Density of positive rank fibers in elliptic fibrations

A question of Mazur asks whether for any non-constant elliptic fibration {Er}_r∈Q, the set {r∈Q:rank(Er(Q))>0}, if infinite, is dense in R (with respect to the Euclidean topology). This has been proved to be true for the family of quadratic twists of a fixed elliptic curve by a quadratic or a cubic polynomial. Here we settle Mazur's question affirmatively for the general quadratic and cubic fibrations. Moreover we show that our method works when Q is replaced by any real number field.     

On the construction of normal wildly ramified extensions over Q p , (p ≠ 2)

Let p be an odd prime number, and let Q _p be the field of rational p-adic numbers.The aim of this work is the determination of the standard form of an Eisenstein polynomial defining a normal wildly ramified extension of Q _p . We prove first the equivalence between normality and cyclicity, give some essential normality conditions for the general case (degree p ⁿ ), then we solve the problem completely for the case (degree p ²) also, we obtain that the normality depends on seven congruences modulo p ^m between the coefficients of the considered polynomial with just m = 2 or 3. Note that the case (degree p) was solved by Öystein Ore (see Math. Annalen 102 (1930), 283–304). Also examples are given.     

Extended quasi-homogeneous polynomial system in {\mathbb{R}^{3}}

In this paper we define an extended quasi-homogeneous polynomial system d x/dt = Q = Q ₁ + Q ₂ + ... + Q _δ , where Q _i are some 3-dimensional quasi-homogeneous vectors with weight α and degree i, i = 1, . . . ,δ. Firstly we investigate the limit set of trajectory of this system. Secondly let Q _T be the projective vector field of Q. We show that if δ ≤ 3 and the number of closed orbits of Q _T is known, then an upper bound for the number of isolated closed orbits of the system is obtained. Moreover this upper bound is sharp for δ = 3. As an application, we show that a 3-dimensional polynomial system of degree 3 (resp. 5) admits 26 (resp. 112) isolated closed orbits. Finally, we prove that a 3-dimensional Lotka-Volterra system has no isolated closed orbits in the first octant if it is extended quasi-homogeneous.     

Cyclic vectors and cellular indecomposable operators on Qp spaces

We identify the functions whose polynomial multiples are weak* dense in Q p spaces and prove that if | f (z) | ≥ | g(z) | and g is cyclic in Q p , then f is cyclic in Q p . We also show that the multiplication operator M z on Q p spaces is cellular indecomposable.     

Deflating quadratic matrix polynomials with structure preserving transformations

Given a pair of distinct eigenvalues (λ₁,λ₂) of an n×n quadratic matrix polynomial Q(λ) with nonsingular leading coefficient and their corresponding eigenvectors, we show how to transform Q(λ) into a quadratic of the form having the same eigenvalue s as Q(λ), with Q_d(λ) an (n-1)×(n-1) quadratic matrix polynomial and q(λ) a scalar quadratic polynomial with roots λ₁ and λ₂. This block diagonalization cannot be achieved by a similarity transformation applied directly to Q(λ) unless the eigenvectors corresponding to λ₁ and λ₂ are parallel. We identify conditions under which we can construct a family of 2n×2n elementary similarity transformations that (a) are rank-two modifications of the identity matrix, (b) act on linearizations of Q(λ), (c) preserve the block structure of a large class of block symmetric linearizations of Q(λ), thereby defining new quadratic matrix polynomials Q₁(λ) that have the same eigenvalue s as Q(λ), (d) yield quadratics Q₁(λ) with the property that their eigenvectors associated with λ₁ and λ₂ are parallel and hence can subsequently be deflated by a similarity applied directly to Q₁(λ). This is the first attempt at building elementary transformations that preserve the block structure of widely used linearizations and which have a specific action.     

The image of Colmez’s Montreal functor

We prove a conjecture of Colmez concerning the reduction modulo p of invariant lattices in irreducible admissible unitary p-adic Banach space representations of GL₂(Q _{?? p} ) with p≥5. This enables us to restate nicely the p-adic local Langlands correspondence for GL₂(Q _{?? p} ) and deduce a conjecture of Breuil on irreducible admissible unitary completions of locally algebraic representations.     

