Algebraische Zahlentheorie

The workshop brought together researchers from Europe, Japan and the US, who reported on various recent developments in algebraic number theory and related fields. Dominant topics were Shimura varieties, automorphic forms and Iwasawa theory

On computing Belyi maps

We survey methods to compute three-point branched covers of the projective line, also known as Belyi maps. These methods include a direct approach, involving the solution of a system of polynomial equations, as well as complex analytic methods, modular forms methods, and p-adic methods. Along the way, we pose several questions and provide numerous examples.Comment: 57 pages, 3 figures, extensive bibliography; English and French abstract; revised according to referee's suggestion

Moduli of Langlands Parameters

Let $F$ be a nonarchimedean local field of residue characteristic $p$, let $\hat{G}$ be a split connected reductive group over $\mathbb{Z}[1/p]$ with an action of $W_F$, and let $^LG$ denote the semidirect product $\hat{G}\rtimes W_F$. We construct a moduli space of Langlands parameters $W_F \to {^LG}$, and show that it is locally of finite type and flat over $\mathbb{Z}[1/p]$, and that it is a reduced local complete intersection. We give parameterizations of the connected components of this space over algebraically closed fields of characteristic zero and characteristic $\ell\neq p$, as well as of the components over $\overline{\mathbb{Z}}_{\ell}$ and (conjecturally) over $\overline{\mathbb{Z}}[1/p]$. Finally we study the functions on this space that are invariant under conjugation by $\hat{G}$ (or, equivalently, the GIT quotient by $\hat{G}$) and give a complete description of this ring of functions after inverting an explicit finite set of primes depending only on $^LG$.Comment: 79 page

Ahlfors circle maps and total reality: from Riemann to Rohlin

This is a prejudiced survey on the Ahlfors (extremal) function and the weaker {\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e. those (branched) maps effecting the conformal representation upon the disc of a {\it compact bordered Riemann surface}. The theory in question has some well-known intersection with real algebraic geometry, especially Klein's ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a gallery of pictures quite pleasant to visit of which we have attempted to trace the simplest representatives. This drifted us toward some electrodynamic motions along real circuits of dividing curves perhaps reminiscent of Kepler's planetary motions along ellipses. The ultimate origin of circle maps is of course to be traced back to Riemann's Thesis 1851 as well as his 1857 Nachlass. Apart from an abrupt claim by Teichm\"uller 1941 that everything is to be found in Klein (what we failed to assess on printed evidence), the pivotal contribution belongs to Ahlfors 1950 supplying an existence-proof of circle maps, as well as an analysis of an allied function-theoretic extremal problem. Works by Yamada 1978--2001, Gouma 1998 and Coppens 2011 suggest sharper degree controls than available in Ahlfors' era. Accordingly, our partisan belief is that much remains to be clarified regarding the foundation and optimal control of Ahlfors circle maps. The game of sharp estimation may look narrow-minded "Absch\"atzungsmathematik" alike, yet the philosophical outcome is as usual to contemplate how conformal and algebraic geometry are fighting together for the soul of Riemann surfaces. A second part explores the connection with Hilbert's 16th as envisioned by Rohlin 1978.Comment: 675 pages, 199 figures; extended version of the former text (v.1) by including now Rohlin's theory (v.2

Harmonic Analysis and the Trace Formula

The purpose of this workshop was to discuss recent results in harmonic analysis that arise in the study of the trace formula. This theme is common to different directions of research on automorphic forms such as representation theory, periods, and families