000 / Revised version: 27 July 2001     

The generic fiber of the universal deformation space associated to a tame Galois representation

We will study the generic fiber over of the universal deformation ring R _Q , as defined by Mazur, for deformations unramified outside a finite set of primes Q of a given Galois representation , E a number field, k a finite field of characteristic l. The main result will be that, if ˉρ is tame and absolutely irreducible, and if one assumes the Leopoldt conjecture for the splitting field E ₀ of , then defines a smooth l-adic analytic variety, near the trivial lift ρ₀ of ˉρ, whose dimension is given by cohomological constraints and as predicted by Mazur. As a corollary it follows that, in the cases considered here, R _Q is a quotient of by an ideal I generated by exactly m equations, where and . Under the above assumptions for and ˉρ odd, using ideas of Coleman, Gouvêa and Mazur it should now be possible to show that modular points are Zariski-dense in the component of , that contains the trivial lift ρ₀, provided this lift satisfies the Artin conjecture and E ₀ satisfies the Leopoldt conjecture. Furthermore, in the Borel case, we show that the Krull dimension of R _Q can exceed any given number, provided Q is chosen appropriately. At the same time, we present some evidence that despite this fact, one might however expect that the dimension of the generic fiber is given by the same cohomological formula as in the tame case. Received: 12 December 1997 / Revised version: 5 February 1998     

<Emphasis Type="Italic">p</Emphasis>-adic interpolation of Taylor coefficients of modular forms

In this paper we show that the Taylor coefficients of a Hecke eigenform at a CM-point, suitably modified, form a sequence of algebraic numbers that satisfy the Kubota–Leopoldt generalization of the Kummer congruences for primes that split in the imaginary quadratic field associated with a CM-point. More generally, we show that these numbers are moments of a certain p-adic measure. In addition, we write down explicitly the “Euler factor” at p in terms of the p th Hecke eigenvalue of the modular form in question and certain data of the CM-point. P. Guerzhoy is supported by NSF grant DMS-0700933.     

Congruence of Siegel modular forms and special values of their standard zeta functions

In this paper, we consider the relationship between the congruence of cuspidal Hecke eigenforms with respect to Sp _n (Z) and the special values of their standard zeta functions. In particular, we propose a conjecture concerning the congruence between Saito-Kurokawa lifts and non-Saito-Kurokawa lifts, and prove it under certain condition. Partially supported by Grant-in-Aid for Scientific Research C-17540003, JSPS.     

On mod <Emphasis Type="Italic">p</Emphasis> properties of Siegel modular forms

We prove two results on mod p properties of Siegel modular forms. First, we use theta series in order to construct of a Siegel modular form of weight p−1 which is congruent to 1 mod p. Second, we define a theta operator on q-expansions and show that the algebra of Siegel modular forms mod p is stable under , by exploiting the relation between and generalized Rankin-Cohen brackets.     

Arithmetic properties of similitude theta lifts from orthogonal to symplectic groups

Adapting the work of Kudla and Millson we obtain a lifting of cuspidal cohomology classes for the symmetric space associated to GO(V) for an indefinite rational quadratic space V of even dimension to holomorphic Siegel modular forms on GSp _n (A). For n =  2 we prove the Thom Lemma for hyperbolic 3-space, which together with results of Kudla and Millson imply an interpretation of the Fourier coefficients of the theta lift as period integrals of the cohomology class over certain cycles, and relates those over infinite geodesics to L-values of cusp forms for GL₂ over imaginary quadratic fields. This allows us to prove, for almost all primes p, the p-integrality of the lift for a particular choice of Schwartz function. We further calculate the Hecke eigenvalues (including for some “bad” places) for this choice in the case of V of signature (3,1).     

Multiplikatorsysteme und Charaktere Hermitescher Modulgruppen

We consider the hermitian modular group U(2n, o),n2, over an arbitrary imaginary-quadratic number field of discriminantd. By K-theoretic methods it is known that SU(2n, o) is generated by elementary matrices. Therefore, it is easy to determine the factor commutator group and all abelian characters of U(2n, o). Apart from characters which come via the determinant from non-trivial units, there is only one other non-trivial character of U(2n, o) in those cases wheren=2,d0 mod 4. This character arises in the same way as for Siegel's modular group Sp(4, ) by an action of U(4, o) on odd theta characteristics.Multiplier systems of 2n are characters, since they must have integral weight. Therefore as a corollary we obtain all multiplier systems of 2n.

     

Uniformity over primes of tamely ramified splittings

We fix a prime p and let f(X) vary over all monic integer polynomials of fixed degree n. Given any possible shape of a tamely ramified splitting of p in an extension of degree n, we prove that there exists a rational function φ(X)∈ℚ(X) such that the density of the monic integer polynomials f(X) for which the splitting of p has the given shape in ℚ[X]/f(X) is φ(p) (here reducible polynomials can be neglected). As a corollary, we prove that, for p≥n, the density of irreducible monic polynomials of degree n in ℤ _p [X] is the value at p of a rational function φ _n (X)∈ℚ(X). All rational functions involved are effectively computable. Received: 15 September 1998 / Revised version: 21 October 1999     

Hyperbolic Coxeter groups of large dimension

We construct examples of Gromov hyperbolic Coxeter groups of arbitrarily large dimension. We also extend Vinbergs theorem to show that if a Gromov hyperbolic Coxeter group is a virtual Poincaré duality group of dimension n, then n 61.Coxeter groups acting on their associated complexes have been extremely useful source of examples and insight into nonpositively curved spaces over last several years. Negatively curved (or Gromov hyperbolic) Coxeter groups were much more elusive. In particular their existence in high dimensions was in doubt.In 1987 Gabor Moussong [M] conjectured that there is a universal bound on the virtual cohomological dimension of any Gromov hyperbolic Coxeter group. This question was also raised by Misha Gromov [G] (who thought that perhaps any construction of high dimensional negatively curved spaces requires nontrivial number theory in the guise of arithmetic groups in an essential way), and by Mladen Bestvina [B2].In the present paper we show that high dimensional Gromov hyperbolic Coxeter groups do exist, and we construct them by geometric or group theoretic but not arithmetic means.     

