On the Torsion Congruence for Zeta Functions of Totally Real Fields

In this note, we study the special values for zeta functions of totally real fields using the Shintani's cone decomposition. We prove certain congruence between the special values for zeta functions under the prime degree field extension. This congruence implies the `torsion congruence' proved by Ritter-Weiss which is crucial in the proof of the noncommutative Iwasawa main conjecture for totally real fields.Comment: 10 page

Numerical evidence toward a 2-adic equivariant ''Main Conjecture''

International audienceWe test a conjectural non abelian refinement of the classical 2-adic Main Conjecture of Iwasawa theory. In the first part, we show how, in the special case that we study, the validity of this refinement is equivalent to a congruence condition on the coefficients of some power series. Then, in the second part, we explain how to compute the first coefficients of this power series and thus numerically check the conjecture in that setting

On λ-invariants attached to cyclic cubic number fields

We describe an algorithm for finding the coefficients of F(X) modulo powers of p, where p ≠2 is a prime number and F(X) is the power series associated to the zeta function of Kubota and Leopoldt. We next calculate the 5-adic and 7-adic λ-invariants attached to those cubic extensions K/Q with cyclic Galois group A₃ (up to field discriminant <10⁷), and also tabulate the class number of K(e2πi/p) for p=5 and p=7. If the λ-invariant is greater than zero, we then determine all the zeros for the corresponding branches of the p-adic L-function and deduce Λ-monogeneity for the class group tower over the cyclotomic Zp-extension of K

p-adic Verification of Class Number Computations

The aim of this thesis is to determine if it is possible, using p-adic techniques, to unconditionally evaluate the p-valuation of the class number h of an algebraic number field K. This is important in many areas of number theory, especially Iwasawa theory. The class group ClK of an algebraic number field K is the group of fractional ideals of K modulo principal ideals. Its cardinality (the class number h) is directly linked to the existence of unique factorisation in K, and hence the class group is of core importance to almost all multiplicative problems concerning number fields. The explicit computation of ClK (and h) is a fundamental task in computational number theory. Despite its importance, existing algorithms cannot obtain the class group unconditionally in a reasonable amount of time if the field has a large discriminant. Although faster, specialised algorithms (focused only on calculating the p-valuation) are limited in the cases with which they can deal. We present two algorithms to verify the p-valuation of h for any totally real abelian number field, with no restrictions on p. Both algorithms are based on the p-adic class number formula and work by computing p-adic L-functions Lp(s,χ) at the value of s = 1. These algorithms came about from two different ways of computing Lp(1,χ), using either a closed or a convergent series formula. We prove that our algorithms compare favourably against existing class group algorithms, with superior complexity for number fields of degree 5 or higher. We also demonstrate that our algorithms are faster in practice. Finally, we present some open questions arising from the algorithms

From the Pell's equation to Gross-Stark units

This thesis is a mathematical journey from the Pell’s equation to Gross–Stark units, centered around the theme of the relationship between leading terms of L-series, and algebraic units. Our story begins with Pell’s equation, and two methods to solve it. In particular, we will focus on the study the fundamental solution of a real quadratic number field k. Then we will move to a general abelian extension of number fields K/k by stating the Stark conjecture. To conclude we will discuss the p-adic analogue of the Stark conjecture, namely the Gross–Stark conjecture. We will state the conjecture for k a real quadratic number field and K its narrow Hilbert class field. We will define the Gross–Stark unit, and compute an explicit exampl

On Iwasawa λ\lambda-invariants for abelian number fields and random matrix heuristics

Following both Ernvall-Mets\"{a}nkyl\"{a} and Ellenberg-Jain-Venkatesh, we study the density of the number of zeroes (i.e. the cyclotomic $\lambda$-invariant) for the $p$-adic zeta-function twisted by a Dirichlet character $\chi$ of any order. We are interested in two cases: (i) the character $\chi$ is fixed and the prime $p$ varies, and (ii) $\text{ord}(\chi)$ and the prime $p$ are both fixed but $\chi$ is allowed to vary. We predict distributions for these $\lambda$-invariants using $p$-adic random matrix theory and provide numerical evidence for these predictions. We also study the proportion of $\chi$-regular primes, which depends on how $p$ splits inside $\mathbb{Q}(\chi)$. Finally in an extensive Appendix, we tabulate the values of the $\lambda$-invariant for every character $\chi$ of conductor $\leq 1000$ and for odd primes $p$ of small size.Comment: 159 pages, 2 figure

Topics in p-adic analysis

Since their introduction by Hensel more than 100 years ago, the p-adic numbers have been playing a fundamental role in mathematics, number theory in particular, and nowadays it is hardly an exaggeration to contend that they are no less important than real and complex counterparts, due to its wide range of connections with other branches like rigid analytic geometry, p-adic Hodge theory, Iwasawa theory, and so on. In this dissertation, we present a total of 3 topics that crucially employ ideas and techniques from p-adic analysis. More precisely, these topics are: - Sum expressions for totally real p-adic L-functions: In Chapter 2 and 3, we establish the infinite sum expressions of totally real p-adic L-functions, generalizing the earlier works of Delbourgo. This uses crucially the computational tool provided by the Amice-Mazur p-adic Fourier transform. As applications, we present a generalization of the Ferrero-Greenberg formula in the totally real setting, as well as a numerical criterion of the vanishing of the mu-invariant à la Iwasawa--Ferrero. - On the BDP Iwasawa main conjecture for modular forms: Under the Heegner hypothesis, Kobayashi--Ota showed that one inclusion of the Iwasawa main conjecture involving the Bertolini--Darmon--Prasanna p-adic L-function holds after tensoring by Q_p. In Chapter 4, under certain hypotheses, we present the joint work with Antonio Lei, in which we show that the same inclusion holds integrally. This uses several modern developments in p-adic analysis, including Perrin-Riou's exponential/logarithm map, Pollack-Sprung's factorization into Coleman maps and its formulation in terms of p-adic Hodge theory by Lei and collaborators. - Note on p-adic local functional equation: Given distinct primes l, p, in the first part of Chapter 5 we record a p-adic valued Fourier theory on a local field K over Q_l, which is developed under the perspective of group schemes. The second part studies the Haar measure of K and befitting Schwartz class functions, and their zeta integrals in a rigid analytic setting. Eventually, following Tate’s original argument, it proves a p-adic local functional equation over K that uncannily resembles the complex one found in Tate thesis