The purpose of this note is to draw attention to the question in the title. If C⊆Kn is an (absolutely) irreducible affine curve, defined by equations over a number field K, an algebraic integer point of C is a point P = (x1, …, xn) with all of x1, …, xn integers of some finite extension L of K. For such an algebraic integer point P to exist, there are obviously necessary local conditions: for every prime p of K there must exist a prime B above p and a corresponding finite extension LB of the completion Kp such that C has a B-adic integer point. We would like to know whether these obviously necessary local conditions are also sufficient.

Let A be a sequence of positive integers. Define P(A) to be the set of all integers representable as a sum of distinct terms of A. Note that if A contains a repeated value, we are free to use it as many times as it occurs in A. We call A complete if every sufficiently large positive integer is in P(A), and entirely complete if every positive integer is in P(A). Completeness properties have received considerable, if somewhat sporadic, attention over the years. See Chapter 6 of [3] for a survey.

I was recently challenged to find all the cases when the sum of three consecutive integral cubes is a square; that is to find all integral solutions x, y of

y2=(x−1)3+x3+(x+1)3

=3x(x2+2)

This is an example of a curve of genus 1. There is an effective procedure for finding all integral points on a given curve of genus 1 ([1, Theorem 4.2], [2]): that is, it can be guaranteed to find all the integral points and to show that no others exist with a finite amount of work. Unlike some effective procedures, which have only logical interest, this one can actually be carried out in practice, at least with the aid of a computer ([3], [5]). There are, however, older methods for dealing with problems of this kind which, while not effective, very often lead more easily to a complete set of solutions (and a proof that it is complete). I solve the problem here by a technique introduced in [4]. It requires only the elementary theory of algebraic number–fields. The motivation is p–adic, but it is simpler not to introduce p–adic theory overtly.

This note points out a new aspect of the well-known relationship between the subjects mentioned in the title. The following result and its generalization in totally real algebraic number fields is central to the discussion. Let denote the Legendre symbol for relatively prime numbers a and b ℇ ℤ and a substitution of the modular subgroup Γ0(4). Then, if γ>0 and b≡1 mod 2,

with

and

According to (1), the Legendre symbol behaves somewhat like a modular function ﹙apart from the known behaviour under and ﹚. (1) follows (see below) from the functional equation

with

provided that

Here we used and always will use the abbreviation

and ℇδ means the absolutely least residue of δ mod 4. In the proof, Hecke [4] assumed γ>0 (see also Shimura [5]).

Number fields such as described in the title play a rôle in the study of Artin L-functions and automorphic forms for the groups SL2 over rings of integers in quadratic extensions of ℚ. They are also of some interest on their own. We have not found many examples in the literature. Lang [4] mentions an unramified A5-extension of a real quadratic number field which is due to E. Artin.

The study of modular forms has been deeply influenced by famous conjectures and hypotheses concerning

where T(n) denotes Ramanujan's function. The fundamental discriminant Δ is a cusp form of weight 12 with respect to the modular group. Its associated Dirichlet series

defines an entire function of s and satisfies the functional equation

The most penetrating statements that have been made on T(n) and LΔ(s)are:

Of these four problems only A1 has been established so far. This was done by Deligne [1] using methods from algebraic geometry and number theory. While B1 trivially holds with ε > 1/2, it was established in [2] for every ε>1/3. Serre [12] proved A2 for a positive proportion of the integers and Hafner [5] showed that LΔ has a positive proportion of its non-trivial zeros on the line σ=6. The proofs of the last three results are largely analytic in nature.

The classical generalizations (already investigated in the second half of last century) of the modular group SL(2, ℤ) are the groups ГK = SL(2, o)(o the principal order of a totally real number field K, [K:ℚ]=n), operating, originally, on a product of n upper half-planes or, for n=2, on the product 1×− of an upper and a lower half-plane by

(where v(i), for v∈K, denotes the jth conjugate of v), and Гn = Sp(n, ℤ), operating on n={Z∣Z=X+iY∈ℂ(n,n),tZ=Z, Y>0} by

Nowadays ГK is called Hilbert's modular group of K and Гn Siegel's modular group of degree (or genus) n. For n=1 we have Гℚ=Г1= SL(2, ℤ). The functions corresponding to modular forms and modular functions for SL(2, ℤ) and its subgroups are holomorphic (or meromorphic) functions with an invariance property of the form

J(L, t) for fixed L (or J(M, Z) for fixed M) denoting a holomorphic function without zeros on ) (or on n). A function J;, defined on ℤK×or ℤn×n to be able to appear in (1.3) with f≢0, has to satisfy certain functional equations (see below, (2.3)–(2.5) for ГK, (5.7)–(5.9) for Гn) and is called an automorphic factor (AF) then. In close analogy to the case n=1, mainly AFs of the following kind have been used:

with a complex number r, the weight of J, and complex numbers v(L), v(M). AFs of this kind are called classical automorphic factors (CAP) in the sequel. If r∉ℤ, the values of the function v on ГK (or Гn) depend on the branch of (…)r. For a fixed choice of the branch (for each L∈ГK or M∈Гn) the functional equations for J, by (1.4), (1.5), correspond to functional equations for v. A function v satisfying those equations is called a multiplier system (MS) of weight r for ГK (or Гn).

