When $p$ is an odd prime, Delbourgo observed that any Kubota-Leopoldt $p$-adic $L$-function, when multiplied by an auxiliary Euler factor, can be written as an infinite sum. We shall establish such expressions without restriction on $p$, and without the Euler factor when the character is nontrivial, by computing the periods of appropriate measures. As an application, we will reprove the Ferrero-Greenberg formula for the derivative $L_p'(0,χ)$. We will also discuss the convergence of sum expressions in terms of elementary $p$-adic analysis, as well as their relation to Stickelberger elements; such discussions in turn give alternative proofs of the validity of sum expressions.
In 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.
Inspired 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.
The p-adic Kummer--Leopoldt constant kappa\_K of a number field K is (assuming the Leopoldt conjecture) the least integer c such that for all n \textgreater{}\textgreater{} 0, any global unit of K, which is locally a p^(n+c)th power at the p-places, is necessarily the p^nth power of a global unit of K. This constant has been computed by Assim \& Nguyen Quang Do using Iwasawa's techniques,after intricate studies and calculations by many authors. We give an elementary p-adic proof and an improvement of these results, then a class field theory interpretation of kappa\_K. We give some applications…
Let $A$ be an abelian variety defined over a number field $F$ with supersingular reduction at all primes of $F$ above $p$. We establish an equivalence between the weak Leopoldt conjecture and the expected value of the corank of the classical Selmer group of $A$ over a $p$-adic Lie extension (not neccesasily containing the cyclotomic $\Zp$-extension). As an application, we obtain the exactness of the defining sequence of the Selmer group. In the event that the $p$-adic Lie extension is one-dimensional, we show that the dual Selmer group has no nontrivial finite submodules. Finally, we show that…
Samit Dasgupta has proved a formula factoring a certain restriction of a 3-variable Rankin-Selberg $p$-adic $L$-function as a product of a 2-variable $p$-adic $L$-function related to the adjoint representation of a Hida family and a Kubota-Leopoldt $p$-adic $L$-function. We prove a result involving Selmer groups that along with Dasgupta's result is consistent with the main conjectures associated to the 4-dimensional representation (to which the 3-variable $p$-adic $L$-function is associated), the $3$-dimensional representation (to which the $2$-variable $p$-adic $L$-function is associated) and…
We compute the $p$-adic $L$-functions of evil Eisenstein series, showing that they factor as products of two Kubota--Leopoldt $p$-adic $L$-functions times a logarithmic term. This proves in particular a conjecture of Glenn Stevens.
In this paper we propose a construction of $p$-adic Euler $\ell$-function using Kubota-Leopoldt's approach and Washington's one. We also compute the derivative of $p$-adic Euler $\ell$-function at $s=0$ and the values of $p$-adic Euler $\ell$-function at positives integers.
We use $\ell$-adic class field theory to take a new view on cyclotomic norms and Leopoldt or Gross generalized conjectures. By the way we recall and complete some classical results. We illustrate the logarithmic approach by various numerical examples and counter-examples obtained with PARI.
By analyzing the coefficients of the power series defining the Kubota--Leopoldt $p$-adic $L$-function associated to the non-trivial character of a real quadratic field, we prove a congruence of Ankeny--Artin--Chowla-type for prime power modulus. Additionally, we show how some classical congruences relating Bernoulli numbers and Wilson quotients fit naturally into the theory of the $p$-adic Riemann zeta function.
We prove that a two-variable p-adic l_q-function has the series p-adic expansion which interpolates a linear combinations of terms of the generalized q-Euler polynomials at non positive integers. The proof of this original construction is due to Kubota and Leopoldt in 1964, although the method given this note is due to Washington
We are studing Galois actions on fundamental groups. Using towers of coverings we construct measures on the products of finite copies of Z_p. Using these measures we can calculate coefficients of Galois representations. In the simplest case of measures on Z_p, we get Kubota_Leopoldt p-adic L-functions and p-adic Hurwitz zeta functions.
We consider the p-adic distributions derived from Eisenstein series in [GMPS], whose Mellin transforms are reciprocals of the Kubota-Leopoldt p-adic L-function. These distributions were shown there to be measures when p is regular. They fail to be measures when p is irregular; in this paper we give quantitative estimates that describe their behavior more precisely.
We construct $p$-adic multiple $L$-functions in several variables, which are generalizations of the classical Kubota-Leopoldt $p$-adic $L$-functions, by using a specific $p$-adic measure. Our construction is from the $p$-adic analytic side of view, and we establish various fundamental properties of these functions.
The conjecture of Leopoldt states that the $p$ - adic regulator of a number field does not vanish. It was proved for the abelian case in 1967 by Brumer, using Baker theory. We prove this conjecture for CM number fields $\K$. The proof uses Iwasawa's methods -- especially Takagi Theory -- for deriving his skew symmetric pairing, together with Kummer- and Class Field Theory.
We generalize a formula of Leopoldt which relates the p-adic regulator modulo p of a real abelian extension of Q with the value of the relative Dedekind zeta function at s=2-p. We use this generalization to give a statement on the non-vanishing modulo p of this relative zeta function at the point s=1 under a mild condition.