Restricting Hecke-Siegel operators to Jacobi modular forms

We compute the action of Hecke operators on Jacobi forms of “Siegel degree” n and m×m index M, provided 1?j?n−m. We find they are restrictions of Hecke operators on Siegel modular forms, and we compute their action on Fourier coefficients. Then we restrict the Hecke-Siegel operators T(p), Tj(p²) (n−m<j?n) to Jacobi forms of Siegel degree n, compute their action on Fourier coefficients and on indices, and produce lifts from Jacobi forms of index M to Jacobi forms of index M^′ where detM|detM^′. Finally, we present an explicit choice of matrices for the action of the Hecke operators on Siegel modular forms, and for their restrictions to Jacobi modular forms.     

Galois actions on Q-curves and winding quotients

We prove two ``large images' results for the Galois representations attached to a degree d Q-curve E over a quadratic field K: if K is arbitrary, we prove maximality of the image for every prime p>13 not dividing d, provided that d is divisible by q (but d≠q) with q=2 or 3 or 5 or 7 or 13. If K is real we prove maximality of the image for every odd prime p not dividing d D, where D= disc(K), provided that E is a semistable Q-curve. In both cases we make the (standard) assumptions that E does not have potentially good reduction at all primes p∤6 and that d is square free. The first author is supported by BFM2003-06092.     

Companion forms over totally real fields

We show that if F is a totally real field in which p splits completely and f is a mod p Hilbert modular form with parallel weight 2 < k < p, which is ordinary at all primes dividing p and has tamely ramified Galois representation at all primes dividing p, then there is a “companion form” of parallel weight k′ := p + 1 − k. This work generalises results of Gross and Coleman–Voloch for modular forms over Q.     

On the value set of the Ramanujan function

We give lower bounds on the number of distinct values of the Ramanujan function τ(n), n ≦ x, and on the number of distinct residues of τ(n), n ≦ x, modulo a prime ℓ. We also show that for any prime ℓ the values τ(n), n ≦ ℓ⁴, form a finite additive basis modulo ℓ. Received: 6 October 2004     

A relative Kuznietsov trace formula on G2

In this paper, we set up the general formulation to study distinguished residual representations of a reductive group G by the relative trace formula approach. This approach simplifies the argument of [JR], which deals with this type of relative trace formula for a special symmetric pair (GL(2n), Sp(2n)) and also works for non-symmetric, spherical pairs. To illustrate our idea and method, we complete our relative trace formula (both the geometric side identity and the spectral side identity) for the case (G ₂, SL(3)). Received: 6 February 1999     

Growth functions for semi-regular tilings of the hyperbolic plane

This paper studies the growth function, with respect to the generating set of edge identifications, of a surface group with fundamental domainD in the hyperbolic plane ann-gon whose angles alternate between /p and /q. The possibilities ofn,p andq for which a torsion-free surface group can have such a fundamental polygon are classified, and the growth functions are computed. Conditions are given for which the denominator of the growth function is a product of cyclotomic polynomials and a Salem polynomial.This work was supported in part by NSF Research Grants.     

Minimal length of two intersecting simple closed geodesics

On a hyperbolic Riemann surface, given two simple closed geodesics that intersect n times, we address the question of a sharp lower bound L _n on the length attained by the longest of the two geodesics. We show the existence of a surface S _n on which there exists two simple closed geodesics of length L _n intersecting n times and explicitly find L _n for . The first author was supported in part by SNFS grant number 2100-065270, the second author was supported by SNFS grant number PBEL2-106180.     

Towards the arithmetic degree of line bundles¶ on abelian varieties

In this paper we analyze the integral of the star-product of (n+1) Green currents associated to (n+1) global sections of an ample line bundle equipped with a translation invariant metric over an n-dimensional, polarized abelian variety. The integral is shown to equal the logarithm of the Petersson norm of a certain Siegel modular form, which is explicitly described in terms of the given data. This result can be interpreted as evaluating an archimedian height on a family of polarized abelian varieties. The key ingredient to the proof of the main formula is a dd ^c -variational formula for the integral under consideration. In the case of dimensions n=1,2,3 explicit examples in terms of classical Riemann theta functions are given. Received: 13 February 1998     

On the index of composition of integers from various sets

Given an integer n ≥ 2, let λ(n) := (log n)/(log γ(n)), where γ(n) = Π _p|n p, stand for the index of composition of n, with λ(1) = 1. We study the distribution function of (λ(n) – 1) log n as n runs through particular sets of integers, such as the shifted primes, the values of a given irreducible cubic polynomial and the shifted powerful numbers. Research supported in part by a grant from NSERC. Research supported by the Applied Number Theory Research Group of the Hungarian Academy of Science and by a grant from OTKA. Professor M.V. Subbarao passed away on February 15, 2006. Received: 3 March 2006 Revised: 28 October 2006