Équations différentielles linéaires et majorations de multiplicités

Bertrand, Daniel; Beukers, Frits

doi:10.24033/asens.1488

Cited by 34 publications

(81 citation statements)

References 9 publications

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“…But this matches exactly the definition of the exponents of the operator L; see [Bertrand and Beukers 1985].…”

Section: The Companion Bundles Of a Differential Equationsupporting

confidence: 72%

Exponents of a meromorphic connection on a compact Riemann surface

Corel¹

2009

Pacific J. Math.

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We give a general definition of the exponents of a meromorphic connection ∇ on a holomorphic vector bundle Ᏹ of rank n over a compact Riemann surface X. We prove that they can be computed as invariants of a vector bundle Ᏹ L canonically attached to Ᏹ, which we construct and call the Levelt bundle of Ᏹ, and whose degree (equal to the sum of the exponents) we estimate by upper and lower bounds (Fuchs' relations). We use this definition to construct, for every linear differential equation on a compact Riemann surface (with regular or irregular singularities), the companion bundle of the equation, a vector bundle endowed with a meromorphic connection that is equivalent to the given equation and has precisely the same singularities and the same set of exponents.

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“…But this matches exactly the definition of the exponents of the operator L; see [Bertrand and Beukers 1985].…”

Section: The Companion Bundles Of a Differential Equationsupporting

confidence: 72%

Exponents of a meromorphic connection on a compact Riemann surface

Corel¹

2009

Pacific J. Math.

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“…It is possible to sharpen this tov = (w + l)! + w + l using the improvement in Brownawell [2] of Corollary 2 of Nesterenko [8] and even further to y = 2m by D. Bertrand and F. Beukers [1]. However this does not affect our discussion of the effectivity of the measure.…”

Section: Effectivity Of Tmentioning

confidence: 95%

Effectivity in independence measures for values ofE-functions

Brownawell

1985

J Aust Math Soc A

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We establish a measure of algebraic independence for values of E-functions which is more nearly effectively computable than the previous one. When the system of equations meets either of two criteria, then the measure becomes entirely effectively computable.1980 Mathematics subject classification (Amer, Math. Soc): 10 F 37. Statement of main resultIn 1929 C. L. Siegel [11] proved the algebraic independence of values of certain functions at an algebraic point. These functions satisfy a system of first order linear differential equations over C(z) and in addition are A7:-functions, so the coefficients of their Maclaurin expansion are of the form ajv\ with the a v from a fixed algebraic number field K (see Section 3 below). This new class of functions included the exponential function (thus generalizing the celebrated theorem of Lindemann) and certain hypergeometric and Bessel functions. In 1949 Siegel [12] formalized this approach for functions satisfying a more general normality condition, whose verification however proved very elusive for further classes.In 1959 A. B. Shidlovsky [9] removed that imperfection in an ingenious way by relating the rank of a certain matrix representing the derivatives of a function to

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“…The proof of Theorems 1 and 2 is based on the strategy of [8]: we apply Siegel's linear independence criterion using a general version of Shidlovsky's lemma (namely Theorem 3, stated in §1.3 and proved in [8] following the approach of Bertrand-Beukers [4] and Bertrand [3]). This makes it necessary to relate the construction to a Padé approximation problem with essentially as many equations as the number of unknowns.…”

Section: Corollarymentioning

confidence: 99%

Irrationality of values of L‐functions of Dirichlet characters

Fischler

2019

J. London Math. Soc.

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In a recent paper with Sprang and Zudilin, the following result was proved: if a is large enough in terms of ε > 0, then at least 2 (1−ε) log a log log a values of the Riemann zeta function at odd integers between 3 and a are irrational. This improves on the Ball-Rivoal theorem, that provides only 1−ε 1+log 2 log a such irrational values -but with a stronger property: they are linearly independent over the rationals.In the present paper we generalize this recent result to both L-functions of Dirichlet characters and Hurwitz zeta function. The strategy is different and less elementary: the construction is related to a Padé approximation problem, and a generalization of Shidlovsky's lemma is used to apply Siegel's linear independence criterion.We also improve the analogue of the Ball-Rivoal theorem in this setting: we obtain 1−ε 1+log 2 log a linearly independent values L(f, s) with s ≤ a of a fixed parity, when f is a Dirichlet character. The new point here is that the constant 1 + log 2 does not depend on f .

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Équations différentielles linéaires et majorations de multiplicités

Cited by 34 publications

References 9 publications

Exponents of a meromorphic connection on a compact Riemann surface

Exponents of a meromorphic connection on a compact Riemann surface

Effectivity in independence measures for values ofE-functions

Irrationality of values of L‐functions of Dirichlet characters

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