P-adic Measures Research Articles

Dirichlet series expansions of p-adic L-functions

We study p-adic L-functions L_p(s,chi ) for Dirichlet characters chi . We show that L_p(s,chi ) has a Dirichlet series expansion for each regularization parameter c that is prime to p and the conductor of chi . The expansion is proved by transforming a known formula for p-adic L-functions and by controlling the limiting behavior. A finite number of Euler factors can be factored off in a natural manner from the p-adic Dirichlet series. We also provide an alternative proof of the expansion using p-adic measures and give an explicit formula for the values of the regularized Bernoulli distribution. The result is particularly simple for c=2, where we obtain a Dirichlet series expansion that is similar to the complex case.

A local to global principle for expected values

This paper constructs a new local to global principle for expected values over free Z-modules of finite rank. In our strategy we use the same philosophy as Ekedahl's Sieve for densities, later extended and improved by Poonen and Stoll in their local to global principle for densities. We show that under some additional hypothesis on the system of p-adic subsets used in the principle, one can use p-adic measures also when one has to compute expected values (and not only densities). Moreover, we show that our additional hypotheses are sharp, in the sense that explicit counterexamples exist when any of them is missing. In particular, a system of p-adic subsets that works in the Poonen and Stoll principle is not guaranteed to work when one is interested in expected values instead of densities. Finally, we provide both new applications of the method, and immediate proofs for known results.

A p-adic variant of Kontsevich–Zagier integral operation rules and of Hrushovski–Kazhdan style motivic integration

Abstract We prove that if two semi-algebraic subsets of ℚ p n {\mathbb{Q}_{p}^{n}} have the same p-adic measure, then this equality can already be deduced using only some basic integral transformation rules. On the one hand, this can be considered as a positive answer to a p-adic analogue of a question asked by Kontsevich–Zagier in the reals (though the question in the reals is much harder). On the other hand, our result can also be considered as stating that over ℚ p {\mathbb{Q}_{p}} , universal motivic integration (in the sense of Hrushovski–Kazhdan) coincides with the usual p-adic integration.

P-adic Measures for Reciprocals of L-functions of Totally Real Number Fields

We generalize the work of Gelbart, Miller, Pantchichkine, and Shahidi on constructing p-adic measures to the case of totally real fields K. This measure is the Mellin transform of the reciprocal of the p-adic L-function which interpolates the special values at negative integers of the Hecke L-function of K. To define this measure as a distribution, we study the non-constant terms in the Fourier expansion of a particular Eisenstein series of the Hilbert modular group of K. Proving the distribution is a measure requires studying the structure of the Iwasawa algebra.

P-adic distribution of CM points and Hecke orbits, I : Convergence towards the Gauss point

We study the asymptotic distribution of CM points on the moduli space of elliptic curves over $\mathbb{C}_p$, as the discriminant of the underlying endomorphism ring varies. In contrast with the complex case, we show that there is no uniform distribution. In this paper we characterize all the sequences of discriminants for which the corresponding CM points converge towards the Gauss point of the Berkovich affine line. We also give an analogous characterization for Hecke orbits. In the companion paper we characterize all the remaining limit measures of CM points and Hecke orbits.

P-Adic measures associated with zeta values and p-adic log multiple gamma functions

We study a relation between two refinements of the rank one abelian Gross–Stark conjecture. For a suitable abelian extension [Formula: see text] of number fields, a Gross–Stark unit is defined as a [Formula: see text]-unit of [Formula: see text] satisfying certain properties. Let [Formula: see text]. Yoshida and the author constructed the symbol [Formula: see text] by using [Formula: see text]-adic [Formula: see text] multiple gamma functions, and conjectured that the [Formula: see text] of a Gross–Stark unit can be expressed by [Formula: see text]. Dasgupta constructed the symbol [Formula: see text] by using the [Formula: see text]-adic multiplicative integration, and conjectured that a Gross–Stark unit can be expressed by [Formula: see text]. In this paper, we give an explicit relation between [Formula: see text] and [Formula: see text] and prove that two refinements are consistent.

Q-Extension of Fubini Numbers

In this paper we define a q-extension of Fubini numbers which we call q-Fubini numbers, and generalized q-Fubini numbers of order r. Using the p-adic Laplace transform and p-adic integration, we obtain these numbers as moments of appropriate p-adicmeasures. Then we establish some identities and congruences for these numbers. We establish also a relationship between generalized q-Fubini numbers of order r and q-Fubini numbers. Further, as done in previous works we introduce a concept of generalized q-Fubini numbers, attached to a continuous plℤp-invariant function ψ defined on ℤp. These numbers are also the moments of appropriate p-adic measures, we obtain identities and congruences which generalize those associated to q-Fubini numbers.

