Kubota Leopoldt P-adic L-functions Research Articles

P-адические L-функции и p-адические кратные дзета значения

Статья посвящена памяти Георгия Вороного. Описываются новые избранные результаты о рядах Эйзенштейна, о (мотивных), (p-адических), (кратных) значениях (круговых) дзета и L-функций, и их приложения, полученные ниже перечисляемыми авторами, а также элементарное введение в эти результаты. Дан краткий обзор новых результатов о (мотивных), (p-адических), (кратных) значениях (круговых) дзета функциях, L-функциях и рядах Эйзенштейна. Статья ориентирована на избранные задачи и не является исчерпывающей. Начало статьи содержит краткое изложение результатов о числах Бернулли, связанных с исследованиями Георгия Вороного. Результаты о кратных значениях дзета функций были представлены Д. Загиром, П. Делинем и А. Гончаровым, А. Гончаровым, Ф. Брауном, К. Глэносом (Glanois) и другими. С. Унвер ("Unver) исследовал кратные p-адические дзета-значения глубины два. Таннакиева интерпретация кратных p-адических дзета-значений дана Х. Фурушо. Краткая история и связи между группами Галуа, фундаментальными группами, мотивами и арифметическими функциями представлены в докладе Ю. Ихара. Результаты о кратных дзета-значениях, группах Галуа и геометрии модулярных многообразий представлены Гончаровым. Интересная унипотентная мотивная фундаментальная группа определена и исследована Делинем и Гончаровым. В данной работе мы кратко упоминаем в рамках (p-адических) L-функций и (p-адических) (кратных) дзета-значений применения подходов Куботы-Леопольдта и Ивасавы, которые основанны на p-адических L-функциях Куботы-Леопольда, и арифметических p-адических L-функциях Ивасавы. Прореферирован ряд недавних работ (и соответствующих результатов): кратные дзета-значения в корнях из единицы, построение семейств мотивных итерированных интегралов с предписанными свойствами по Глэносу (Glanois); явные выражения для круговых p-адических кратных дзета-значений глубины два по Унверу (Unver); связи арифметических степеней циклов Кудлы-Рапопорта на интегральной модели многообразия Шимуры, соответствующей унитарной группе сигнатуры (1,1), с коэффициентами Фурье центральных производных рядов Эйзенштейна рода 2 по Санкарану (Sankaran). Более полно с содержанием статьи можно ознакомиться по приводимому ниже оглавлению: Введение. 1. Сравнения типа Вороного для чисел Бернулли. 2. Римановы дзета-значения. 3. О группах классов колец с теорией дивизоров. Мнимые квадратичные и круговые поля. 4. Ряды Эйзенштейна. 5. Группы классов, поля классов и дзета-функции. 6. Кратные дзета-значения. 7. Элементы неархимедовых локальных полей и неархимедова анализа. 8. Итерированные интегралы и (кратные) дзета-значения. 9. Формальные и p-делимые группы. 10. Мотивы и (p-адические) (кратные) дзета-значения. 11. О рядах Эйзенштейна, ассоциированных с многообразиями Шимуры. Разделы 1-9 и подраздел 11.1 (О некоторых многообразиях Шимуры и модулярных формах Зигеля) можно рассматривать как элементарное введение в результаты раздела 10 и подраздела 11.2 (О несобственном пересечении дивизоров Кудлы-Рапопорта и рядах Эйзенштейна).Я глубоко признателен Н. М. Добровольскому за помощь и поддержку в процессе подготовки статьи к печати.

Hida duality and the Iwasawa main conjecture

The central result of this paper is a refinement of Hida’s duality theorem between ordinary $${\varLambda }$$ -adic modular forms and the universal ordinary Hecke algebra. In particular, we give a sufficient condition for this duality to be integral with respect to particular submodules of the space of ordinary $${\varLambda }$$ -adic modular forms. This refinement allows us to give a simple proof that the universal ordinary cuspidal Hecke algebra modulo Eisenstein ideal is isomorphic to the Iwasawa algebra modulo an ideal related to the Kubota-Leopoldt p-adic L-function. The motivation behind these results stems from a proof of the Iwasawa main conjecture over $${\mathbb {Q}}$$ by Ohta. While simple and elegant, Ohta’s proof requires some restrictive hypotheses which we are able to remove using our results.

