Periods and special values of the hypergeometric series

1989 ◽  
Vol 106 (3) ◽  
pp. 389-401 ◽  
Author(s):  
Matthias Flach
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Modular Forms ◽  
Hypergeometric Series ◽  
Joint Paper ◽  
Special Values ◽  
Precise Statement ◽  
Image Position

The aim of this paper is to complement results by Wolfart [14] about algebraic values of the classical hypergeometric seriesfor rational parameters a, b, c and algebraic arguments z. Wolfart essentially determines the set of a, b, c ∈ ℚ,z ∈ ℚ for which F(a, b, c; z) ∈ ℚ and indicates, in a joint paper with F. Beukers[1], that some of these values can be expressed in terms of special values of modular forms. This method yields a few strikingly explicit identities likebut it does not give general statements about the nature of the algebraic values in question. In this paper we identify F(a, b, c; z) as a generator of a Kummer extension of a certain number field depending on z, which in particular bounds its degree as an algebraic number in terms of the degree of z. Our theorem in §2 seems to be the most precise statement one can make in general but sometimes improvements are possible as we point out at the end of §2.

A Periodic Jacobi-Perron Algorithm

1965 ◽  
Vol 17 ◽  
pp. 933-945
Author(s):  
Leon Bernstein
Keyword(s):  
Algebraic Equation ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Image Position

In the first part of this paper I shall demonstrate that one irrational root of the algebraic equationcreates an algebraic number field, out of which n — 1 irrationals can be chosen so that they yield a periodic Jacobi-Perron algorithm. The coefficients in (1) are subject to certain restrictions which will be elaborated below.

scholarly journals On the divisibility of r2(n)

1977 ◽  
Vol 18 (1) ◽  
pp. 109-111 ◽  
Author(s):  
E. J. Scourfield
Keyword(s):  
Modular Form ◽  
Algebraic Number ◽  
Modular Forms ◽  
Congruence Subgroup ◽  
Algebraic Number Field ◽  
The Past ◽  
Divisibility Property ◽  
Integral Weight ◽  
Image Position ◽  
Positive Integers

During the past few years, some papers of P. Deligne and J.-P. Serre (see [2], [9], [10] and other references cited there) have included an investigation of certain properties of coefficients of modular forms, and in particular Serre [10] (see also [11]) obtained the divisibility property (1) below. Letbe a modular form of integral weight k ≧ 1 on a congruence subgroup of SL2(Z), and suppose that each cn belongs to the ring RK of integers of an algebraic number field K finite over Q. For c ∈ RK and m ≧ 1 an integer, write c ≡ 0 (mod m) if c ∈ m RK and c ≢ 0 (mod m) otherwise. Then Serre showed that there exists α > 0 such thatas x → ∞, where throughout this note N(n ≦ x: P) denotes the number of positive integers n ≦ x with the property P.

Special values of the hypergeometric series

1991 ◽  
Vol 109 (2) ◽  
pp. 257-261 ◽  
Author(s):  
G. S. Joyce ◽  
I. J. Zucker
Keyword(s):  
Gamma Function ◽  
Hypergeometric Series ◽  
Special Values ◽  
Image Position

Recently, several authors [1, 3, 9] have investigated the algebraic and transcendental values of the hypergeometric seriesfor rational parameters a, b, c and algebraic arguments z. This work has led to some interesting new identities such asand where Γ(x) denotes the gamma function.

Uniform bounds for the number of solutions to Yn = f(X)

1986 ◽  
Vol 100 (2) ◽  
pp. 237-248 ◽  
Author(s):  
J.-H. Evertse ◽  
J. H. Silverman
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Solutions To Equations ◽  
Number Of Solutions ◽  
Explicit Result ◽  
Uniform Bounds ◽  
The Past ◽  
Image Position ◽  
Integral Solutions

Let K be an algebraic number field and f(X) ∈ K[X]. The Diophantine problem of describing the solutions to equations of the formhas attracted considerable interest over the past 60 years. Siegel [12], [13] was the first to show that under suitable non-degeneracy conditions, the equation (+) has only finitely many integral solutions in K. LeVeque[7] proved the following, more explicit, result. Letwhere a ∈ K* and αl,…,αk are distinct and algebraic over K. Then (+) has only finitely many integral solutions unless (nl,…,nk) is a permutation of one of the n-tuples

How to solve a quadratic equation in integers

1981 ◽  
Vol 89 (1) ◽  
pp. 1-5 ◽  
Author(s):  
Fritz J. Grunewald ◽  
Daniel Segal
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Quadratic Equation ◽  
Effective Procedure ◽  
Image Position

In answer to a question posed by J. L Britton in (2), we sketch in this note an effective procedure to decide whether an arbitrary quadratic equationwith rational coefficients, has a solution in integers. A similar procedure will in fact decide whether such an equation over a (suitably specified) algebraic number field k has a solution in any (suitably specified) order in k; but we shall not burden the exposition by giving chapter and verse for this claim.

