Global L -functions over function fields

Böckle, Gebhard

doi:10.1007/s002080200325

Cited by 20 publications

(32 citation statements)

References 15 publications

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“…(see e.g., [Bo1,Bo2,Go2]) and all such functions must be studied. fact, without his construction in the rank 1 case almost no arithmetic in general is even possible.…”

Section: Introductionmentioning

confidence: 99%

On the L -series of F. Pellarin

Goss

2013

Journal of Number Theory

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The calculation, by L. Euler, of the values at positive even integers of the Riemann zeta function, in terms of powers of π and rational numbers, was a watershed event in the history of number theory and classical analysis. Since then many important analogs involving L-values and periods have been obtained. In analysis in finite characteristic, a version of Euler's result was given by L. Carlitz (1937) [Ca2], (1940) [Ca3] in the 1930s which involved the period of a rank 1 Drinfeld module (the Carlitz module) in place of π . In a very original work (Pellarin, 2011 [Pe2]), F. Pellarin has quite recently established a "deformation" of Carlitz's result involving certain L-series and the deformation of the Carlitz period given in Anderson and Thakur (1990) [AT1]. Pellarin works only with the values of this L-series at positive integral points. We show here how the techniques of Goss (1996) [Go1] also allow these new Lseries to be analytically continued -with associated trivial zeroes -and interpolated at finite primes.

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“…(see e.g., [Bo1,Bo2,Go2]) and all such functions must be studied. fact, without his construction in the rank 1 case almost no arithmetic in general is even possible.…”

Section: Introductionmentioning

confidence: 99%

On the L -series of F. Pellarin

Goss

2013

Journal of Number Theory

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“…Moreover, this logarithmic growth, when combined with the deep a priori estimates of Amice [Am1], actually provides the analytic continuation of these L-series at all places of k. It thus became reasonable to also expect such logarithmic growth for the degrees of special polynomials associated to L-series of general rank Drinfeld modules, t-modules, etc. And, indeed, such a basic result was recently established by Böckle in [Boc1] as a stepping stone in his analytic continuation of such L-series.…”

Section: Introductionmentioning

confidence: 86%

“…Let L(s) be the L-series of a -sheaf of the type shown to be entire in [Boc1] (see, e.g., Theorem 2 below). In this paper, we show how the logarithmic growth of the degrees of L(x − j) is enough to establish the analytic continuation and logarithmic growth of any partial L-series (see Definition 25) associated to L(s).…”

Section: Introductionmentioning

confidence: 99%

“…The approach to characteristic p L-series in [Boc1] is via cohomology. Due to the labors of Drinfeld, Anderson, Taguchi, Wan, Pink and Böckle, the basic object of characteristic p arithmetic has evolved from Drinfeld modules to " -sheaves" which are simply coherent sheaves over the product of a base scheme X with Spec(A) equipped with a Frobenius-linear morphism (see Definition 12).…”

Section: Introductionmentioning

confidence: 99%

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Applications of non-Archimedean integration to the L-series of τ-sheaves

Goss

2005

Journal of Number Theory

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Let F be a -sheaf. Building on previous work of Drinfeld, Anderson, Taguchi, and Wan, Böckle and Pink (A cohomological theory of crystals over function fields, in preparation), develop a cohomology theory for F. Böckle (Math. Ann. 323 (2002) 737) uses this theory to establish the analytic continuation of the L-series associated to F (which is a characteristic pvalued "Dirichlet series") and the logarithmic growth of the degrees of its special polynomials. In this paper, we shall show that this logarithmic growth is all that is needed to analytically continue the original L-series as well as all associated partial L-series. Moreover, we show that the degrees of the special polynomials attached to the partial L-series also grow logarithmically. Our tools are Böckle's original results, non-Archimedean integration, and the very strong estimates of Amice (Bull. Soc. Math. France 92 (1964) 117). Along the way, we define certain natural modules associated with non-Archimedean measures (in the characteristic 0 case as well as in characteristic p). ଁ This work is dedicated to the memory of Arnold Ross; a wonderful colleague who had an enormous positive influence on the mathematics community.

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“…Due to the recent work of G. Böckle [Boc2], [Go4] we now know that these functions give rise to L-series via "Hecke operators" just as with classical modular forms. Moreover, the standard formalism used to construct L-series mentioned above carries over very readily to Drinfeld modules, again leading to a theory of L-series [Boc1] and gamma functions [Th1], [ABP1] involving only finite characteristic analysis; an analog of Euler's formula on ζ(2n) is easily established in this context. While some general results are known about these functions and their zeroes [Wa1], [Go3], most of the general principles of the theory are still unknown.…”

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confidence: 99%

Book Review: Number theory in function fields

Goss¹

2003

Bull. Amer. Math. Soc.

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We begin with a short (obviously incomplete) historical summary related to the book being reviewed. A complex number α is said to be "algebraic" if there is a polynomial 0 = p(x) with integer coefficients such that p(α) = 0; otherwise α is said to be "transcendental". The first recorded instance of mathematicians encountering an "irrational" algebraic number appears to be the ancient Greeks. Indeed, if one considers a square with unit sides, then, of course, by the Pythagorean Theorem a diagonal must have length √ 2. To Pythagoras (and his followers) is also due the result that √ 2 cannot be expressed as the quotient of integers. One story has it that Hippasus, the person who revealed this irrationality, died at sea in a shipwreck, "struck by the wrath of the gods."As history evolved, so did mathematicians' views about numbers. During the sixteenth century cubic equations were discussed by the Italian algebraists G. Cardano and R. Bombelli ([v1], Part C §2). In particular a very curious phenomenon appeared with equations like x 3 = 15x + 4 . Here one readily finds that x = 4 is a root and, after division by x−4, that the other two roots are also real. However, the Italian school had developed methods to handle such an equation directly. These methods do give the correct real roots but only through the use of quantities like 3 2 + √ −121, which are obviously non-real. This caused Cardano to have serious

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Global L -functions over function fields

Cited by 20 publications

References 15 publications

On the L -series of F. Pellarin

On the L -series of F. Pellarin

Applications of non-Archimedean integration to the L-series of τ-sheaves

Book Review: Number theory in function fields

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