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$p$ -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 heights for semi-stable abelian varieties

John W. Jones (1990)

Compositio Mathematica

$p$ -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 representations arising from descent on abelian varieties

Michael Harris (1979)

Compositio Mathematica

$p$ -adic representations of a local field. (Représentations $p$ -adiques d'un corps local.)

Colmez, Pierre (1998)

Documenta Mathematica

P. V. éléments dans un corps de nombres algébriques

Marthe Grandet-Hugot (1972)

Acta Arithmetica

$p$ -адические ряды Гекке мнимых квадратичных полей

М.М. Вишик, Ю.И. Манин (1974)

Matematiceskij sbornik

p-adic Distributions Associated to Heegner Points on Modular Curves.

Winfried Kohnen (1987)

Mathematische Zeitschrift

p-Adic Height Pairings I.

Peter Schneider (1982)

Inventiones mathematicae

P-Adic Homogeneous Spaces and Moduli Problems.

Ulrich Stuhler (1986)

Mathematische Zeitschrift

p-adic L-functions and p-adic periods of modular forms.

Glenn Stevens, Ralph Greenberg (1993)

Inventiones mathematicae

Parallélogrammes galoisiens infinis

Emmanuel Andréo, Richard Massy (2001)

Annales mathématiques Blaise Pascal

Parameterized families of quadratic number fields with 3-rank at least 2

Carl Erickson, Nathan Kaplan, Neil Mendoza, Allison M. Pacelli, Todd Shayler (2007)

Acta Arithmetica

Parametric form of an eight class field

Harvey Cohn, George Cooke (1976)

Acta Arithmetica

Paramétrisation de structures algébriques et densité de discriminants

Karim Belabas (2003/2004)

Séminaire Bourbaki

La composition de Gauss donne une structure de groupe aux orbites de formes quadratiques binaires entières de discriminant $D$ , sous l’action de ${SL}_{2}$ par changement de variable, essentiellement le groupe des classes de l’ordre quadratique de discriminant $D$ . Les domaines fondamentaux associés permettent calculs explicites et évaluation d’ordres moyens. Je présenterai les lois de composition supérieures découvertes par M. Bhargava à partir de la classification des espaces vectoriels préhomogènes réguliers,...

Partial fraction decompositions and Kummer congruences for the normalized coefficients of the Laurent expansion of elliptic functions parametrizing a nonsingular cubic curve in Hessian normal form.

Chip Snyder (1979)

Journal für die reine und angewandte Mathematik

Partitions and indefinite quadratic forms.

G.E. Andrews, F. Dyson, D. Hickerson (1988)

Inventiones mathematicae

Patterns and periodicity in a family of resultants

Kevin G. Hare, David McKinnon, Christopher D. Sinclair (2009)

Journal de Théorie des Nombres de Bordeaux

Given a monic degree $N$ polynomial $f (x) \in ℤ [x]$ and a non-negative integer $ℓ$ , we may form a new monic degree $N$ polynomial $f_{ℓ} (x) \in ℤ [x]$ by raising each root of $f$ to the $ℓ$ th power. We generalize a lemma of Dobrowolski to show that if $m < n$ and $p$ is prime then $p^{N (m + 1)}$ divides the resultant of $f_{p^{m}}$ and $f_{p^{n}}$ . We then consider the function $(j, k) \mapsto Res (f_{j}, f_{k}) mod p^{m}$ . We show that for fixed $p$ and $m$ that this function is periodic in both $j$ and $k$ , and exhibits high levels of symmetry. Some discussion of its structure as a union of lattices is also given.

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