• Volume 97, Issue 1-3

December 1987,   pages  1-324

• Foreword

• The characters of supercuspidal representations as weighted orbital integrals

Weighted orbital integrals are the terms which occur on the geometric side of the trace formula. We shall investigate these distributions on ap-adic group. We shall evaluate the weighted orbital integral of a supercuspidal matrix coefficient as a multiple of the corresponding character.

• On the proof of the reciprocity law for arithmetic Siegel modular functions

Earlier we obtained a new proof of Shimura’s reciprocity law for the special values of arithmetic Hilbert modular functions. In this note we show how from this result one may derive Shimura’s reciprocity law for special values of arithmetic Siegel modular functions. To achieve this we use Shimura’s classification of the special points of the Siegel space, Satake’s classification of the equivariant holomorphic imbeddings of Hilbert-Siegel modular spaces into a larger Siegel space, and, finally, a corrected version of some of Karel’s results giving an action of the Galois group Gal(Qab/Q) on arithmetic Siegel modular forms.

• On some generalizations of Ramanujan’s continued fraction identities

In this note we establish continued fraction developments for the ratios of the basic hypergeometric function2ϕ1(a,b;c;x) with several of its contiguous functions. We thus generalize and give a unified approach to establishing several continued fraction identities including those of Srinivasa Ramanujan.

• On the set of discrete subgroups of bounded covolume in a semisimple group

In this noteG is a locally compact group which is the product of finitely many groups Gs(ks)(s∈S), where ks is a local field of characteristic zero and Gs an absolutely almost simpleks-group, ofks-rank ≥1. We assume that the sum of the rs is ≥2 and fix a Haar measure onG. Then, given a constantc &gt; 0, it is shown that, up to conjugacy,G contains only finitely many irreducible discrete subgroupsL of covolume ≥c (4.2). This generalizes a theorem of H C Wang for real groups. His argument extends to the present case, once it is shown thatL is finitely presented (2.4) and locally rigid (3.2).

• Explicit Ramanujan-type approximations to pi of high order

We combine previously developed work with a variety of Ramanujan’s higher order modular equations to make explicit, in very simple form, algebraic approximations to π which converge with orders including 7, 11, 15 and 23.

• Almost poised basic hypergeometric series

Given a basic hypergeometric series with numerator parametersa1,a2, ...,ar and denominator parametersb2, ...,br, we say it isalmost poised ifbi, =a1qδ,iaii = 0, 1 or 2, for 2 ≤ir. Identities are given for almost poised series withr = 3 andr = 5 when a1, =q−2n.

• On Whittaker models and the vanishing of Fourier coefficients of cusp forms

The purpose of this paper is to construct examples of automorphic cuspidal representations which possess a ψ-Whittaker model even though their ψ-Fourier coefficients vanish identically. This phenomenon was known to be impossible for the groupGL(n), but in general remained an open problem. Our examples concern the metaplectic group and rely heavily upon J L Waldspurger’s earlier analysis of cusp forms on this group.

• On prime representing polynomials

A heuristic method is presented to determine the number of primesp ≤x, represented by an irreducible polynomialf(n), without non-trivial fixed factor (f(y)&lt;∈Z[y]; n∈Z. The method is applied to two specific polynomials and the results are compared with those of the heuristic approach of Hardy and Littlewood.

• (GLn, GLm)-duality and symmetric plethysm

In [7] the author has given an exposition of the theory of invariants of binary forms in terms of a particular version of Classical Invariant Theory. Reflection shows that many aspects of the development apply also ton-ary forms. The purpose of this paper is to make explicit this more general application. The plethysms S’(Sp(ℂn)) are computed quite explicitly forl = 2, 3 and 4.

• The area within a curve

The area of a simple closed convex curve can be estimated in terms of the number of points of a square lattice that lie within the curve. We obtain the usual error bound without integration using a form of the Hardy—Littlewood—Ramanujan circle method, and also present simple estimates for the mean square error.

• On the nonvanishing of someL-functions

The non-vanishing, at the centre of symmetry, of theL-function attached to an automorphic representation of GL(2) or its twists by quadratic characters has been extensively investigated, in particular by Waldspurger. The purpose of this paper is to outline a new proof of Waldspurger’s results. The automorphic representations of GL(2) and its metaplectic cover are compared in two different ways; one way is by means of a “relative trace formula”; the relative trace formula presented here is actually a generalization of the work of Iwaniec.

• On exponential sums involving the Ramanujan function

Let τ(n) be the arithmetical function of Ramanujan, α any real number, and x≥2. The uniform estimate$$\mathop \Sigma \limits_{n \leqslant x} \tau (n)e(n\alpha ) \ll x^6 \log x$$ is a classical result of J R Wilton. It is well known that the best possible bound would be ≪x6. The validity of this hypothesis is proved.

• Approximation of exponential sums by shorter ones

A new theorem on approximation of exponential sum by shorter one is proved.

