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$P_{2}$ in short intervals

Henryk Iwaniec, M. Laborde (1981)

Annales de l'institut Fourier

For any sufficiently large real number $x$ , the interval $[x, x + x^{0, 45}]$ contains at least one integer having at most two prime factors .

$p$ -adic Abelian Stark conjectures at $s = 1$

David Solomon (2002)

Annales de l’institut Fourier

A $p$ -adic version of Stark’s Conjecture at $s = 1$ is attributed to J.-P. Serre and stated (faultily) in Tate’s book on the Conjecture. Building instead on our previous paper (and work of Rubin) on the complex abelian case, we give a new approach to such a conjecture for real ray-class extensions of totally real number fields. We study the coherence of our $p$ -adic conjecture and then formulate some integral refinements, both alone and in combination with its complex analogue. A ‘Weak Combined Refined’ version...

$p$ -adic analysis and Bell numbers of two variables. (Analyse $p$ -adique et nombres de Bell à deux variables.)

Mazouz, Abdelhak (1996)

Bulletin of the Belgian Mathematical Society - Simon Stevin

$p$ -adic analytic spaces.

Berkovich, Vladimir G. (1998)

Documenta Mathematica

$p$ -adic chaos and random number generation.

Woodcock, Christopher F., Smart, Nigel P. (1998)

Experimental Mathematics

$p$ -adic Clifford algebras

Bertin Diarra (1994)

Annales mathématiques Blaise Pascal

$p$ -adic Differential Operators on Automorphic Forms on Unitary Groups

Ellen E. Eischen (2012)

Annales de l’institut Fourier

The goal of this paper is to study certain $p$ -adic differential operators on automorphic forms on $U (n, n)$ . These operators are a generalization to the higher-dimensional, vector-valued situation of the $p$ -adic differential operators constructed for Hilbert modular forms by N. Katz. They are a generalization to the $p$ -adic case of the $C^{\infty}$ -differential operators first studied by H. Maass and later studied extensively by M. Harris and G. Shimura. The operators should be useful in the construction of certain $p$ -adic...

$p$ -adic dynamical systems and formal groups

Hua-Chieh Li (1996)

Compositio Mathematica

$p$ -adic estimates for exponential sums and the theorem of Chevalley-Warning

Alan Adolphson, Steven Sperber (1987)

Annales scientifiques de l'École Normale Supérieure

$p$ -adic Fourier theory.

Schneider, P., Teitelbaum, J. (2001)

Documenta Mathematica

$p$ -adic heights for semi-stable abelian varieties

John W. Jones (1990)

Compositio Mathematica

$p$ -adic interpolation of convolutions of Hilbert modular forms

Volker Dünger (1997)

Annales de l'institut Fourier

In this paper we construct $p$ -adic measures related to the values of convolutions of Hilbert modular forms of integral and half-integral weight at the negative critical points under the assumption that the underlying totally real number field $F$ has class number $h_{F} = 1$ . This extends the result of Panchishkin [Lecture Notes in Math., 1471, Springer Verlag, 1991 ] who treated the case that both modular forms are of integral weight. In order to define the measures, we need to introduce the twist operator...

$p$ -adic interpolation of logarithmic derivatives associated to certain Lubin-Tate formal groups

John L. Boxall (1986)

Annales de l'institut Fourier

The purpose of this paper is to generalize, to certain commutative formal groups of dimension one and height greater than one defined over the ring of integers of a finite extension of $Q_{p}$ , some results on $p$ -adic interpolation developed by Kubota, Leopoldt, Iwasawa, Mazur, Katz and others notably for the multiplicative group ${\hat{G}}_{m}$ , and which they used to construct $p$ -adic $L$ -functions.

$p$ -adic interpolation of special values of Hecke L-functions

Pierre Colmez, Leila Schneps (1992)

Compositio Mathematica

$p$ -adic $L$ -functions attached to characters of $p$ -power order

Kenneth A. Ribet (1977/1978)

Séminaire Delange-Pisot-Poitou. Théorie des nombres

$p$ -adic $L$ -functions for elliptic curves with complex multiplication I

Pierrette Cassou-Noguès (1980)

Compositio Mathematica

$p$ -adic $L$ -functions for modular forms

Shai Haran (1987)

Compositio Mathematica

$p$ -adic $L$ -functions for unitary Shimura varieties. I: Construction of the Eisenstein measure.

Harris, Michael, Li, Jian-Shu, Skinner, Christopher M. (2007)

Documenta Mathematica

$p$ -adic $L$ -functions of Hilbert modular forms

Andrzej Dabrowski (1994)

Annales de l'institut Fourier

We construct $p$ -adic $L$ -functions (in general case unbounded) attached to “motivic" primitive Hilbert cusp forms as a non-archimedean Mellin transform of the corresponding admissible measure. In order to prove the growth conditions of the appropriate complex-valued distributions we represent them as Rankin type representation and use Atkin–Lehner theory and explicit form of Fourier coefficients of Eisenstein series.

$p$ -adic measures attached to Siegel modular forms

Siegfried Böcherer, Claus-Günther Schmidt (2000)

Annales de l'institut Fourier

We study the critical values of the complex standard- $L$ -function attached to a holomorphic Siegel modular form and of the twists of the $L$ -function by Dirichlet characters. Our main object is for a fixed rational prime number $p$ to interpolate $p$ -adically the essentially algebraic critical $L$ -values as the Dirichlet character varies thus providing a systematic control of denominators of critical values by generalized Kummer congruences. In order to organize this information we prove the existence of...

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