In this paper we prove a formula, much in the spirit of one due to Rubin, which expresses the leading coefficients of various p-adic L-functions in the presence of an exceptional zero in terms of Nekovar's p-adic height pairings on his extended Selmer groups. In a particular case, the Rubin-style formula we prove recovers a p-adic Kronecker limit formula. In a disjoint case, we observe that our computations with Nekovar's heights agree with the Ferrero- Greenberg formula (more generally, Gross' conjectural formula) for the leading coefficient of the Kubota-Leopoldt p-adic L-function (resp., th…
The aim of this paper is to investigate the trivial zeros of the Katz $p$-adic $L$-functions by the CM congruence. We prove the existence of trivial zeros of the Katz $p$-adic $L$-functions for general CM fields and establish a first derivative formula of the cyclotomic $p$-adic $L$-functions at trivial zeros under some Leopoldt hypothesis. The crucial ingredients in our proof are a special case of $p$-adic Kronecker limit formula for CM fields and a leading term formula of anticyclotomic $p$-adic $L$-functions at trivial zeros via the explicit congruences between CM and non-CM Hida families o…
This paper deals with a multiple version of zeta- and L-functions both in the complex case and in the p-adic case: [I] Our motivation in the complex case is to find suitable rigorous meaning of the values of multivariable multiple zeta-functions (MZFs) at non-positive integer points. (a) We reveal that MZFs turn to be entire on the whole space after taking the desingularization. Further we show that the desingularized function is given by a suitable finite linear combination of MZFs with some arguments shifted. It is also shown that specific combinations of Bernoulli numbers attain the special…
In this paper, we define the p-adic Euler L-functions using the fermionic p-adic integral on Zp. By computing the values of the p-adic Euler L-functions at negative integers, we show that for Dirichlet characters with odd conductor, this definition is quivalent to the previous definition following Kubata-Leopoldt and Washington's approach. We also study the behavior of p-adic Euler L-functions at positive integers. An interesting thing is that most of the results in Section 11.3.3 of Cohen's book [H. Cohen, Number Theory Vol. II: Analytic and Modern Tools, Graduate Texts in Mathematics, 240. S…
The conjecture of Leopoldt states that the $p$ - adic regulator of a number field does not vanish. It was proved for the abelian case in 1967 by Brumer, using Baker theory. A conjecture, due to Gross and Kuz'min will be shown here to be in a deeper sense a dual of Leopoldt's conjecture with respect to the Iwasawa involution. We prove both conjectures for arbitrary number fields $\K$. The main ingredients of the proof are the Leopoldt reflection, the structure of quasi - cyclic $\Z_p[ \Gal(\K/\Q) ]$ - modules of some of the most important $Λ[ \Gal(\K/\Q) ]$ - modules occurring ($T$ acts on them…
The use of overconvergent cohomology in constructing $p$-adic $L$-functions, initiated by Stevens and Pollack--Stevens in the setting of classical modular forms, has now been established in a number of settings. The method is compatible with constructions of eigenvarieties by Ash--Stevens, Urban and Hansen, and is thus well-adapted to non-ordinary situations and variation in $p$-adic families. In this note, we give an exposition of the ideas behind the construction of $p$-adic $L$-functions via overconvergent cohomology. Conditional on the non-abelian Leopoldt conjecture, we illustrate them by…
We investigate a question of Burns and Sano concerning the structure of the module of Euler systems for a general $p$-adic representation. Assuming the weak Leopoldt conjecture, and the vanishing of $μ$-invariants of natural Iwasawa modules, we obtain an Iwasawa-theoretic classification criterion for Euler systems which can be used to study this module. This criterion, taken together with Coleman's conjecture on circular distributions, leads us to pose a refinement of the aforementioned question for which we provide strong, and unconditional, evidence. We furthermore answer this question in th…
The article is dedicated to the memory of George Voronoi. It is concerned with ($p$-adic) $L$-functions (in partially ($p$-adic) zeta functions) and cyclotomic ($p$-adic) (multiple) zeta values. The beginning of the article contains a short summary of the results on the Bernoulli numbers associated with the studies of George Voronoi. Results on multiple zeta values have presented by D. Zagier, by P. Deligne and A.Goncharov, by A. Goncharov, by F. Brown, by C. Glanois and others. S. Ünver have investigated p-adic multiple zeta values in the depth two. Tannakian interpretation of p-adic multiple…
On the factorisation of the $p$-adic Rankin-Selberg $L$-function in the supersingular case
Warwick Mathematics Institute, University of Warwick（华威大学华威数学研究所）；Equinor ASA（Equinor公司）；Faculty of Mathematics & Computer Science, UniDistance Suisse（瑞士开放大学数学与计算机科学系）
The Euler--Riemann zeta function is a largely studied numbertheoretic object, and the birthplace of several conjectures, such as the Riemann Hypothesis. Different approaches are used to study it, including $p$-adic analysis : deriving information from $p$-adic zeta functions. A generalized version of $p$-adic zeta functions (Riemann zeta function) are $p$-adic $L$-functions (resp. Dirichlet $L$-functions). This paper describes formalization of $p$-adic $L$-functions in an interactive theorem prover Lean 3. Kubota--Leopoldt $p$-adic $L$-functions are meromorphic functions emerging from the spec…
Working in the p-adic analog of the complex numbers, we'll define a line integral on a small arc of a circle. This allows new versions of the Residue Theorem, the Cauchy-Goursat Theorem on discs with and without holes, Cauchy's Integral Formula and the Z-P Theorem. In contrast to results in complex analysis, these integrals allow the points on a boundary circle, the bulk of a p-adic disc, to be treated the same as points interior to the boundary circle. The theory of the integral is developed, especially for functions holomorphic on an open disc, and integrals will be calculated for rational f…
