Note on g-Hilbertian fields and nonrational curves¶with the Hilbert property

In a recent paper, Fried and Jarden prove the existence, for all integers g, of non-Hilbertian fields K which cannot be covered by a finite number of sets of the form ϕ (X(K)), where X is a curve of genus ≤g and ϕ is a rational function on X of degree ≥ 2. (If no bound is given on the genus we recover the notion of Hilbertian field.) This generalizes the case g=0, obtained previously by Corvaja and Zannier with a more elementary method. By a suitable modification of that method, we give here a new proof of the result of Fried and Jarden which avoids the use of deep group theoretical results. By a somewhat related construction we give an example of a curve X/Q of any prescribed genus and a Hilbertian field K⊂ˉQ such that X/K has the Hilbert property, i.e. the set of rational points X(K) is not thin. Received: 10 March 1998 / Revised version: 20 April 1998     

Approximation d’un nombre réel par des unités algébriques

 In this paper, we prove that for any real number ξ, which is not an algebraic number of degree , there exist infinitely many real algebraic units α of degree n + 1 such that . We also show how the flexibility of H. Davenport and W. M. Schmidt’s method allows to replace, with the same exponent of approximation, units of degree over Z (i.e. elements α with both α and integral over Z) by units of degree over a finite intersection .

(Received 14 March 2000; in revised form 16 November 2000)     

On Sums of Roots of Unity

 Suppose that for some root of unity ζ of order Q with and all coefficients a _i belonging to a number field L. We bound Q in terms of k and . This generalizes a result of Conway and Jones for the case of rational coefficients. Moreover, we give an application to linear relations among characteristic functions of arithmetical progressions. (Received 18 January 1999; in revised from 14 June 1999)     

Convergence of Rational Interpolants with Preassigned Poles

Let Ω be a domain in the extended complex plane such that ∞∈Ω . Further, let K= C / Ω and, for each n , let Q _n be a monic polynomial of degree n with all its zeros in K . This paper is concerned with whether (Q _n ) can be chosen so that, if f is any holomorphic function on Ω and P _n is the polynomial part of the Laurent expansion of Q _n f at ∞ , then (P _n /Q _n ) converges to f locally uniformly on Ω . It is shown that such a sequence (Q _n ) can be chosen if and only if either K has zero logarithmic capacity or Ω is regular. January 21, 1999. Date accepted: August 17, 1999.     

Class number and class group problems for some non-normal totally real cubic number fields

Let \lcub;K _m } _{m ≥ 4} be the family of non-normal totally real cubic number fields associated with the Q-irreducible cubic polynomials P _m (x) =x ³−mx ²−(m+1)x− 1, m≥ 4. We determine all these K _m 's with class numbers h _m ≤ 3: there are 14 such K _m 's. Assuming the Generalized Riemann hypothesis for all the real quadratic number fields, we also prove that the exponents e _m of the ideal class groups of these K _m go to infinity with m and we determine all these K _m 's with ideal class groups of exponents e _m ≤ 3: there are 6 suchK _m with ideal class groups of exponent 2, and 6 such K _m with ideal class groups of exponent 3. Received: 16 November 2000 / Revised version: 16 May 2001     

A remark on the least n with χ (n) ≠ 1

Suppose that a > 2. We prove that the number of positive integers q ≦ Q such that there exists a primitive character χ modulo q with χ (n) = 1 for all n ≦ (log Q)^a is O(Q^1/(1-a)+ε). Received: 7 December 2004     

Cycles for Rational Maps of Good Reduction Outside a Prescribed Set

Let K be a number field and S a fixed finite set of places of K containing all the archimedean ones. Let R _S be the ring of S-integers of K. In the present paper we study the cycles in for rational maps of degree ≥2 with good reduction outside S. We say that two ordered n-tuples (P ₀, P ₁,… ,P _n−1) and (Q ₀, Q ₁,… ,Q _n−1) of points of are equivalent if there exists an automorphism A ∈ PGL₂(R _S ) such that P _i = A(Q _i ) for every index i∈{0,1,… ,n−1}. We prove that if we fix two points , then the number of inequivalent cycles for rational maps of degree ≥2 with good reduction outside S which admit P ₀, P ₁ as consecutive points is finite and depends only on S and K. We also prove that this result is in a sense best possible.     

