Of Problems and Numbers

An epsilon constant conjecture for higher dimensional representations

The equivariant local epsilon constant conjecture was formulated in various forms by Fontaine and Perrin-Riou, Benois and Berger, Fukaya and Kato and others. If $N/K$ is a finite Galois extension of $p$-adic fields and $V$ a $p$-adic representation of $G_K$, then the above conjecture describes the epsilon constants attached to $V$ in terms of the Galois cohomology of $T$, where $T$ is a $G_K$-stable $\mathbb Z_p$-sublattice $T$ such that $V=T\otimes_{\mathbb Z_p}\mathbb Q_p$.
Here we will discuss the case when $N/K$ is at most weakly ramified (this includes the case of tame ramification) and $T=\mathbb Z_p^r(\chi^\mathrm{cyc})(\rho^\mathrm{nr})$, i.e. the $\mathbb Z_p$-module $\mathbb Z_p^r$ with the trivial action of $G_K$ twisted by the cyclotomic character and by an unramified representation $\rho^\mathrm{nr}:G_K\to\mathrm{Gl}_r(\mathbb Z_p)$. The main results generalize previous work by Izychev, Venjakob, Bley and the author. This is a joint work with Werner Bley.

On the local-to-global principle for value sets

Given a finite morphism $f: X \to Y$ between algebraic curves over a number field $k$, we study the set of rational points $y\in Y(k)$ such that for every place $p$ of $k$, there exists a $p$-adic point $x_p\in X(k_p)$ such that $f(x)=y$. In the particular case in which $X$ and $Y$ are elliptic curves, this problem was investigated by Dvornicich and Zannier.

Heights of units, higher order invariants

We shall discuss recent results obtained jointly with F. Amoroso which build on earlier work of R. Dvornicich and F. Amoroso on heights of units and related higher order invariants.

TBA

Fitting ideals of modules in cyclotomic Iwasawa theory

When exploring the Galois module structure of class groups, we are led to the study of modules over the Iwasawa algebra $\Lambda[G]$, where $G$ is the group of a given Galois extension $L/K$ of algebraic number fields. It is probably impossible to describe these modules $X$, which arise as projective limits of $p$-parts of class groups, with absolute precision, that is up to isomorphism over the Iwasawa algebra. A much more tractable substitute for the isomorphism class of $X$, which still contains a lot of information on $X$, is afforded by the Fitting ideal of $X$ over $\Lambda[G]$. (Note that in the Iwasawa theoretic setting, the module $X$ is usually infinite, so there is no class number, contrary to the situation over a number field.) The main arithmetical ingredient in the description of these Fitting ideals is a suitable (equivariant and $p$-adic) L-function, which lives in (a localization of) the Iwasawa algebra. However, in general one cannot expect the Fitting ideal in question to be generated by this function, because there is a deep obstruction coming from class field theory which usually prevents the Fitting ideal from being principal. The point is that $X$ rarely has projective dimension 1 over $\Lambda[G]$. Thus there is more to the complete determination of the Fitting ideal than just the equivariant L-function. We will try to explain this algebraic aspect as well; in this part of the story, the technique of complexes and resolutions is very helpful.
In this talk I will be reporting on recent joint work with M. Kurihara, H. Tokio, and T. Kataoka.

Asymptotics of the Norm of a Sum of Roots of Unity

Thanks to Mann's Theorem and its refinements we understand when a sum of roots of unity of fixed length vanishes. However, it is not well understood how small the archimedean absolute value of a non-zero fixed-length sum of roots of unity can be. The product of these absolute values over all Galois conjugates is the field norm. In this joint work with Vesselin Dimitrov we investigate the asymptotic behavior of this norm. For example, if the vector of roots of unity appear in an equidistributed sequence, the relative norm approaches a Mahler measure. Our result has connections to work of Dvornicich and Zannier on sums of roots of unity that vanish modulo a prime. Our method is inspired by work of Duke

Unlikely intersections for algebraic curves in products of Carlitz modules.

In the last two decades there has been much study of what happens when an algebraic curve in $n$-space is intersected with two multiplicative relations $$x_1^{a_1} \cdots x_n^{a_n}~=~x_1^{b_1} \cdots x_n^{b_n}~=~1 \tag*{($\times$)}$$ for $(a_1, \ldots ,a_n),(b_1,\ldots, b_n)$ linearly independent in ${\bf Z}^n$. Usually the intersection with the union of all $(\times)$ is at most finite, at least in zero characteristic. Recently there have been a number of advances in positive characteristic, even for additive relations $$\alpha_1x_1+\cdots+\alpha_nx_n~=~\beta_1x_1+\cdots+\beta_nx_n~=~0 \tag*{(+)}$$ provided some extra structure of Drinfeld type is supplied. After briefly reviewing the zero characteristic situation, I will describe work with Dale Brownawell for $(+)$ and Carlitz modules.

Recent results in local-global divisibility

We explain the link between the local-global divisibility problem and a classical question posed by Cassels on the divisibility of the elements of the Tate-Shafarevich group in the Weil-Châtelet group. We give an overview of the results achieved for both problems since their formulations. In particular we show the most recent results achieved in local-global divisibility.

TBA

My time with Roberto at the IMOs

I first met Roberto at the IMO (International Mathematical Olympiad) held in 1988 in Australia, and we have been good friends ever since then.
In the talk I will recall some personal memories, and also give a glimpse of some competition mathematics.

Universality theorems for L-functions

After a brief survey of earlier value-distribution results for the Riemann zeta function, we discuss the ideas behind Voronin's universality theorem (1975). Then we present a recent universality result in the eigenvalue aspect for a family of Maass L-functions (joint work with G. Cherubini).

The permutation group method: a survey

In an old joint paper (R. Dvornicich & C. Viola, Some remarks on Beukers' integrals, Colloquia Math. Soc. János Bolyai 51 (1987), 637-657) Roberto and I gave the first improvements upon Apéry's irrationality measures of $\zeta(2)$ and $\zeta(3)$. We obtained these results using analytic estimates of suitable linear combinations of Beukers' integrals. Since the appearance of our paper, several improvements concerning the diophantine approximations of $\zeta(2)$, $\zeta(3)$, the dilogarithms of suitable rational numbers etc. were obtained by various authors using the permutation group method introduced in 1996 by G. Rhin and myself. I will sketch the most recent results in this field.

Irreducibility over cyclotomic fields

We shall illustrate some results obtained jointly with Dvornicich, concerning "explicit" irreducibility theorems over cyclotomic fields.