Search ResultsShowing 1-10 of 10
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Arithmetic of positive characteristic L-series values in Tate algebras
Comments: final versionCategories: math.NTThe second author has recently introduced a new class of L-series in the arithmetic theory of function fields over finite fields. We show that the value at one of these L-series encode arithmetic informations of certain Drinfeld modules defined over Tate algebras. This enables us to generalize Anderson's log-algebraicity Theorem and Taelman's Herbrand-Ribet Theorem.
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Universal Gauss-Thakur sums and L-series
Comments: Corrected several typos and an error in the proof of Proposition 21 Section 3. Improved the general presentation of the paperCategories: math.NTIn this paper we study the behavior of the function omega of Anderson-Thakur evaluated at the elements of the algebraic closure of the finite field with q elements F_q. Indeed, this function has quite a remarkable relation to explicit class field theory for the field K=F_q(T). We will see that these values, together with the values of its divided derivatives, generate the maximal abelian extension of K which is tamely ramified at infinity. We will also see that omega is, in a way that we will explain in detail, an universal Gauss-Thakur sum. We will then use these results to show the existence of functional relations for a class of L-series introduced by the second author. Our results will be finally applied to obtain a new class of congruences for Bernoulli-Carlitz fractions, and an analytic conjecture is stated, implying an interesting behavior of such fractions modulo prime ideals of A=F_q[T].
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arXiv:1210.1660 (Published 2012-10-05)
Arithmetic of Units in F_q[T]
Categories: math.NTKeywords: arithmetic, taelmans unit moduleWe study in this note the arithmetic of Taelman's unit module for the ring F_q[T]
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arXiv:0903.4350 (Published 2009-03-25)
On the linear independence of p-adic L-functions modulo p
Categories: math.NTInspired by Warren Sinnott 's method we prove a linear independence result modulo p for the Iwasawa power series associated to Kubota-Leopoldt p-adic L-functions.
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arXiv:0802.1645 (Published 2008-02-12)
On Jacobi Sums in $\mathbb Q(ζ_p)$
Categories: math.NTWe study the p-adic behavior of Jacobi Sums for $\mathbb Q(\zeta_p)$ and link this study to the p-Sylow subgroup of the ideal class group of $\mathbb Q(\zeta_p\`a^+$
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arXiv:0709.2838 (Published 2007-09-18)
On the p-adic Leopoldt Transform of a power series
DOI: 10.4064/aa134-4-5Categories: math.NTKeywords: p-adic leopoldt transform, iwasawa lambda invariant, abelian number field, p-adic l-functions, iwaswa power seriesTags: journal articleIn this paper we give a bound for the Iwasawa lambda invariant of an abelian number field attached to the cyclotomic Z_p-extension of that field. We also give some properties of Iwaswa power series attached to p-adic L-functions.
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arXiv:math/0606332 (Published 2006-06-14)
On Weil Numbers in Cyclotomic Fields
Categories: math.NTIn this note, we investigate the p-adic behavior of Weil numbers in the cyclotomic $\mathbb Z\_p$-extension of $\mathbb Q(\zeta\_p).$ We determlne the characteristic ideal of the quotient of semi-local units by Weil numbers in terms of the characteristic ideals of some classical modules that appear in Iwasawa Theory.
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arXiv:math/0512015 (Published 2005-12-01)
A Note on a result of Iwasawa
Categories: math.NTWe recover a result of Iwasawa on the p-adic logarithm of principal units with the use of the value at 1 of p-adic L-functions. We deduce an Iwasawa-like result in the odd part of principal units.
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arXiv:math/0510293 (Published 2005-10-14)
On some p-adic power series attached to the arithmetic of $\mathbb Q(ζ\_p).$
Categories: math.NTIn this paper, we prove that the derivative of the Iwasawa power series associated to p-adic L-functions of $\mathbb Q(\zeta\_p)$ are not divisible by p. This extends previous results obtained by Ferrero and Washington in 1979.
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arXiv:math/0502130 (Published 2005-02-07)
On L-functions of cyclotomic function fields
Categories: math.NTWe study two criterions of cyclicity for divisor class groups of functions fields, the first one involves Artin L-functions and the second one involves "affine" class groups. We show that, in general, these two criterions are not linked.