An application of L-functions of imaginary quadratic fields

Let F₁(x, y),…, F_2h+1(x, y) be the representatives of equivalent classes of positive definite binary quadratic forms of discriminant ?q (q is a prime such that q ≡ 3 mod 4) with integer coefficients, then the number of integer solutions of F_i(x, y) = n (i = 1,…, 2h + 1) can be calculated for each natural number n using L-functions of imaginary quadratic field $\text{Q}$((?q)^1/2).     

On Diophantine equations over the ring of all algebraic integers

For each odd prime q an integer NH_q (NH₃ = ?1, NH₅ = ?1, NH₇ = 97, NH₁₁ = ?243, …) is defined as the norm from L to $\text{Q}$ of the Heilbronn sum H_q = Tr_I^{$\text{Q}$(ζ)}(ζ), where ζ is a primitive q²th root of unity and L ?- $\text{Q}$(ζ) the subfield of degree q. Various properties are proved relating the congruence properties of H_q and NH_q modulo p (p ≠ q prime) to the Fermat quotient $\text{(p}{}^{\mathrm{q\; ?\; 1}}\text{? 1)}\text{q}\text{(}\text{mod}\text{q)}$; in particular, it is shown that NH_q is even iff 2^{q ? 1} ≡ 1 (mod q²).     

Rational compositum genus for a pure cubic field

In non-Abelian fields, to which genus theory does not ordinarily apply, many of the diophantine by-products are still available. For instance, the representability of primes as norms of principal ideals in different fields of the same degree will bear interrelations if the fields belong to a small compositum as do the four subfields of $\text{Q}$(m_∞^1/3, m₀^1/3). This is a generalization of a classical phenomenon for quadratic fields. It is particularly effective when just one prime l ≡ 1 (mod 3) divides m₀m_∞.     

Residue properties of certain quadratic units

Explicit formulas are given for the quadratic and quartic characters of units of certain quadratic fields in terms of representations by positive definite binary quadratic forms, as conjectured by Leonard and Williams (Pacific J. Math.71 (1977), Rocky Mountain J. Math.9 (1979)), and by Lehmer (J. Reine Angew. Math.268/69 (1974)). For example, if p and a are primes such that p≡1 (mod 8), q≡5 (mod 8) and the Legendre symbol $\text{(}\text{q}\text{p}\text{)=1}$, and if ε is the fundamental unit of $\text{Q}$(√q), then $\text{(}\text{ε}\text{p}\text{)}{}_4\text{=(?1)}{}^{\mathrm{b+d}}$, where p=a²+16b² and p^k=c²+16qd² with k odd.     

A Good Method of Combining Codes

Let q be an odd prime power, and suppose q?1 (mod8), Let C(q) and C(q)¹ be the two extended binary quadratic residue codes (QR codes) of length q+1, and let

$\text{T(q)={(a+x;b+x;a+b+x):a,b∈C(q),x∈C(q)}{}^1\text{}}$

. We establish a square root bound on the minimum weight in T(q). Since the same type of bound applies to C(q) and C(q)¹, this is a good method of combining codes.     

A family of difference sets in non-cyclic groups

A construction is given for difference sets in certain non-cyclic groups with the parameters $\text{v = q}{}^{\mathrm{s+1}}\text{{[}\text{(q}{}^{\mathrm{s+1}}\text{? 1)}\text{(q ? 1)}\text{] + 1}}$, $\text{k = q}{}^{\mathrm{s}}\text{(q}{}^{\mathrm{s+1}}\text{? 1)}\text{(q ? 1)}$, $\text{λ = q}{}^{\mathrm{s}}\text{(q}{}^{\mathrm{s}}\text{? 1)}\text{(q ? 1)}$, n = q^2s for every prime power q and every positive integer s. If q^s is odd, the construction yields at least $\text{1}\text{2}\text{(q}{}^{\mathrm{s}}\text{+ 1)}$ inequivalent difference sets in the same group. For q = 5, s = 2 a difference set is obtained with the parameters (v, k, λ, n) = (4000, 775, 150, 625), which has minus one as a multiplier.     

