($p$-adic) $L$-functions and ($p$-adic) (multiple) zeta values

Nagoya Mathematical Journal, 1977

Letpbe a prime. If one adjoins toQallpn-th roots of unity forn= 1,2,3, …, then the resulting field will contain a unique subfieldQ∞such thatQ∞is a Galois extension ofQwith Gal (Q∞/Q)Zp, the additive group ofp-adic integers. We will denote Gal (Q∞/Q) byΓ. In a previous paper [6], we discussed a conjecture relatingp-adicL-functions to certain arithmetically defined representation spaces forΓ. Now by using some results of Iwasawa, one can reformulate that conjecture in terms of certain other representation spaces forΓ. This new conjecture, which we believe may be more susceptible to generalization, will be stated below.

Journal of Number Theory, 1976

An elementary proof is given for the existence of the Kubota-Leopoldt padic L-functions. Also, an explicit formula is obtained for these functions, and a relationship between the values of the padic and classical L-functions at positive integers is discussed.

Nagoya Mathematical Journal, 1975

Let p be a prime. If one adjoins to Q all pn -th roots of unity for n = 1, 2, 3, …, then the resulting field will contain a unique subfield Q ∞ such that Q ∞ is a Galois extension of Q with Gal the additive group of p-adic integers. We will denote Gal(Q ∞/Q ) by Γ.

2014

Introduction 1 1. Action of the absolut Galois group on fundamental groups 9 2. Measures associated to towers of projective lines 10 3. Inclusions 16 4. Inversion 18 5. Measures K 1 (z) 21 6. Congruences between coefficients 25 7. ℓ-adic poly-multi-zeta functions? 28 8. ℓ-adic L-functions of Kubota-Leopoldt 28 9. ℓ-adic functions associated to measure K 1 (−1) 31 10. Hurwitz zeta functions and Dirichlet L-series 33 11. ℓ-adic L-functions of Z[1/m] 39 References 41

2015

Let z ∈ Q and let γ be an `-adic path on P1Q̄\{0, 1,∞} from → 01 to z. For any σ ∈ Gal(Q̄/Q), the element x−κ(σ)fγ(σ) ∈ π1(P1Q̄ \{0, 1,∞}, → 01)pro−`. After the embedding of π1 into Q{{X,Y }} we get the formal power series ∆γ(σ) ∈ Q{{X,Y }}. We shall express coefficients of ∆γ(σ) as integrals over (Z`) with respect to some measures Kr(z). The measures Kr(z) are constructed using the tower ( P1Q̄ \({0,∞}∪μ`n ) n∈N of coverings of P 1 Q̄ \{0, 1,∞}. Using the integral formulas we shall show congruence relations between coefficients of the formal power series ∆γ(σ). The congruence relations allow the construction of `-adic functions of non-Archimedean analysis, which however rest mysterious. Only in the special case of the measures K1( → 10) and K1(−1) we recover the familiar Kubota-Leopoldt `-adic L-functions. We recover also `-adic analogues of Hurwitz zeta functions. Hence we get also `-adic analogues of L-series for Dirichlet characters.

2006

: The object of this paper is to give several properties and applications of the multiple p-adic q-L-function of two variables L (r) p,q (s, z, χ). The explicit formulas relating higher order qBernoulli polynomials, which involve sums of products of higher order q-zeta function and higher order Dirichlet q-L-function are given. The value of higher order Dirichlet p-adic q-L-function for positive integers is also calculated. Furthermore, the Kummer-type congruences for multiple generalized q-Bernoulli polynomials are derived by making use of the difference theorem of higher order Dirichlet p-adic q-L-function.

2016

In this paper, we present a unifying approach to the general theory of abelian Stark conjectures. To do so we define natural notions of 'zeta element', of 'Weil-étale cohomology complexes' and of 'integral Selmer groups' for the multiplicative group Gm over finite abelian extensions of number fields. We then conjecture a precise connection between zeta elements and Weil-étale cohomology complexes, we show this conjecture is equivalent to a special case of the equivariant Tamagawa number conjecture and we give an unconditional proof of the analogous statement for global function fields. We also show that the conjecture entails much detailed information about the arithmetic properties of generalized Stark elements including a new family of integral congruence relations between Rubin-Stark elements (that refines recent conjectures of Mazur and Rubin and of the third author) and explicit formulas in terms of these elements for the higher Fitting ideals of the integral Selmer groups of Gm, thereby obtaining a clear and very general approach to the theory of abelian Stark conjectures. As first applications of this approach, we derive, amongst other things, a proof of (a refinement of) a conjecture of Darmon concerning cyclotomic units, a proof of (a refinement of) Gross's 'Conjecture for Tori' in the case that the base field is Q, a proof of new cases of the equivariant Tamagawa number conjecture in situations in which the relevant p-adic L-functions have trivial zeroes, explicit conjectural formulas for both annihilating elements and, in certain cases, the higher Fitting ideals (and hence explicit structures) of ideal class groups, a reinterpretation of the p-adic Gross-Stark Conjecture in terms of the properties of zeta elements and a strong refinement of many previous results (of several authors) concerning abelian Stark conjectures.

2006

The object of this paper is to give several properties and applications of the multiple p-adic q-L-function of two variables L (r) p,q (s, z, χ). The explicit formulas relating higher order q- Bernoulli polynomials, which involve sums of products of higher order q-zeta function and higher order Dirichlet q-L-function are given. The value of higher order Dirichlet p-adic q-L-function for positive integers is also calculated. Furthermore, the Kummer-type congruences for multiple generalized q-Bernoulli polynomials are derived by making use of the difference theorem of higher order Dirichlet p-adic q-L-function.

Our main aim in this series of articles is to present a clear new view, generalization and refinement of a range of well-known results and conjectures concerning the arithmetic properties of zeta elements. In this first article we study the L-functions that are attached to the multiplicative group over a finite abelian extension of global fields.

4th year dissertation.