Overconvergent cohomology of Hilbert modular varieties and p-adic L-functions

For each Hilbert modular form of non-critical slope we construct a p-adic distribution on the Galois group of the maximal abelian extension unramified outside p and ∞ of the totally real field. We prove that the distribution is admissible and interpolates the critical values of the complex L-function of the form. This construction is based on the study of the overconvergent cohomology of Hilbert modular varieties and certain cycles on these varieties.

Construction of a set of p-adic distributions

In this paper adapting to $p$-adic case some methods of real valued Gibbs measures on Cayley trees we construct several $p$-adic distributions on the set $\mathbb{Z}_p$ of $p$-adic integers. Moreover, we give conditions under which these $p$-adic distributions become $p$-adic measures (i.e. bounded distributions).

Formula omitted]-Adic distributions attached to twisted tensor L-functions for GL2 over CM fields

Let F be a totally real field. Let E be a quadratic extension that is totally imaginary over F. Let π be a cuspidal automorphic representation of cohomological type for GL2 over E. The properties of the twisted tensor L-function L(s,π,As) associated to π are studied in [6]. Special values of such functions are studied in [8] for F=Q and [9] for general F. Fix a prime p in F. In this article we study special values of the twisted tensor L-function, L(s,π,χ,As), that is twisted by Hecke characters χ over F having p-power conductor and χ∞=1. Specifically, we construct p-adic distributions that interpolate these special values at certain critical points. This is proved under a certain non-vanishing hypothesis at infinity.Our results here generalize earlier work by the author and Eknath Ghate for E an imaginary quadratic extension (see [2]).

A p-adic interpretation of some integral identities for Hall–Littlewood polynomials

If one restricts an irreducible representation Vλ of GL2n to the orthogonal group (respectively the symplectic group), the trivial representation appears with multiplicity one if and only if all parts of λ are even (resp. the conjugate partition λ′ is even). One can rephrase this statement as an integral identity involving Schur functions, the corresponding characters. Rains and Vazirani considered q,t-generalizations of such integral identities, and proved them using affine Hecke algebra techniques. In a recent paper, we investigated the q=0 limit (Hall–Littlewood), and provided direct combinatorial arguments for these identities; this approach led to various generalizations and a finite-dimensional analog of a recent summation identity of Warnaar. In this paper, we reformulate some of these results using p-adic representation theory; this parallels the representation-theoretic interpretation in the Schur case. The nonzero values of the identities are interpreted as certain p-adic measure counts. This approach provides a p-adic interpretation of these identities (and a new identity), as well as independent proofs. As an application, we obtain a new Littlewood summation identity that generalizes a classical result due to Littlewood and Macdonald. Finally, our p-adic method also leads to a generalized integral identity in terms of Littlewood–Richardson coefficients and Hall polynomials.

P-adic and Motivic Measure on Artin n-stacks

AbstractWe define a notion of p-adic measure on Artin n-stacks that are of strongly finite type over the ring of p-adic integers. p-adic measure on schemes can be evaluated by counting points on the reduction of the scheme modulo pn. We show that an analogous construction works in the case of Artin stacks as well if we count the points using the counting measure defined by Toën. As a consequence, we obtain the result that the Poincaré and Serre series of such stacks are rational functions, thus extending Denef's result for varieties. Finally, using motivic integration we show that as p varies, the rationality of the Serre series of an Artin stack defined over the integers is uniform with respect to p.

A note on p-adic q-ζ-functions II

We show that p-adic q-ζ-function constructed by Koblitz [7] (see also Dąbrowski [4]) can be obtained as Γ-transform of some p-adic measure coming from Lubin–Tate formal group.

On periodic p-adic distributions

In this paper we introduce a notion of periodic p-adic distribution defined on ℤp — the set of p-adic integers. This periodicity depends on a partition of ℤp. For several concrete partitions we describe corresponding periodic p-adic distributions. Moreover, we construct a periodic p-adic measure.