The quartic characters of certain quadratic units

Criteria are obtained for the quartic residue character of the fundamental unit of the real quadratic field $\text{Q((2q)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$, where q is prime and either q ≡ 7(mod 8), or q ≡ 1(mod 8) and X² ? 2qY² = ?2 is solvable in integers X and Y.     

The Schur group of cyclotomic fields

We compute the Schur group of the cyclotomic fields Q(?_m) and real quadratic fields $\text{Q(d}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ where d is a product of an even number of primes congruent to three modulo four. Some results are also given about the Schur group of certain subfields of cyclotomic fields.     

Automorphism of orthogonal groups in characteristic 2

Let V be a nondefective quadratic space over a field F of characteristic 2. Assume that V has dimension at least ten and that F has more than two elements. Let Δ be one of the groups O(V), O⁺(V), O′(V), or $\text{Ω(V)}$ (the full orthogonal group, the rotation group, the spinorial kernel, or the commutator subgroup of O(V), respectively). Then Λ is an automorphism of Λ if and only if Λ(σ) = gσg^?1 for all σ in Δ where g is a semilinear automorphism of V that preserves the quadratic structure of V in the sense that Q(gx) = αQ(x)^u for all x in V where Q is the quadratic form, α is some nonzero element of F, and u is the field automorphism of F associated to g.     

Nonexistence of Complete (st−t/s)-Arcs in Generalized Quadrangles of Order (s, t), I

Let $\text{S}$ be a finite generalized quadrangle (GQ) of order (s, t), s≠1≠t. A k-arc $\text{K}$ is a set of k mutually non-collinear points. For any k-arc of $\text{S}$ we have k⩽st+1; if k=st+1, then $\text{K}$ is an ovoid of $\text{S}$. A k-arc is complete if it is not contained in a k′-arc with k′>k. In S. E. Payne and J. A. Thas, Finite Generalized Quadrangles, Pitman, Boston, 1984, it is proved that an (st−m)-arc, where −1⩽m<t/s, is always contained in a uniquely defined ovoid, hence it is a natural question to ask whether or not complete (st−t/s)-arcs exist. In this note, we prove that the classical GQ H(4, q²) has no complete (q⁵−q)-arcs. We also show that a GQ $\text{S}$ of order s with a regular point has no complete (s²−1)-arcs, except when s=2, i.e. $\text{S}$≅$\text{Q}$(4, 2), and in that case there is a unique example. As a by-product there follows that no known GQ of even order s with s>2 can have complete (s²−1)-arcs. Also, we prove that a GQ of order (s, s²), s≠1, cannot have complete (s³−s)-arcs unless s=2, i.e., $\text{S}$≅$\text{Q}$(5, 2), in which case there is a unique example (up to isomorphism).     

Product formulas for semigroups with elliptic generators

Real constant coefficient nth order elliptic operators, Q, which generate strongly continuous semigroups on L²($\text{R}$^k) are analyzed in terms of the elementary generator,

$\text{A = (?n)}{}^{\mathrm{<mtext>(</mtext><mtext>n</mtext><mtext>2</mtext><mtext>\; ?\; 1</mtext>}}\text{)(n!)}{}^{\mathrm{?1}}\text{∑}\text{k}\text{j = 1}\text{?}{}^{\mathrm{n}}\text{?x}{}_{\mathrm{j}}{}^{\mathrm{n}}$

, for n even. Integral operators are defined using the fundamental solutions p_n(x, t) to u_t = Au and using real polynomials q_l,…, q_k on $\text{R}$^m by the formula, for q = (q_l,…, q_k),

$\text{(F(t)?)(x) = ∫}$

_$\text{R}$m

$\text{?(x + q(z)) P}{}_{\mathrm{n}}\text{(z, t)dz}$

. It is determined when, strongly on L²($\text{R}$^k),

$\text{e}{}^{\mathrm{tQ}}\text{=}\text{lim}\text{j → ∞}\text{F}\text{t}\text{j}{}^{\mathrm{j}}$

. If n = 2 or k = 1, this can always be done. Otherwise the symbol of Q must have a special form.     