Stable lattices in modular Galois representations and Hida deformation

In this paper, we discuss the variation of the numbers of the isomorphic classes of stable lattices when the weight and the level vary in a Hida deformation by using the Kubota–Leopoldt p-adic L-function. Then in Corollary 1.7, we give a sufficient condition for the numbers of the isomorphic classes of stable lattices in Hida deformation to be infinite.

Unified presentation of p-adic L-functions associated with unification of the special numbers

By using partial differential equations (PDEs) of the generating functions for the unification of the Bernoulli, Euler and Genocchi polynomials and numbers, we derive many new identities and recurrence relations for these polynomials and numbers. In [33], Srivastava et al. defined a unified presentation of certain meromorphic functions related to the families of the partial zeta type functions. By using these functions, we construct p-adic functions which are related to the partial zeta type functions. By applying these p-adic function, we construct unified presentation of p-adic L-functions. These functions give us generalization of the Kubota–Leopoldt p-adic L-functions, which are related to the Bernoulli numbers and the other p-adic L-functions, which are related to the Euler numbers and polynomials. We also give some remarks and comments on these functions.

Image of Λ-adic Galois representations modulo p

Let p≥5 be a prime. If an irreducible component of the spectrum of the ‘big’ ordinary Hecke algebra does not have complex multiplication, under mild assumptions, we prove that the image of its mod p Galois representation contains an open subgroup of for the canonical “weight” variable T. This fact appears to be deep, as it is almost equivalent to the vanishing of the μ-invariant of the Kubota–Leopoldt p-adic L-function and the anticyclotomic Katz p-adic L-function. Another key ingredient of the proof is the anticyclotomic main conjecture proven by Rubin/Mazur–Tilouine.

Height pairings, exceptional zeros and Rubin’s formula: the multiplicative group

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., the Deligne-Ribet p-adic L-function) at s = 0.

Divisibility of anticyclotomic L-functions and theta functions with complex multiplication

The divisibility properties of Dirichlet L-functions in infinite families of characters have been studied by Iwasawa, Ferrero and Washington. The families considered by them are obtained by twisting an arbitrary Dirichlet character with all characters of p-power conductor for some prime p. One has to distinguish divisibility by p (the case considered by Iwasawa and FerreroWashington [FeW]) and by a prime � � p (considered by Washington [W1], [W2]). Ferrero and Washington proved the vanishing of the Iwasawa µ-invariant of any branch of the Kubota-Leopoldt p-adic L-function. This means that each of the power series, which p-adically interpolate the nontrivial L-values of twists of a fixed Dirichlet character by characters of p-power conductor, has some coefficient that is a p-adic unit. In the case � � p Washington [W2] obtained the following theorem on divisibility of L-values by � : given an integer n ≥ 1 and a Dirichlet character χ, for all but finitely many Dirichlet characters ψ of p-power conductor with χψ(−1) = (−1) n ,

Irrationality of Certain p-adic Periods for Small p

ery’s proof [13] of the irrationality of ζ(3) is now over 25 years old, and it is perhaps surprising that his methods have not yielded any significant new results (although further progress has been made on the irrationality of zeta values [1, 14]). Shortly after the initial proof, Beukers produced two elegant reinterpretations of Ap´ ery’s arguments; the first using iterated integrals and Legendre polynomials [2], and the second using modular forms [3]. It is this second argument that we will apply to study the irrationality of certain p-adic periods, in particular, the p-adic analogues of ζ(3) and Catalan’s constant for small p. To relate certain classical periods to modular forms, Beukers considers various integrals of holomorphic modular forms that themselves satisfy certain functional equations (analogous to the functional equation for the nonholomorphic Eisenstein series of weight two). That these integrals satisfy functional equations is a consequence of the theory of Eichler integrals. The periods arise as coefficients of the associated period polynomials. In our setting, these auxiliary functional equations are replaced by the notion of overconvergent p-adic modular forms [6, 7, 10]. In this guise, our p-adic periods will occur as coefficients of overconvergent Eisenstein series of negative integral weight. These p-adic periods are equal to special values of Kubota–Leopoldt p-adic L-functions. Thus ζ(3) is replaced by ζp(3) = Lp(3,id) and Catalan’s constant