• On endomorphisms of degree two

LetR be a commutative ring, Δ∈R and letRΔ be the set of conjugacy classes ofR-module endomorphismsf satisfyingf ∘ f = Δ·id. Using a certain subspace of the tensor product of two endomorphisms a commutative and associative product on Rx0394; can be defined. ForR = ℤ a generalization of the composition of quadratic forms arises as a special case.

• Traces of Eichler—Brandt matrices and type numbers of quaternion orders

LetA be a totally definite quaternion algebra over a totally real algebraic number fieldF andM be the ring of algebraic integers ofF. For anyM-orderG ofA we derive formulas for the massm(G) and the type numbert(G) of G and for the trace of the Eichler-Brandt matrixB(G, J) ofG and any integral idealJ ofM in terms of genus invariants ofG and of invariants ofF andJ. Applications to class numbers of quaternion orders and of ternary quadratic forms are indicated.

• The Hecke-algebras related to the unimodular and modular group over the Hurwitz order of integral quaternions

In the present paper the elementary divisor theory over the Hurwitz order of integral quaternions is applied in order to determine the structure of the Hecke-algebras related to the attached unimodular and modular group of degreen. In the casen = 1 the Hecke-algebras fail to be commutative. Ifn &gt; 1 the Hecke-algebras prove to be commutative and coincide with the tensor product of their primary components. Each primary component turns out to be a polynomial ring inn resp.n + 1 resp. 2n resp. 2n+1 algebraically independent elements. In the case of the modular group of degreen, the law of interchange with the Siegel ϕ-operator is described. The induced homomorphism of the Hecke-algebras is surjective except for the weightsr = 4n-4 andr = 4n-2.

• Poincaré series forSO(n, 1)

A theory of Poincaré series is developed for Lobachevsky space of arbitrary dimension. For a general non-uniform lattice a Selberg-Kloosterman zeta function is introduced. It has meromorphic continuation to the plane with poles at the corresponding automorphic spectrum. When the lattice is a unit group of a rational quadratic form, the Selberg-Kloosterman zeta function is computed explicitly in terms of exponential sums. In this way a non-trivial Ramanujan-like bound analogous to “Selberg’s 3/16 bound” is proved in general.

• Fluctuations in the mean of Euler’s phi function

We consider the error term in the mean value estimate of Euler’s phi function ψ(n), and show that it is Ω+- (x(log log x)1/2). This improves on the earlier results of Pillai and Chowla, and of Erdös and Shapiro.

• On the supersingular reduction of elliptic curves

Let a∈Q and denote byEa the curvey2 = (x2+ l)(x + a). We prove thatEa(Fp) is cyclic for infinitely many primesp. This fact was known previously only under the assumption of the generalized Riemann hypothesis.

• The Manin—Drinfeld theorem and Ramanujan sums

The Manin—Drinfeld theorem asserts the finiteness of the cuspidal divisor class group of a modular curve corresponding to a congruence subgroup. The purpose of the note is to draw attention to the connection between this theorem and Ramanujan sums, and to the question of what happens for non-congruence subgroups.

• On Ramanujan’s modular identities

For Ramanujan’s modular identities connected with his well-known partition congruences for the moduli 5 or 7, we had given, in an earlier paper, natural and uniform proofs through the medium of modular forms. Analogous (modular) identities corresponding to the (more difficult) case of the modulus 11 are provided here, with the consequent partition congruences; the relationship with relevant results of N J Fine is also sketched.

• Hypergeometric series and continued fractions

Ramanujan’s results on continued fractions are simple consequences of three-term relations between hypergeometric series. Theirq-analogues lead to many of the continued fractions given in the ‘Lost’ notebook in particular the famous one considered by Andrews and others.

• Multiplicative properties of the partition function

A lower bound for the number of multiplicatively independent values ofp(n) forN ≤n &lt;N + R is given. The proof depends on the Hardy-Ramanujan formula and is of an elementary nature.

• The states of the character ring of a compact group

Deligne’s generalization of the Hadamard—Vallée Poussin method in classical number theory is formulated as the representability of certain states of the character ring of a compact group, and the determination of all the representable states is carried out.

• On an approximate identity of Ramanujan

In his second notebook, Ramanujan says that$$\frac{q}{{x + }}\frac{{q^4 }}{{x + }}\frac{{q^8 }}{{x + }}\frac{{q^{12} }}{{x + }} \cdots = 1 - \frac{{qx}}{{1 + }}\frac{{q^2 }}{{1 - }}\frac{{q^3 x}}{{1 + }}\frac{{q^4 }}{{1 - }} \cdots$$ “nearly” forq andx between 0 and 1. It is shown in what senses this is true. In particular, asq → 1 the difference between the left and right sides is approximately exp −c(x)/(l-q) wherec(x) is a function expressible in terms of the dilogarithm and which is monotone decreasing with c(0) = π2/4,c(1) = π2/5; thus the difference in question is less than 2· l0−85 forq = 0·99 and allx between 0 and 1.

• # Proceedings – Mathematical Sciences

Current Issue
Volume 129 | Issue 3
June 2019