Integral Kirillov model

We prove that an irreducible cuspidal Q̄_ℓ-representation of GL(n, Q_p) with a central character with values in Z̄_ℓ^* has a unique Z̄_ℓ-integral structure, given by the Kirillov Z̄_ℓ-representation.     

Congruences modulo 8 for the class numbers of Q(√±p), p ≡ 3(mod 4) a prime

A congruence modulo 8 is proved relating the class numbers of the quadratic fields Q(√p) and Q(√?p), where p is a prime congruent to 3 modulo 4.     

Asymptotic behavior of number fields with prescribed ℓ-class numbers

Let K be a cyclic Galois extension of the rational numbers Q of degree ?, where ? is a prime number. Let h_? denote the order of the Sylow ?-subgroup of the ideal class group of K. If h_? = ?^s(s ≥ 0), it is known that the number of (finite) primes that ramify in K/Q is at most s + 1 (or s + 2 if K is real quadratic). This paper shows that “most” of these fields K with h_? = ?^s have exactly s + 1 ramified primes (or s + 2 ramified primes if K is real quadratic). Furthermore the Sylow ?-subgroup of the ideal class group is elementary abelian when h_? = ?^s and there are s + 1 ramified primes (or s + 2 ramified primes if K is real quadratic).     

Factoring Polynomials over Special Finite Fields

We exhibit a deterministic algorithm for factoring polynomials in one variable over finite fields. It is efficient only if a positive integer k is known for which Φ_k(p) is built up from small prime factors; here Φ_k denotes the kth cyclotomic polynomial, and p is the characteristic of the field. In the case k=1, when Φ_k(p)=p−1, such an algorithm was known, and its analysis required the generalized Riemann hypothesis. Our algorithm depends on a similar, but weaker, assumption; specifically, the algorithm requires the availability of an irreducible polynomial of degree r over Z/pZ for each prime number r for which Φ_k(p) has a prime factor l with l≡1 mod r. An auxiliary procedure is devoted to the construction of roots of unity by means of Gauss sums. We do not claim that our algorithm has any practical value.     

Simple components of group algebras Q[G] central over real quadratic fields

Let k be a real quadratic field, and $\text{U}$ a central division quaternion algebra over k. In this paper sufficient conditions are given to insure that $\text{U}$ appears in a simple component of the group algebra Q[G] of some finite group G over the rational field Q. In particular, when k is assumed to be Q(√2) or Q(√5), the necessary and sufficient conditions for $\text{U}$ to appear in some Q[G] are given.     

The minimal generating space of a polynomial

Given a polynomial $\text{P(X}{}_1\text{,…,X}{}_{\mathrm{N}}\text{)∈}\text{R}\text{[}\text{X}\text{]}$, we calculate a subspace G_p of the linear space 〈X〉 generated by the indeterminates which is minimal with respect to the property $\text{P∈}\text{R}\text{[}\text{G}{}_{\mathrm{p}}\text{]}$ (the algebra generated by G_p, and prove its uniqueness. Furthermore, we use this result to characterize the pairs (P,Q) of polynomials P(X₁,…,X_n) and Q(X₁,…,X_n) for which there exists an isomorphism T:〈X〉 →〈X〉 that “separates P from Q,” i.e., such that for some k(1<k<n) we can write P and Q as $\text{P}{}^1\text{(Y}{}_1\text{,…,Y}{}_{\mathrm{k}}\text{)}$ and $\text{Q}{}^1\text{(Y}{}_{\mathrm{<mtext>k+</mtext><mtext>1</mtext>}}\text{,…,Y}{}_{\mathrm{n}}\text{)}$ respectively, where $\text{Y}\text{=T}\text{X}$.     

Lucas sequences and quadratic orders

We consider the Lucas sequences (U _n ) _{n ≥ 0} defined by U ₀ = 0, U ₁ = 1, and U _n =  PU _n–1 – QU _n–2 for non-zero integral parameters P, Q such that Δ = P ² – 4Q is not a square. We use the arithmetic of the quadratic order with discriminant Δ to investigate the zeros and the period length of the sequence (U _n ) _{n ≥ 0} modulo a positive integer d coprime to Q. For a prime p not dividing Q, we give precise formulas for p-powers, we determine the p-adic value of U _n , and we connect the results with class number relations for quadratic orders.     

