An elliptic analogue of the Franklin-Schneider theorem (original) (raw)
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Visnyk Lvivskogo Universytetu. Seriya Mekhaniko-Matematychna
Dedicated to the 60th birthday of M. M. Zarichnyi Yaroslav KHOLYAVKA, Olga MYLYO svn prnko xtionl niversity of vvivD niversytetsk trFD ID UWHHHD vvivD krine eEmilsX ykholdfrnkoFlvivFuD olgFmylyodgmilFom Let ℘(z), sn z be algebraically independent Weierstrass and Jacobi elliptic functions with algebraic invariants and algebraic elliptic module, (2ω 1 , 2ω 3) and (4K, 2iK) be the main periods of ℘(z) and sn z respectively, α be an algebraic number dierent from the poles of ℘(z) and sn z. We estimate from below the simultaneous approximation of sn(2ω 1), sn(α), ℘(4K), and ℘(α).
Some theorems on diophantine approximation
Transactions of the American Mathematical Society, 1966
Introduction. The study of the values at rational points of transcendental functions defined by linear differential equations with coefficients in Q[z] (2) can be traced back to Hurwitz [1] who showed that if ,. , 1 z 1 z2 Az) = l+-b-lT+WTa)2l +where « is a positive integer, b is an integer, and b\a is not a negative integer, then for all nonzero z in Q((-1)1/2) the number y'(z)jy(z) is not in g((-1)1/2). Ratner [2] proved further results. Then Hurwitz [3] generalized his previous results to show that if nZ) ,+g(0) 1! +g(0)-g(l)2! + where f(z) and g(z) are in Q[z\, neither f(z) nor g(z) has a nonnegative integral zero, and degree (/(z)) < degree (g(z)) = r, then for all nonzero z in the imaginary quadratic field Q((-n)1'2) two of the numbers y(z),y(l)(z),-,yir\z) have a ratio which is not in Q((-n)112). Perron [4], Popken [5], C. L. Siegel [6], and K. Mahler [7] have obtained important results in this area. In this paper we shall use a generalization of the method which was developed by Mahler [7] to study the approximation of the logarithms of algebraic numbers by rational and algebraic numbers. Definition. Let K denote the field Q((-n)i/2) for some nonnegative integer «. Definition. For any monic 0(z) in AT[z] of degree k > 0 and such that 6(z) has no positive integral zeros we define the entire function oo d f(z)= £-. d^O d n oc«)
On some applications of diophantine approximations
Proceedings of the National Academy of Sciences, 1984
Siegel's results [Siegel, C. L. (1929) Abh. Preuss. Akad. Wiss. Phys.-Math. Kl. 1] on the transcendence and algebraic independence of values of E -functions are refined to obtain the best possible bound for the measures of irrationality and linear independence of values of arbitrary E -functions at rational points. Our results show that values of E -functions at rational points have measures of diophantine approximations typical to “almost all” numbers. In particular, any such number has the “2 + ε” exponent of irrationality: ǀΘ - p / q ǀ > ǀ q ǀ -2-ε for relatively prime rational integers p,q , with q ≥ q 0 (Θ, ε). These results answer some problems posed by Lang. The methods used here are based on the introduction of graded Padé approximations to systems of functions satisfying linear differential equations with rational function coefficients. The constructions and proofs of this paper were used in the functional (nonarithmetic case) in a previous paper [Chudnovsky, D. V. &...
Rational points on certain elliptic surfaces
Acta Arithmetica, 2007
\ Q, and let us assume that deg f ≤ 4. In this paper we prove that if deg f ≤ 3, then there exists a rational base change t → ϕ(t) such that there is a non-torsion section on the surface E f •ϕ . A similar theorem is valid in case when deg f = 4 and there exists t 0 ∈ Q such that infinitely many rational points lie on the curve Et 0 : y 2 = x 3 + f (t 0 )x. In particular, we prove that if deg f = 4 and f is not an even polynomial, then there is a rational point on E f . Next, we consider a surface E g : y 2 = x 3 + g(t), where g ∈ Q[t] is a monic polynomial of degree six. We prove that if the polynomial g is not even, there is a rational base change t → ψ(t) such that on the surface E g•ψ there is a non-torsion section. Furthermore, if there exists t 0 ∈ Q such that on the curve E t 0 : y 2 = x 3 +g(t 0 ) there are infinitely many rational points, then the set of these t 0 is infinite. We also present some results concerning diophantine equation of the form x 2 − y 3 − g(z) = t, where t is a variable.
On rational approximation of algebraic functions
Advances in Mathematics, 2006
We construct a new scheme of approximation of any multivalued algebraic function f (z) by a sequence {rn(z)} n∈N of rational functions. The latter sequence is generated by a recurrence relation which is completely determined by the algebraic equation satisfied by f (z). Compared to the usual Padé approximation our scheme has a number of advantages, such as simple computational procedures that allow us to prove natural analogs of the Padé Conjecture and Nuttall's Conjecture for the sequence {rn(z)} n∈N in the complement CP 1 \ D f , where D f is the union of a finite number of segments of real algebraic curves and finitely many isolated points. In particular, our construction makes it possible to control the behavior of spurious poles and to describe the asymptotic ratio distribution of the family {rn(z)} n∈N . As an application we settle the so-called 3-conjecture of Egecioglu et al dealing with a 4-term recursion related to a polynomial Riemann Hypothesis.
On applications of diophantine approximations
Proceedings of the National Academy of Sciences, 1984
This paper is devoted to the study of the arithmetic properties of values of G -functions introduced by Siegel [Siegel, C. L. (1929) Abh. Preuss. Akad. Wiss. Phys.-Math. Kl. 1]. One of the main results is a theorem on the linear independence of values of G -functions at rational points close to the origin. In this theorem, no conditions are imposed on the p -adic convergence of a G -function at a generic point. The theorem finally realizes Siegel's program on G -function values outlined in his paper.
A refinement of a conjecture of Gross, Kohnen, and Zagier
Contemporary mathematics, 2018
AMS: Number theory-Arithmetic algebraic geometry (Diophantine geometry)-Arithmetic aspects of modular and Shimura varieties. msc | Algebraic geometry-Arithmetic problems. Diophantine geometry-Rational points. msc | Algebraic geometry-Arithmetic problems. Diophantine geometry-Applications to coding theory and cryptography. msc | Number theory-Discontinuous groups and automorphic forms-Automorphic forms on. msc | Number theory-Algebraic number theory: global fields-Class field theory. msc | Number theory-Zeta and L-functions: analytic theory-None of the above, but in this section. msc | Algebraic geometry-Arithmetic problems. Diophantine geometry-Modular and Shimura varieties. msc | Number theory-Arithmetic algebraic geometry (Diophantine geometry)-L-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture. msc | Number theory-Arithmetic algebraic geometry (Diophantine geometry)-Curves of arbitrary genus or genus = 1 over global fields.
The Birch and Swinnerton-Dyer Conjecture
2006
A polynomial relation f (x, y) = 0 in two variables defines a curve C 0 . If the coefficients of the polynomial are rational numbers then one can ask for solutions of the equation f (x, y) = 0 with x, y ∈ Q, in other words for rational points on the curve. The set of all such points is denoted C 0 (Q). If we consider a non-singular projective model C of the curve then topologically C is classified by its genus, and we call this the genus of C 0 also. Note that C 0 (Q) and C(Q) are either both finite or both infinite. Mordell conjectured, and in 1983 Faltings proved, the following deep result Theorem [F1]. If the genus of C 0 is greater than or equal to two, then C 0 (Q) is finite.