Congruence modules related to Eisenstein series

The purpose of this paper is to study the structure of congruence modules (or modules of congruences) associated with Eisenstein series in various contexts in the Λ-adic theory of elliptic modular forms. Under some assumptions, we explicitly describe such modules in terms of Kubota–Leopoldt p-adic L-functions.

Cuspidal groups, ordinary Eisenstein series, and Kubota–Leopoldt p-adic L-functions

Cuspidal groups, ordinary Eisenstein series, and Kubota–Leopoldt p-adic L-functions

Computation of the Zeros of p-Adic L-Functions. II

The authors have carried out a computational study of the zeros of Kubota-Leopoldt p-adic L-functions. Results of this study have appeared recently in a previous article. The present paper is a sequel to that article, dealing with the computation of the zeros under certain conditions that complicate the original situation.

Computation of the zeros of 𝑝-adic 𝐿-functions. II

The authors have carried out a computational study of the zeros of Kubota-Leopoldt p-adic L-functions. Results of this study have appeared recently in a previous article. The present paper is a sequel to that article, dealing with the computation of the zeros under certain conditions that complicate the original situation.

Computation of the zeros of $p$-adic $L$-functions

The authors have computed the zeros of the Kubota-Leopoldt p-adic L-functions ${L_p}(s,\chi )$ for some small odd primes p and for a number of Dirichlet characters $\chi$. The zeros of the corresponding Iwasawa power series ${f_\theta }(T)$ are also computed. The characters $\chi$ (associated with quadratic extensions of the pth cyclotomic field) are chosen so as to cover as many different splitting types of ${f_\theta }(T)$ as possible. The $\lambda$-invariant of this power series, equal to its number of zeros, assumes values up to 8. The article is a report on these computations and their results, including the required theoretical background. Much effort is devoted to a study of the accuracy of the computed approximations.

Congruences dyadiques entre nombres de classes de corps quadratiques

We prove here a general mod 16 congruence between the class numbers of the two fields $$\mathbb{Q}(\sqrt d )$$ and $$\mathbb{Q}(\sqrt { - d} )$$ , for any square free positive integer d. Similarly we obtain congruences mod 64 relating the class numbers of the quadratic extensions corresponding to the integers d, 2d, -d, -2d. We use the p-adic analytic class number formula, which leads to study the difference of the values at 0 and 1 of the KUBOTA-LEOPOLDT's p-adic L-function (for p=2, and for the quadratic character associated to the real quadratic field). We use then IWASAWA's series, conveniently described for our purpose by Barsky. This gives a new proof for mod 16 congruences formally or simultaneously obtained by various authors. Moreover we get new results mod 16 and mod 64. Part A) presents the method and results, Part B) the proofs. All the details can be found in [4].

Euler factors forp‐adicL‐functions

MathematikaVolume 25, Issue 1 p. 68-75 Research Article Euler factors for p-adic L-functions Lawrence C. Washington, Lawrence C. Washington Department of Mathematics, University of Maryland, College Park, Maryland, 20742 U.S.A.Search for more papers by this author Lawrence C. Washington, Lawrence C. Washington Department of Mathematics, University of Maryland, College Park, Maryland, 20742 U.S.A.Search for more papers by this author First published: 01 June 1978 https://doi.org/10.1112/S002557930000927XCitations: 1AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Citing Literature Volume25, Issue1June 1978Pages 68-75 RelatedInformation

A note on p-adic L-functions

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