Polynomials with Dp as Galois group

A useful criterion characterizing a monic irreducible polynomial over $\text{Q}$ with Galois group D_p (the dihedral group of order 2p, p: prime) is given by making use of the geometry of D_p, i.e., D_p is the symmetry group of the regular p-gon. We derive explicit numerical examples of polynomials with dihedral Galois groups D₅ and D₇.     

Generalization of Kummer's criterion for divisibility of class numbers

We give a necessary and sufficient condition for the relative class number of an imaginary field contained in Q(e^2πi/p?) to be divisible by p. We also give a sufficient condition for the class number of a real field contained in Q(e^2πi/p?) not to be divisible by p.     

Extension of a theorem of Cauchy and Jacobi

Let q and p be prime with q = a² + b² ≡ 1 (mod 4), a ≡ 1 (mod 4), and p = qf + 1. In the nineteenth century Cauchy (Mém. Inst. France17 (1840), 249–768) and Jacobi (J. für Math.30 (1846), 166–182) generalized the work of earlier authors, who had determined certain binomial coefficients (mod p) (see H. J. S. Smith, “Report on the Theory of Numbers,” Chelsea, 1964), by determining two products of factorials given by Π_kkf! (mod p = qf + 1) where k runs through the quadratic residues and the quadratic non-residues (mod q), respectively. These determinations are given in terms of parameters in representations of p^h or of 4p^h by binary quadratic forms. A remarkable feature of these results is the fact that the exponent h coincides with the class number of the related quadratic field. In this paper C. R. Mathews' (Invent. Math.54 (1979), 23–52) recent explicit evaluation of the quartic Gauss sum is used to determine four products of factorials (mod p = qf + 1, q ≡ 5 (mod 8) > 5), given by Π_kkf! where k runs through the quartic residues (mod q) and the three cosets which may be formed with respect to this subgroup. These determinations appear to be considerably more difficult. They are given in terms of parameters in representations of 16p^h by quaternary quadratic forms. Stickelberger's theorem is required to determine the exponent h which is shown to be closely related to the class number of the imaginary quartic field Q(i√2q + 2a√q), q = a² + b² ≡ 5 (mod 8), a odd.     

Special values of L-functions of elliptic curves over Q and their base change to real quadratic fields

For an elliptic curve E over Q, and a real quadratic extension F of Q, satisfying suitable hypotheses, we study the algebraic part of certain twisted L-values for E/F. The Birch and Swinnerton-Dyer conjecture predicts that these L-values are squares of rational numbers. We show that this question is related to the ratio of Petersson inner products of a quaternionic form on a definite quaternion algebra over Q and its base change to F.     

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 convex polynomial that is not sos-convex

A multivariate polynomial p(x)?=?p(x ₁, . . . , x _n ) is sos-convex if its Hessian H(x) can be factored as H(x)?= M ^T (x) M(x) with a possibly nonsquare polynomial matrix M(x). It is easy to see that sos-convexity is a sufficient condition for convexity of p(x). Moreover, the problem of deciding sos-convexity of a polynomial can be cast as the feasibility of a semidefinite program, which can be solved efficiently. Motivated by this computational tractability, it is natural to study whether sos-convexity is also a necessary condition for convexity of polynomials. In this paper, we give a negative answer to this question by presenting an explicit example of a trivariate homogeneous polynomial of degree eight that is convex but not sos-convex.     

Nouvelle preuve d'un théorème de Yoccoz

If θ is an irrational real number such that ∑(lnq_n+1)/q_n=+∞ where p_n/q_n are the convergents of θ, then the quadratic polynomial $\text{P}{}_{\mathrm{\theta }}\text{(z)=}\text{e}{}^{\mathrm{<mtext>i</mtext><mtext>2\pi \theta </mtext>}}\text{z+z}{}^2$ is not linearizable at 0. This theorem has been proved in 1988 by J.C. Yoccoz, who first constructs a nonlinearizable germ by inverting a renormalisation procedure, and then proves universality of the quadratic family for that question. We give an alternative proof, based on the study of the explosion of parabolic cycles. To cite this article: A. Chéritat, C. R. Acad. Sci. Paris, Ser. I 338 (2004).