Ce texte est consacré au système d’Euler de Kato, construit à partir des unités modulaires, et à son image par l’application exponentielle duale (loi de réciprocité explicite de Kato). La présentation que nous en donnons est sensiblement différente de la présentation originelle de Kato.

Special values of certain $L$ functions of the type $L(M,s)$ are studied where $M$ is a motive over a totally real field $F$ with coefficients in another field $T$, and$$L(M,s)=\prod _{𝔭}{L}_{𝔭}(M,𝒩{𝔭}^{-s})$$is an Euler product $𝔭$ running through maximal ideals of the maximal order ${𝒪}_F$ of $F$ and$$\begin{array}{cc}\hfill L_{𝔭}(M,X{)}^{-1}& =(1-{\alpha }^{\left(1\right)}(𝔭\left)X\right)·(1-{\alpha }^{\left(2\right)}(𝔭\left)X\right)·...·(1-\alpha (d\left)\right(𝔭\left)X\right)\hfill \\ & =1+A_1\left(𝔭\right)X+...+A_d\left(𝔭\right)X^d\hfill \end{array}$$being a polynomial with coefficients in $T$. Using the Newton and the Hodge polygons of $M$ one formulate a conjectural criterium for the existence of a $p$-adic analytic continuation of the special values. This conjecture is verified in a number of cases related to...

Let $E$ be a semistable elliptic curve over $\mathbb{Q}$. We prove weak forms of Kato’s $K_1$-congruences for the special values $L\left(1,E/\mathbb{Q}({\mu }_{p^n},\sqrt[p^n]{\Delta })\right).$ More precisely, we show that they are true modulo $p^{n+1}$, rather than modulo $p^{2n}$. Whilst not quite enough to establish that there is a non-abelian $L$-function living in $K_1\left({\mathbb{Z}}_p\left[\left[\mathrm{Gal}(\mathbb{Q}({\mu }_{{\mathrm{p}}^{\infty }},\sqrt[{\mathrm{p}}^{\infty }]{\Delta })/\mathbb{Q})\right]\right]\right)$, they do provide strong evidence towards the existence of such an analytic object. For example, if $n=1$ these verify the numerical congruences found by Tim and Vladimir Dokchitser.

Let $E$ be a modular elliptic curve over $\mathbb{Q}$$L^{\left(n\right)}...$

This paper studies a two-variable zeta function $Z_K(w,s)$ attached to an algebraic number field $K$, introduced by van der Geer and Schoof, which is based on an analogue of the Riemann-Roch theorem for number fields using Arakelov divisors. When $w=1$ this function becomes the completed Dedekind zeta function ${\widehat{\zeta }}_K\left(s\right)$ of the field $K$. The function is a meromorphic function of two complex variables with polar divisor $s(w-s)$, and it satisfies the functional equation $Z_K(w,s)=Z_K(w,w-s)$. We consider the special case $K=\mathbb{Q}$, where for $w=1$ this function...

We consider the problem of determining whether a given prime p is a congruent number. We present an easily computed criterion that allows us to conclude that certain primes for which congruency was previously undecided, are in fact not congruent. As a result, we get additional information on the possible sizes of Tate-Shafarevich groups of the associated elliptic curves. We also present a related criterion for primes p such that $16$ divides the class number of the imaginary quadratic field ℚ(√-p)....