Let ℬ = {bi:b1 <b2<…} be an infinite sequence of positive integers that exceed 1 and are pairwise coprime, so that

Assume also that

Let A=Aℬ denote the sequence of ℬ-free numbers, that is, of positive integers divisible by no element of ℬ. This concept, generalizing square-free and k-free numbers, derives from Erdös [2] who proved in 1966 that there exists a constant c, 0<c<l, independent of ℬ, such that the interval (x, x+xc) contains elements of A provided only that x is large enough. This result of Erdös was shown by Szemeredi [7] in 1973 to hold with c=½+ε, if x≥xo(ε, ℬ), and quite recently Bantle and Grupp [1] have sharpened Szemeredi's result to c=9/20+ε.

Let Q(x) = Q(x1, …, xn)∈ℤ[x1, …, xn] be a quadratic form. We investigate the size of the smallest non-zero solution of the congruence Q(x)≡0 (mod q). We seek a bound Bn(q), independent of Q, such that there is always a non-zero solution satisfying

The form gives the trivial lower bound Bn(q)≥(q/n)½ for all q and n, since if x≠0 and q∣ Q(x), then Q(x)≥q.

There are not a few situations in the theory of numbers where it is desirable to have as sharp an estimate as possible for the number r(n) of representations of a positive integer n by an irreducible binary cubic form

A variety of approaches are available for this problem but, as they stand, they are all defective in that they introduce unwanted factors into the estimate. For instance, an estimate involving the discriminant of f(x, y) is obtained if we adopt the Lagrange procedure [5] of using congruences of the type f(σ, 1)≡0, mod n, to reduce the problem to one where n=1. Alternatively, following Oppenheim (vid. [2]), Greaves [3], and others, we may appeal to the theory of factorization of ideals, which leads to unwanted logarithmic factors owing to the involvement of algebraic units. Having had need, however, in some recent work on quartic forms [4] for an estimate without such extraneous imperfections, we intend in the present note to prove that

uniformly with respect to the coefficients of f(x, y), where ds(n) denotes the number of ways of expressing n as a product of s factors.

Let Δ denote the Laplace operator acting on the space L2(Г/H) of automorphic functions with respect to a congruence group Г, square integrable over the fundamental domain F=Г/H. It is known that Δ has a point spectrum

with (Weyl's law)

and it has a purely continuous spectrum on [¼,∞) of finite multiplicity equal to the number of inequivalent cusps. The eigenpacket of the continuous spectrum is formed by the Eisenstein series Ea(z, s) on s = ½+it where a ranges over inequivalent cusps. The eigenfunctions ui(z) with positive eigenvalues are Maass cusp forms.

If α is a real irrational number, there exist infinitely many reduced rational fractions p/q for which

and √5 is the best constant possible. This result is due to A. Hurwitz. The following generalization was proposed in [2]. Let Г be a finitely generated fuchsian group acting on H+, the upper half of the complex plane. Let ℒ be the limit set of Г P and the set of cusps (parabolic vertices). Assume ∞∊P. Then if α∊ℒ–P, we have

for infinitely many p/q∊Г(∞), where k depends only on Г. Attention centers on

k running over the set for which (1.2) holds. We call hthe Hurwitz constant for Г. When Г=Г(1), the modular group, (1.2) reduces to (1.1) and h(Г(l))=√5. A proof of (1.2) when Г is horocyclic (i.e., ℒ=ℝ, the real axis) was furnished by Rankin [4]; he also found upper and lower bounds for h. See also [3, pp. 334–5], where the theorem is proved for arbitrary finitely generated Г.

The concept of “marked polygon”, made explicit in this paper, is implicit in all studies of the relationships between the edges and vertices of a fundamental polygon for Fuchsian group, as well as in the topology of surfaces. Once the matching of the edges under the action of the group is known, one can deduce purely combinatorially the distribution of the vertices into equivalence classes, or cycles. Knowing a little more, the order of the rotation group fixing a vertex in each cycle, we can write down a presentation for the group.

We define the nth cyclotomic polynomial Φn(z) by the equation

and we write

where ϕ is Euler's function.

Erdös and Vaughan [3] have shown that

uniformly in n as m-→∞, where

and that for every large m

The title is somewhat misleading, since the classical modular group Г= PSL(2, ℤ) is certainly not a subgroup of GL(2, ℤ). What is meant of course are the faithful representations of F as a subgroup of GL(2, ℤ), where Г is to be thought of as the free product of a cyclic group of order 2 and a cyclic group of order 3. No such representation is possible as a subgroup of SL(2, ℤ); it is necessary to have matrices of determinant −1 as well.

Let ω be a primitive cube root of unity. We define the cubic residue symbol (Legendre symbol) on ℤ[ω] as follows. Let πεℤ[ω] be a prime, (3, π)=1. For α ε ℤ[ω] such that (α, π)=1 we let be that third root of unity so that

Suppose that is a newform of weight k on Г1(N). Thus f is in particular a cusp form on Г1(N), satisfying

for all n≥1. Associated with f is a Dirichlet character

such that

for all, .

Let T(N) be the least integer such that one can assign ±l's to any N points in the unit square so that the sum of these values in any rectangle with sides parallel to those of the square have absolute value at most T(N). In [1] Beck showed, among other results, that (for N≥2)

.

La fonction η de Dedekind est définie par

où , Im(z)>0. C'est une forme modulaire parabolique de poids 1/2. Si r est un entier, la puissance r–ième de η s'écrit;

où les coefficients pr(n) sone définis par l'identité

.

On présente des exemples de représentations de de dimension 2, de déterminant pair, qui sont de type diédral (I) ou de conducteur premier et de type quelconque (II), en imitant la construction de Tate (Serre [11]) de représentations de déterminant impair.