On Artin's conjecture

Let $\text{F}$ be a family of number fields which are normal and of finite degree over a given number field K. Consider the lattice L(scF) spanned by all the elements of $\text{F}$. The generalized Artin problem is to determine the set of prime ideals of K which do not split completely in any element H of L(scF), H≠K. Assuming the generalized Riemann hypothesis and some mild restrictions on $\text{F}$, we solve this problem by giving an asymptotic formula for the number of such prime ideals below a given norm. The classical Artin conjecture on primitive roots appears as a special case. In another case, if $\text{F}$ is the family of fields obtained by adjoining to $\text{Q}$ the q-division points of an elliptic curve E over $\text{Q}$, the Artin problem determines how often E($\text{F}$_p) is cyclic. If E has complex multiplication, the generalized Riemann hypothesis can be removed by using the analogue of the Bombieri-Vinogradov prime number theorem for number fields.     

Über asymmetrische diophantische approximationen

For irrational numbers θ define α(θ) = lim sup{1/(q(p ? qθ))|p ∈ $\text{Z}$, q ∈ $\text{N}$, p ? qθ > 0} and α(θ) = 0 for rationals. Put $\text{α}\text{(θ) = max{α(θ), α(?0)}}$. Then $\text{U}$ = α($\text{R}$β$\text{Q}$) is an asymmetric analogue to the Lagrange spectrum $\text{U}\text{=}\text{α}\text{(RβQ)}$. Our results concerning $\text{U}$ partly contrast the known properties of $\text{U}$. In fact, $\text{U}$ is a perfect set, each element of which is a condensation point of the spectrum and has continuously many preimages. $\text{U}$ is the closure of its rational elements and of its elements of the form p√m (p ∈ $\text{Q}$), as well. The arbitrarily well approximable numbers form a G_δ-set of 2. category. One has, roughly speaking, $\text{α}\text{→ ∞}$ for α → 1. Finally, the well-known Markov sequence which constitutes the lower Lagrange and Markov spectrum is proved to be a (small) subset of $\text{U}$?[√5,3).     

Class numbers of definite binary quadratic lattices over algebraic function fields

Let k be an algebraic function field of one variable X having a finite field GF(q) of constants with q elements, q odd. Confined to imaginary quadratic extensions $\text{K}\text{k}$, class number formulas are developed for both the maximal and nonmaximal binary quadratic lattices L on (K, N), where N denotes the norm from K to k. The class numbers of L grow either with the genus g(k) of k (assuming the fields under consideration have bounded degree) or with the relative genus $\text{g(}\text{K}\text{k}\text{)}$ (assuming the lattices under consideration have bounded scale). In contrast to analogous theorems concerning positive definite binary quadratic lattices over totally real number fields, k is not necessarily totally real.     

On quasigroups rich in associative triples

Let G be a group and $\text{G(1)}$ a quasigroup on the same underlying set. Let dist($\text{G, G(1)}$) denote the number of pairs (x, y) ?G² such that $\text{xy ≠ x 1 y}$. For a finite quasigroup Q, n = card(Q), let t = dist(Q) = min dist(G, Q), where G runs through all groups with the same underlying set, and s = s(Q) the number of non-associative triples. Then 4tn?2t²?24t?s?4tn. If 1 ? s < 3n²/32, then 3tn < s holds as well. Let n ? 168 be an even integer and let σ = min s(Q), where Q runs through all non-associative quasigroups of order n. Then σ = 16n?64.     

