102 research outputs found
Analytic properties of mirror maps
We consider a multi-parameter family of canonical coordinates and mirror maps
o\ riginally introduced by Zudilin [Math. Notes 71 (2002), 604-616]. This
family includes many of the known one-variable mirror maps as special cases, in
particular many of modular origin and the celebrated example of Candelas, de la
Ossa, Green and\
Parkes [Nucl. Phys. B359 (1991), 21-74] associated to the quintic
hypersurface in . In [Duke Math. J. 151 (2010),
175-218], we proved that all coeffi\ cients in the Taylor expansions at 0 of
these canonical coordinates (and, hence, of the corresponding mirror maps) are
integers. Here we prove that all coefficients in the Taylor expansions at 0 of
these canonical coordinates are positive. Furthermore, we provide several
results pertaining to the behaviour of the canonical coordinates and mirror
maps as complex functions. In particular, we address analytic continuation,
points of singularity, and radius of convergence of these functions. We present
several very precise conjectures on the radius of convergence of the mirror
maps and the sign pattern of the coefficients in their Taylor expansions at 0.Comment: AmS-LaTeX; 40 page
On the values of G-functions
Let f be a G-function (in the sense of Siegel), and x be an algebraic number;
assume that the value f(x) is a real number. As a special case of a more
general result, we show that f(x) can be written as g(1), where g is a
G-function with rational coefficients and arbitrarily large radius of
convergence. As an application, we prove that quotients of such values are
exactly the numbers which can be written as limits of sequences a(n)/b(n),
where the generating series of both sequences are G-functions with rational
coefficients. This result provides a general setting for irrationality proofs
in the style of Apery for zeta(3), and gives answers to questions asked by T.
Rivoal in [Approximations rationnelles des valeurs de la fonction Gamma aux
rationnels : le cas des puissances, Acta Arith. 142 (2010), no. 4, 347-365]
Arithmetic theory of E-operators
In [S\'eries Gevrey de type arithm\'etique I Th\'eor\'emes de puret\'e et de
dualit\'e, Annals of Math. 151 (2000), 705--740], Andr\'e has introduced
E-operators, a class of differential operators intimately related to
E-functions, and constructed local bases of solutions for these operators. In
this paper we investigate the arithmetical nature of connexion constants of
E-operators at finite distance, and of Stokes constants at infinity. We prove
that they involve values at algebraic points of E-functions in the former case,
and in the latter one, values of G-functions and of derivatives of the Gamma
function at rational points in a very precise way. As an application, we define
and study a class of numbers having certain algebraic approximations defined in
terms of E-functions. These types of approximations are motivated by the
convergents to the number e, as well as by recent constructions of
approximations to Euler's constant and values of the Gamma function. Our
results and methods are completely different from those in our paper [On the
values of G-functions, Commentarii Math. Helv., to appear], where we have
studied similar questions for G-functions
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