This paper concerns the arithmetic of certain $p$-adic families of elliptic modular forms. We relate, using a formula of Rubin, some Iwasawa-theoretic aspects of the three items in the title of this paper. In particular, we examine several conjectures, three of which assert the non-triviality of an Euler system, a $p$-adic regulator, and the derivative of a $p$-adic $L$-function. We investigate sufficient conditions for the first conjecture to hold and show that, under additional assumptions, the first...

We prove formulas for the k-higher Mahler measure of a family of rational functions with an arbitrary number of variables. Our formulas reveal relations with multiple polylogarithms evaluated at certain roots of unity.

We present an algorithm for computing discriminants and prime ideal decomposition in number fields. The algorithm is a refinement of a $p$-adic factorization method based on Newton polygons of higher order. The running-time and memory requirements of the algorithm appear to be very good.

Let $B$ be a quaternion algebra over a number field $k$. To a pair of Hilbert symbols $\{a,b\}$ and $\{c,d\}$ for $B$ we associate an invariant $\rho ={\rho }_R\left(\left[𝒟\right(a,b\left)\right],\left[𝒟\right(c,d\left)\right]\right)$ in a quotient of the narrow ideal class group of $k$. This invariant arises from the study of finite subgroups of maximal arithmetic kleinian groups. It measures the distance between orders $𝒟(a,b)$ and $𝒟(c,d)$ in $B$ associated to $\{a,b\}$ and $\{c,d\}.$ If $a=c$, we compute ${\rho }_R(\left[𝒟(a,b)\right],\left[𝒟(c,d)\right])$ by means of arithmetic in the field $k\left(\sqrt{a}\right).$ The problem of extending this algorithm to the general case leads to studying a finite graph associated...

Let $p$ be a prime number. We say that a number field $F$ satisfies the condition $\left(H_{p^n}^{\prime }\right)$ when any abelian extension $N/F$ of exponent dividing $p^n$ has a normal integral basis with respect to the ring of $p$-integers. We also say that $F$ satisfies $\left(H_{p^{\infty }}^{\prime }\right)$ when it satisfies $\left(H_{p^n}^{\prime }\right)$ for all $n\ge 1$. It is known that the rationals $\mathbb{Q}$ satisfy $\left(H_{p^{\infty }}^{\prime }\right)$ for all prime numbers $p$. In this paper, we give a simple condition for a number field $F$ to satisfy $\left(H_{p^n}^{\prime }\right)$ in terms of the ideal class group of $K=F\left({\zeta }_{p^n}\right)$ and a “Stickelberger ideal” associated to the Galois group...