Linear independence of Gamma values in positive characteristic

Brownawell, W. Dale; Papanikolas, Matthew A.

doi:10.1515/crll.2002.068

Cited by 24 publications

(46 citation statements)

References 23 publications

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“…When the action of d8(t) is scalar on Lie(G), then Q [d,(t)]-linear dependency coincides with Q -linear dependency. That is often the case in examples, e.g., [5].…”

Section: Basic Principlementioning

confidence: 81%

Minimal Extensions of Algebraic Groups and Linear Independence

Brownawell

2001

Journal of Number Theory

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We introduce the notion of a minimal extension of t-groups. Linear independence of the coordinates of the logarithm of an algebraic point in a minimal extension of t-groups follows naturally from linear independence of the coordinates of the image in the tangent space of the base t-group. We illustrate this principle through a leisurely parade of examples. In particular, we establish a general theorem about divided derivatives for t-modules. Minimal extensions turn out to correspond to Frattini covers for t-groups. Academic Press

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“…When the action of d8(t) is scalar on Lie(G), then Q [d,(t)]-linear dependency coincides with Q -linear dependency. That is often the case in examples, e.g., [5].…”

Section: Basic Principlementioning

confidence: 81%

Minimal Extensions of Algebraic Groups and Linear Independence

Brownawell

2001

Journal of Number Theory

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“…This Gamma function is meromorphic over the field C. It satisfies similar relations as the standard relations (20) which are satisfied by the Euler Gamma function. Also, in finite characteristic, the counterpart of the relations of Deligne-Koblitz-Ogus has been obtained by Deligne, Anderson and Thakur (see [14]). …”

Section: Finite Characteristicmentioning

confidence: 86%

“…The extension of Chowla-Selberg formula by G. Shimura to Abelian varieties of CM type gives rise to the relations of Deligne-Koblitz-Ogus on Gamma function (see [14] [65]. He proved a generalization of Baker's Theorem 4 to commutative algebraic groups.…”

Section: Then One At Least Of the Two Numbers ℘(U) Au + Bζ(u) Is Tramentioning

confidence: 99%

“…In 1992 G. Anderson introduced the notion of soliton, which, according to [14], is a higher dimensional version of the shtuka function for rank 1 Drinfeld modules. The meromorphic functions which were used by R. Coleman in his study of Frobenius endomorphisms on the curves of Fermat and Artin-Schreier had been interpreted by D. Thakur in terms of his Gamma function.…”

Section: Finite Characteristicmentioning

confidence: 99%

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Transcendence of Periods: The State of the Art

Waldschmidt¹

2006

Pure and Applied Mathematics Quarterly

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The set of real numbers and the set of complex numbers have the power of continuum. Among these numbers, those which are "interesting", which appear "naturally", which deserve our attention, form a countable set. Starting from this point of view we are interested in the periods as defined by M. Kontsevich and D. Zagier. We give the state of the art on the question of the arithmetic nature of these numbers: to decide whether a period is a rational number, an irrational algebraic number or else a transcendental number is the object of a few theorems and of many conjectures. We also consider the approximation of such numbers by rational or algebraic numbers. AcknowledgmentThis is an english updated version of the paper in french: Michel Waldschmidt

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“…In analogy with de Rham cohomology for elliptic curves, in the late 1980's Anderson, Deligne, Gekeler, and Yu developed a de Rham cohomology theory for Drinfeld modules (for more details, see [5,8,14,18]). Fix a rank r Drinfeld satisfying the functional equation…”

Section: Introductionmentioning

confidence: 99%

Algebraic relations among periods and logarithms of rank 2 Drinfeld modules

Papanikolas¹

2011

ajm

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Abstract. For any rank 2 Drinfeld module ρ defined over an algebraic function field, we consider its period matrix P ρ , which is analogous to the period matrix of an elliptic curve defined over a number field. Suppose that the characteristic of F q is odd and ρ is without complex multiplication. We show that the transcendence degree of the field generated by the entries of P ρ over F q (θ) is 4. As a consequence, we show also the algebraic independence of Drinfeld logarithms of algebraic functions which are linearly independent over F q (θ).

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Linear independence of Gamma values in positive characteristic

Abstract: Abstract. We investigate the arithmetic nature of special values of Thakur's function field Gamma function at rational points. Our main result is that all linear dependence relations over the field of algebraic functions are consequences of the Anderson-Deligne-Thakur bracket relations.

Cited by 24 publications

References 23 publications

Minimal Extensions of Algebraic Groups and Linear Independence

Minimal Extensions of Algebraic Groups and Linear Independence

Transcendence of Periods: The State of the Art

Algebraic relations among periods and logarithms of rank 2 Drinfeld modules

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