Counting Bounded Elements of a Number Field

We estimate, in a number field, the number of elements and the maximal number of linearly independent elements, with prescribed bounds on their valuations. As a by-product, we obtain new bounds for the successive minima of ideal lattices. Our arguments combine group theory, ramification theory, and the geometry of numbers.

The p-adic Kummer–Leopoldt constant: Normalized p-adic regulator

The [Formula: see text]-adic Kummer–Leopoldt constant [Formula: see text] of a number field [Formula: see text] is (assuming the Leopoldt conjecture) the least integer [Formula: see text] such that for all [Formula: see text], any global unit of [Formula: see text], which is locally a [Formula: see text]th power at the [Formula: see text]-places, is necessarily the [Formula: see text]th power of a global unit of [Formula: see text]. This constant has been computed by Assim and Nguyen Quang Do using Iwasawa’s techniques, after intricate studies and calculations by many authors. We give an elementary [Formula: see text]-adic proof and an improvement of these results, then a class field theory interpretation of [Formula: see text]. We give some applications (including generalizations of Kummer’s lemma on regular [Formula: see text]th cyclotomic fields) and a natural definition of the normalized [Formula: see text]-adic regulator for any [Formula: see text] and any [Formula: see text]. This is done without analytical computations, using only class field theory and especially the properties of the so-called [Formula: see text]-torsion group [Formula: see text] of Abelian [Formula: see text]-ramification theory over [Formula: see text].

Reduction of dynatomic curves

In this paper, we make partial progress on a function field version of the dynamical uniform boundedness conjecture for certain one-dimensional families ${\mathcal{F}}$ of polynomial maps, such as the family $f_{c}(x)=x^{m}+c$, where $m\geq 2$. We do this by making use of the dynatomic modular curves $Y_{1}(n)$ (respectively $Y_{0}(n)$) which parametrize maps $f$ in ${\mathcal{F}}$ together with a point (respectively orbit) of period $n$ for $f$. The key point in our strategy is to study the set of primes $p$ for which the reduction of $Y_{1}(n)$ modulo $p$ fails to be smooth or irreducible. Morton gave an algorithm to construct, for each $n$, a discriminant $D_{n}$ whose list of prime factors contains all the primes of bad reduction for $Y_{1}(n)$. In this paper, we refine and strengthen Morton’s results. Specifically, we exhibit two criteria on a prime $p$ dividing $D_{n}$: one guarantees that $p$ is in fact a prime of bad reduction for $Y_{1}(n)$, yet this same criterion implies that $Y_{0}(n)$ is geometrically irreducible. The other guarantees that the reduction of $Y_{1}(n)$ modulo $p$ is actually smooth. As an application of the second criterion, we extend results of Morton, Flynn, Poonen, Schaefer, and Stoll by giving new examples of good reduction of $Y_{1}(n)$ for several primes dividing $D_{n}$ when $n=7,8,11$, and $f_{c}(x)=x^{2}+c$. The proofs involve a blend of arithmetic and complex dynamics, reduction theory for curves, ramification theory, and the combinatorics of the Mandelbrot set.

Logarithmic Ramifications of Étale Sheaves by Restricting to Curves

In this article, we prove that the Swan conductor of an étale sheaf on a smooth variety defined by Abbes and Saito’s logarithmic ramification theory can be computed by its classical Swan conductors after restricting it to curves. It extends the main result of Barrientos [7] for rank $1$ sheaves. As an application, we give a logarithmic ramification version of generalizations of Deligne and Laumon’s lower semi-continuity property for Swan conductors of étale sheaves on relative curves to higher relative dimensions in a geometric situation.