On integer and fractional parts of powers of Salem numbers

Let N be a positive rational integer and let P be the set of powers of a Salem number of degree d. We prove that for any α∈P the fractional parts of the numbers , when n runs through the set of positive rational integers, are dense in the unit interval if and only if N≦ 2d − 4. We also show that for any α∈P the integer parts of the numbers αⁿ are divisible by N for infinitely many n if and only if N≦ 2d − 3. Received: 27 April 2005     

On the composition length of finite primitive linear groups

LetG be a finite primitive linear group over a fieldK, whereK is a finite field or a number field. We bound the composition length ofG in terms of the dimension of the underlying vector space and of the degree ofK over its prime subfield. As a byproduct, we prove a result of number theory which bounds the number of prime factors (counting multiplicities) ofq ⁿ−1, whereq, n>1 are integers, improving a result of Turull and Zame [6].     

Lipschitz classes on local fields

The Lipschitz class Lipαon a local field K is defined in this note,and the equivalent relationship between the Lipschitz class Lipαand the Holder type space C~α(K)is proved.Then,those important characteristics on the Euclidean space R~n and the local field K are compared,so that one may interpret the essential differences between the analyses on R~n and K.Finally,the Cantor type fractal functionθ(x)is showed in the Lipschitz class Lip(m,K),m＜(ln 2/ln 3).     

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     

Randomness of the square root of 2 and the Giant Leap,Part 1

We prove that the “quadratic irrational rotation” exhibits a central limit theorem. More precisely, let α be an arbitrary real root of a quadratic equation with integer coefficients; say, α = $ \sqrt 2 $ \sqrt 2 . Given any rational number 0 < x < 1 (say, x = 1/2) and any positive integer n, we count the number of elements of the sequence α, 2α, 3α, …, nα modulo 1 that fall into the subinterval [0, x]. We prove that this counting number satisfies a central limit theorem in the following sense. First, we subtract the “expected number” nx from the counting number, and study the typical fluctuation of this difference as n runs in a long interval 1 ≤ n ≤ N. Depending on α and x, we may need an extra additive correction of constant times logarithm of N; furthermore, what we always need is a multiplicative correction: division by (another) constant times square root of logarithm of N. If N is large, the distribution of this renormalized counting number, as n runs in 1 ≤ n ≤ N, is very close to the standard normal distribution (bell shaped curve), and the corresponding error term tends to zero as N tends to infinity. This is the main result of the paper (see Theorem 1.1). The proof is rather complicated and long; it has many interesting detours and byproducts. For example, the exact determination of the key constant factors (in the additive and multiplicative norming), which depend on α and x, requires surprisingly deep algebraic tools such as Dedeking sums, the class number of quadratic fields, and generalized class number formulas. The crucial property of a quadratic irrational is the periodicity of its continued fraction. Periodicity means self-similarity, which leads us to Markov chains: our basic probabilistic tool to prove the central limit theorem. We also use a lot of Fourier analysis. Finally, I just mention one byproduct of this research: we solve an old problem of Hardy and Littlewood on diophantine sums.     

The best Lipschitz constants of Bernstein polynomials and Bezier nets over a given triangle

This paper proved the following three facts about the Lipschitz continuous property of Bernstein polynomials and Bezier nets defined on a triangle: suppose f(P) is a real valued function defined on a triangle T, (1) If f(P) satisfies Lipschitz continuous condition, i.e. f(P)∃Lip_Aα, then the corresponding Bernstein Bezier net f_n∃Lip _Asec ^αφα, here φ is the half of the largest angle of triangle T; (2) If Bernstein Bezier net f_n∃Lip _Bα, then its elevation Bezier net Ef_n∃Lip _Bα; and (3) If f(P)∃Lip _Aα, then the corresponding Bernstein polynomials B_n(f;P)∃Lip _Asec ^αφα, and the constant Asec^αφ is best in some sense. Supported by NSF and SF of National Educational Committee     

On the indices of curves over local fields

Fix a non-negative integer g and a positive integer I dividing 2g − 2. For any Henselian, discretely valued field K whose residue field is perfect and admits a degree I cyclic extension, we construct a curve C _/K of genus g and index I. This is obtained via a systematic analysis of local points on arithmetic surfaces with semistable reduction. Applications are discussed to the corresponding problem over number fields.     