Simultaneous rational approximations to certain algebraic numbers

1967 ◽  
Vol 63 (3) ◽  
pp. 693-702 ◽  
Author(s):  
A. Baker
Keyword(s):  
Positive Integer ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Infinitely Many Solutions ◽  
Rational Approximations ◽  
Algebraic Numbers ◽  
Famous Theorem ◽  
Image Position

It is generally conjectured that if α1, α2 …, αk are algebraic numbers for which no equation of the formis satisfied with rational ri not all zero, and if K > 1 + l/k, then there are only finitely many sets of integers p1, p2, …, pkq, q > 0, such thatThis result would be best possible, for it is well known that (1) has infinitely many solutions when K = 1 + 1/k. † If α1, α2, …, αk are elements of an algebraic number field of degree k + 1 the result can be deduced easily (see Perron (11)). The famous theorem of Roth (13) asserts the truth of the conjecture in the case k = 1 and this implies that for any positive integer k, (1) certainly has only finitely many solutions if K > 2. Nothing further in this direction however has hitherto been proved.‡

scholarly journals The Brauer group of cubic surfaces

1993 ◽  
Vol 113 (3) ◽  
pp. 449-460 ◽  
Author(s):  
Sir Peter Swinnerton-Dyer
Keyword(s):  
Finite Group ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Rational Surface ◽  
Hasse Principle ◽  
Weak Approximation ◽  
Brauer Group ◽  
Cubic Surfaces ◽  
Image Position

1. Let V be a non-singular rational surface defined over an algebraic number field k. There is a standard conjecture that the only obstructions to the Hasse principle and to weak approximation on V are the Brauer–Manin obstructions. A prerequisite for calculating these is a knowledge of the Brauer group of V; indeed there is one such obstruction, which may however be trivial, corresponding to each element of Br V/Br k. Because k is an algebraic number field, the natural injectionis an isomorphism; so the first step in calculating the Brauer–Manin obstruction is to calculate the finite group H1 (k), Pic .

An Independent System of Units in Certain Algebraic Number Fields

1985 ◽  
Vol 37 (4) ◽  
pp. 644-663
Author(s):  
Claude Levesque
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Algebraic Number Fields ◽  
Independent System ◽  
System Of Units ◽  
Image Position

For Kn = Q(ω) a real algebraic number field of degree n over Q such thatwith D ∊ N, d ∊ Z, d|D2, and D2 + 4d > 0, we proved in [5] (by using the approach of Halter-Koch and Stender [6]) that ifwiththenis an independent system of units of Kn.

scholarly journals Mahler Measures as Linear Combinations of L–values of Multiple Modular Forms

2015 ◽  
Vol 67 (2) ◽  
pp. 424-449 ◽  
Author(s):  
Detchat Samart
Keyword(s):  
Modular Forms ◽  
Hypergeometric Series ◽  
Mahler Measure ◽  
Laurent Polynomials ◽  
Algebraic Numbers ◽  
Quadratic Character ◽  
Linear Combinations ◽  
Special Values

AbstractWe study the Mahler measures of certain families of Laurent polynomials in two and three variables. Each of the known Mahler measure formulas for these families involves L–values of at most one newform and/or at most one quadratic character. In this paper we show, either rigorously or numerically, that the Mahler measures of some polynomials are related to L–values of multiple newforms and quadratic characters simultaneously. The results suggest that the number of modular L–values appearing in the formulas significantly depends on the shape of the algebraic value of the parameter chosen for each polynomial. As a consequence, we also obtain new formulas relating special values of hypergeometric series evaluated at algebraic numbers to special values of L–functions.

ON THE POLYNOMIAL xd + ax + b OVER 𝔽q AND GAUSSIAN HYPERGEOMETRIC SERIES

2013 ◽  
Vol 09 (07) ◽  
pp. 1753-1763 ◽  
Author(s):  
RUPAM BARMAN ◽  
GAUTAM KALITA
Keyword(s):  
Modular Forms ◽  
Hypergeometric Series ◽  
American Mathematical Society ◽  
Polynomial Equation ◽  
Mathematical Society ◽  
Number Of Solutions ◽  
Special Values ◽  
Regional Conference ◽  
Gaussian Hypergeometric Series ◽  
Cbms Regional Conference Series

For d ≥ 2, denote by Pd(x) the polynomial over 𝔽q given by [Formula: see text]. We explicitly find the number of solutions in 𝔽q of the polynomial equation Pd(x) = 0 in terms of special values of dFd-1 and d-1Fd-2 Gaussian hypergeometric series with characters of orders d and d - 1 as parameters. This solves a problem posed by K. Ono (see p. 204 in [Web of Modularity : Arithmetic of the Coefficients of Modular Forms and q-Series, CBMS Regional Conference Series in Mathematics, No. 102 (American Mathematical Society, Providence, RI, 2004)]) on special values of n+1Fn Gaussian hypergeometric series for n > 2.

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