An integral transform in L2-cohomology for the ladder representations of U(p, q)

Starting from the realization of the Fock space as L²-cohomology of $\text{C}$^{p + q}, $\text{H}$^0,p($\text{C}$^{p + q}) = ⊕_m?$\text{Z}$$\text{H}$_m^0,p($\text{C}$^{p + q}), an integral transform is constructed which is a direct-image mapping from $\text{H}$_m^{0,p($\text{C}$p + q) into the space of holomorphic sections of some vector bundle E_m over M ≈ U(p, q)/(U(q) × U(p)), m ? 0}. The transform intertwines the natural actions of U(p, q) and is injective if m ? 0, so it provides a geometric realization of the ladder representations of U(p, q). The sections in the image of the transform satisfy certain linear differential equations, which are explicitly described. For example, Maxwell's equations are of this form if p = q = 2 and m = 2. Thus, this transform is analogous to the Penrose correspondence.     

On pre-periods of discrete influence systems

We investigate mappings of the form $\text{g = ƒA}$ where ƒ is a cyclically monotonous mapping of finite range and A is a linear mapping given by a symmetric matrix. We give some upper bounds on the pre-period of g, i.e. the maximum q for which all g(x),g²(x),…,g^q(x) are distinct.     

A third-order optimum property of the maximum likelihood estimator

Let θ⁽ⁿ⁾ denote the maximum likelihood estimator of a vector parameter, based on an i.i.d. sample of size n. The class of estimators θ⁽ⁿ⁾ + n^?1q(θ⁽ⁿ⁾), with q running through a class of sufficiently smooth functions, is essentially complete in the following sense: For any estimator T⁽ⁿ⁾ there exists q such that the risk of θ⁽ⁿ⁾ + n^?1q(θ⁽ⁿ⁾) exceeds the risk of T⁽ⁿ⁾ by an amount of order o(n^?1) at most, simultaneously for all loss functions which are bounded, symmetric, and neg-unimodal. If $\text{q}{}^1$ is chosen such that $\text{θ}{}^{\mathrm{(n)}}\text{+ n}{}^{\mathrm{?1}}\text{q}{}^1\text{(θ}{}^{\mathrm{(n)}}\text{)}$ is unbiased up to $\text{o(n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{)}$, then this estimator minimizes the risk up to an amount of order o(n^?1) in the class of all estimators which are unbiased up to $\text{o(n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{)}$.The results are obtained under the assumption that T⁽ⁿ⁾ admits a stochastic expansion, and that either the distributions have—roughly speaking—densities with respect to the lebesgue measure, or the loss functions are sufficiently smooth.     

Spectral properties of zeroth-order pseudodifferential operators

Let Q be a self-adjoint, classical, zeroth order pseudodifferential operator on a compact manifold X with a fixed smooth measure dx. We use microlocal techniques to study the spectrum and spectral family, {E_S}_{S∈$\text{R}$} as a bounded operator on L²(X, dx).Using theorems of Weyl (Rend. Circ. Mat. Palermo, 27 (1909), 373–392) and Kato (“Perturbation Theory for Linear Operators,” Springer-Verlag, 1976) on spectra of perturbed operators we observe that the essential spectrum and the absolutely continuous spectrum of Q are determined by a finite number of terms in the symbol expansion. In particular Spec_ESSQ = range(q(x, ξ)) where q is the principal symbol of Q. Turning the attention to the spectral family {E_S}_{S∈$\text{R}$}, it is shown that if $\text{dE}\text{ds}$ is considered as a distribution on $\text{R}$×X×X it is in fact a Lagrangian distribution near the set $\text{{σ=0}?}\text{T}{}^1\text{(}\text{R}\text{×X×X)}\text{0}$ where (s, x, y, σ, ξ,η) are coordinates on T¹($\text{R}$×X×X) induced by the coordinates (s, x, y) on $\text{R}$×X×X. This leads to an easy proof that$\text{?(Q)}$ is a pseudodifferential operator if ?∈C^∞($\text{R}$) and to some results on the microlocal character of E_s. Finally, a look at the wavefront set of $\text{dE}\text{ds}$ leads to a conjecture about the existence of absolutely continuous spectrum in terms of a condition on q(x, ξ).