ANOTHER LOOK AT IWASAWA λ-INVARIANTS OF p-ADIC MEASURES ON $\mathbb {Z}_{p}^n$ AND Γ-TRANSFORMS

In [Iwasawa λ-invariants of p-adic measures on [Formula: see text] and their Γ-transforms, J. Number Theory132(10) (2012) 2258–2266; Iwasawa λ-invariants and Γ-transforms of p-adic measure on [Formula: see text], Int. J. Number Theory6(8) (2010) 1819–1829], the author and Saikia defined Iwasawa λ-invariants for multi-variable power series and proved a relation between the Iwasawa λ-invariant of a p-adic measure on [Formula: see text] and its Γ-transform. In this paper, we give two new definitions of λ-invariants for multi-variable power series which generalize the λ-invariant of single variable power series and prove that the new λ-invariants also satisfy the same relation. We also give algebraic interpretations of the new λ-invariants and discuss an application of our main results.

A Dynamical System Approach to Phase Transitions for p-Adic Potts Model on the Cayley Tree of Order Two

In the present paper, we introduce a new kind of p-adic measures for (q + 1)-state Potts model, called generalized p-adic quasi Gibbs measure. For such a model, we derive a recursive relations with respect to boundary conditions. We employ a dynamical system approach to establish phase transition phenomena for the given model. Namely, using the derived recursive relations we define a one-dimensional fractional p-adic dynamical system. We show that if q is divisible by p, then such a dynamical system has two repelling and one attractive fixed points. In this case, there exists a strong phase transition. If q is not divisible by p, then the fixed points are neutral, and this yields the existence of a quasi phase transition.

On Dynamical Systems and Phase Transitions for q + 1-state p-adic Potts Model on the Cayley Tree

In the present paper, we study a new kind of p-adic measures for q + 1-state Potts model, called p-adic quasi Gibbs measure. For such a model, we derive a recursive relations with respect to boundary conditions. Note that we consider two mode of interactions: ferromagnetic and antiferromagnetic. In both cases, we investigate a phase transition phenomena from the associated dynamical system point of view. Namely, using the derived recursive relations we define a fractional p-adic dynamical system. In ferromagnetic case, we establish that if q is divisible by p, then such a dynamical system has two repelling and one attractive fixed points. We find basin of attraction of the fixed point. This allows us to describe all solutions of the nonlinear recursive equations. Moreover, in that case there exists the strong phase transition. If q is not divisible by p, then the fixed points are neutral, and this yields that the existence of the quasi phase transition. In antiferromagnetic case, there are two attractive fixed points, and we find basins of attraction of both fixed points, and describe solutions of the nonlinear recursive equation. In this case, we prove the existence of a quasi phase transition.

Lebesgue-Radon-Nikodým theorem with respect to fermionic p-adic invariant measure on ℤ p

In the paper, we derive an analog of the Lebesgue-Radon-Nikodým theorem with respect to fermionic p-adic invariant measure on ℤ p .

Operational calculus on the ring of p-adic integers

We develop a one-dimensional Mikusinski type Operational Calculus on the ring of p-adic integers ℤ p . The Calculus utilizes the space C(ℤ p ) of continuous functions with values in the field of p-adic numbers ℚ p . C(ℤ p ) is equipped with a convolution, which turns it into an integral domain. A hyperfunction is by definition an element of a fraction field of C(ℤ p ). We refer to this fraction field as the p-adic Mikusinski field, prove its separability and incompleteness. Hyperfunctions corresponding to operations of shift, taking difference, indefinite summation, differentiation and integration are considered. A generalization of the p-adic exponential function a x is constructed for all a∈ℚ p . Also a variant of the Mikusinski field is considered that allows to treat continuous functions and p-adic measures simultaneously.

POWER SERIES EXPANSIONS OF MODULAR FORMS AND THEIR INTERPOLATION PROPERTIES

We define a power series expansion of an holomorphic modular form f in the p-adic neighborhood of a CM point x of type K for a split good prime p. The modularity group can be either a classical conguence group or a group of norm 1 elements in an order of an indefinite quaternion algebra. The expansion coefficients are shown to be closely related to the classical Maass operators and give p-adic information on the ring of definition of f. By letting the CM point x vary in its Galois orbit, the rth coefficients define a p-adic K×-modular form in the sense of Hida. By coupling this form with the p-adic avatars of algebraic Hecke characters belonging to a suitable family and using a Rankin–Selberg type formula due to Harris and Kudla along with some explicit computations of Watson and of Prasanna, we obtain in the even weight case a p-adic measure whose moments are essentially the square roots of a family of twisted special values of the automorphic L-function associated with the base change of f to K.