4,011 research outputs found
Five squares in arithmetic progression over quadratic fields
We give several criteria to show over which quadratic number fields
Q(sqrt{D}) there should exists a non-constant arithmetic progressions of five
squares. This is done by translating the problem to determining when some genus
five curves C_D defined over Q have rational points, and then using a
Mordell-Weil sieve argument among others. Using a elliptic Chabauty-like
method, we prove that the only non-constant arithmetic progressions of five
squares over Q(sqrt{409}), up to equivalence, is 7^2, 13^2, 17^2, 409, 23^2.
Furthermore, we give an algorithm that allow to construct all the non-constant
arithmetic progressions of five squares over all quadratic fields. Finally, we
state several problems and conjectures related to this problem.Comment: To appear in Revista Matem\'atica Iberoamerican
Propri\'et\'es multiplicatives des entiers friables translat\'es
An integer is said to be -friable if its greatest prime factor P(n) is
less than . In this paper, we study numbers of the shape when
and . One expects that, statistically, their
multiplicative behaviour resembles that of all integers less than .
Extending a result of Basquin, we estimate the mean value over shifted friable
numbers of certain arithmetic functions when for some
positive , showing a change in behaviour according to whether tends to infinity or not. In the same range in , we prove
an Erd\"os-Kac-type theorem for shifted friable numbers, improving a result of
Fouvry and Tenenbaum. The results presented here are obtained using recent work
of Harper on the statistical distribution of friable numbers in arithmetic
progressions.Comment: 14 pages. In French, English abstrac
Probability Theory Compatible with the New Conception of Modern Thermodynamics. Economics and Crisis of Debts
We show that G\"odel's negative results concerning arithmetic, which date
back to the 1930s, and the ancient "sand pile" paradox (known also as "sorites
paradox") pose the questions of the use of fuzzy sets and of the effect of a
measuring device on the experiment. The consideration of these facts led, in
thermodynamics, to a new one-parameter family of ideal gases. In turn, this
leads to a new approach to probability theory (including the new notion of
independent events). As applied to economics, this gives the correction, based
on Friedman's rule, to Irving Fisher's "Main Law of Economics" and enables us
to consider the theory of debt crisis.Comment: 48p., 14 figs., 82 refs.; more precise mathematical explanations are
added. arXiv admin note: significant text overlap with arXiv:1111.610
On number fields with given ramification
Let E/F be a CM field split above a finite place v of F, let l be a rational
prime number which is prime to v, and let S be the set of places of E dividing
lv. If E_S denotes a maximal algebraic extension of E unramified outside S, and
if u is a place of E dividing v, we show that any field embedding E_S \to
\bar{E_u} has a dense image.
The "unramified outside S" number fields we use are cut out from the l-adic
cohomology of the "simple" Shimura varieties studied by Kottwitz and
Harris-Taylor. The main ingredients of the proof are then the local Langlands
correspondence for GL_n, the main global theorem of Harris-Taylor, and the
construction of automorphic representations with prescribed local behaviours.
We explain how stronger results would follow from the knowledge of some
expected properties of Siegel modular forms, and we discuss the case of the
Galois group of a maximal algebraic extension of Q unramified outside a single
prime p and infinity.Comment: Extended version (18 pages), new sections added (construction of
automorphic forms with prescribed properties, speculations on generalizations
of the main theorem
An adjustment inventory for primary grades
Thesis (M.A.)--Boston Universit
Coverings of curves of genus 2
We shall discuss the idea of finding all rational points on a curve C by first finding an associated collection of curves whose rational points cover those of C. This classical technique has recently been given a new lease of life by being combined with descent techniques on Jacobians of curves, Chabauty techniques, and the increased power of software to perform algebraic number theory. We shall survey recent applications during the last 5 years which have used Chabauty techniques and covering collections of curves of genus 2 obtained from pullbacks along isogenies on their Jacobians
- …