Approximation d’un nombre réel par des unités algébriques

 In this paper, we prove that for any real number ξ, which is not an algebraic number of degree , there exist infinitely many real algebraic units α of degree n + 1 such that . We also show how the flexibility of H. Davenport and W. M. Schmidt’s method allows to replace, with the same exponent of approximation, units of degree over Z (i.e. elements α with both α and integral over Z) by units of degree over a finite intersection .     

Every finitely generated regular field extension has a stable transcendence base

A fieldK is called stable if every finitely generaed regular field extensionF/K has a transcendence basex ₁, …,x _n with the following properties: The field extensionF/K(x ₁,…,x _n ) is separable and the Galois hull ofF/K(x ₁,…,x _n ) remains regular overK, i.e.K is algebraically closed in . We prove in this paper thatevery field is stable. This generalizes results from [FJ1] and [GJ] which prove that fields of characteristic 0 and infinite perfect fields are stable, respectively. [G] showed that finite fields are stable in dimension 1, i.e. every finitely generated regular field extension of transcendence degree 1 over a finite field has a stable transcendence base. In the last section of this paper we apply the theorem to the construction of PAC fields with additional properties. A fieldK is called PAC if every absolutely irreducible variety overK has at least oneK-rational point.     

Linear independence measures for infinite products

Let f be an entire transcendental function with rational coefficients in its power series about the origin. Further, let f satisfy a functional equation f(qz)= (z−c)f(z)+Q(z) with and some particular c∈ℚ. Then the linear independence of 1,f(α), f(−α) over ℚ for non-zero α∈ℚ is proved, and a linear independence measure for these numbers is given. Clearly, for Q= 0 the function f can be written as an infinite product. Received: 19 September 2000 / Revised version: 14 March 2001     

Potenzieren und Wurzelziehen in rationalen Quaternionenalgebren

LetF be a field not of characteristic 2 andQ =F +F i +F j +F k the quaternion algebra overF whereij = -ji =k andi ² = α andj ² = β with 0 ≠ α, β ∈F fixed. (IfF = ℝ and α = β = - 1 thenQ is the division algebra of the Hamilton quaternions.) IfF = ℚ and Q is a division algebra then by embedding certain quadratic number fields inQ we derive an efficient formula to compute the powers of any quaternion. This formula is even true in general and reads as follows. If a, a₁, a₂, a₃ ∈F andn ∈ ℕ then where ω ig a square root of αa₁ ² + βa ₂ ² - αβa ₃ ² in or overF and andA ₀ =na ^n-1. With the help of this formula and related ones we are able to solve the equationX ⁿ =q for arbitrary quaternionsq and positive integers n in case ofF = ℝ and hence in case ofF ⊂ ℝ as well. IfF = ℝ then the total number of all solutions equals 0, 1, 2, 4,n or ∞. (4 is possible even whenn < 4.) In case ofF = ℚ, which we are primarily interested in, there are always either at most six or infinitely many solutions. Further, for everyq ≠ 0 there is at most one solution provided thatn is odd and not divisible by 3. The questions when there are infinitely many solutions and when there are none can always be decided by checking simple conditions on the radicandq ifF = ℝ. ForF = ℚ the two questions are comprehensively investigatet in a natural connection with ternary and quaternary quadratic rational forms. Finally, by applying some of our theorems on powers and roots of quate-rions we also obtain several nice results in matrix theory. For example, for every k ∈ ℤ the mappingA ↦A ^k on the group of all nonsingular 2-by-2 matrices over ℚ is injective if and only ifk is odd and not divisible by 3.

     

The Banach-Mazur distance to the cube and the Dvoretzky-Rogers factorization

We show the existence of a sequence (λ _n ) of scalars withλ _n =o(n) such that, for any symmetric compact convex bodyB ⊂R ⁿ , there is an affine transformationT satisfyingQ ⊂T(B) ⊂λ _n Q, whereQ is then-dimensional cube. This complements results of the second-named author regarding the lower bound on suchλ _n . We also show that ifX is ann-dimensional Banach space andm=[n/2], then there are operatorsα:l ₂ ^m →X andβ:X→l _∞ ^m with ‖α‖·‖β‖≦C, whereC is a universal constant; this may be called “the proportional Dvoretzky-Rogers factorization”. These facts and their corollaries reveal new features of the structure of the Banach-Mazur compactum. Research performed while this author was visiting IHES. Supported in part by the NSF Grant DMS-8702058 and the Sloan Research Fellowship.     

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.