We provide proofs of Theorems 4.1 and 5.1 of Khare and Wintenberger (Invent. Math., doi:10.1007/s00222-009-0205-7, 2009).

/pdf/serre-s-modularity-conjecture-ii-3zrnk0wv09.pdf

Serre’s modularity conjecture (II)

We attach p-adic L-functions to critical modular forms and study them. We prove that those L-functions fit in a two-variables p-adic L-function defined locally everywhere on the eigencurve.

Critical p-adic L-functions

We discuss two arithmetical problems, at first glance unrelated: 

1)

The properties of the multiple ζ-values 

$$\zeta ({n_{1, \ldots,}}{n_m}): = \sum\limits_{0 1$$

(1)

and their generalizations, multiple polylogarithms at N-th roots of unity.




2)

The action of the absolute Galois group on the pro-l completion 

$$\pi _1^{(l)}({X_N}): = \pi _1^{(l)}({\mathbb{P}^1}\backslash \{ 0{,_{\mu N}},\infty \},\upsilon )$$

of the fundamental group of X N := ℙ1\{0, ∞ and all N-th roots of unity}.

/pdf/multiple-z-values-galois-groups-and-geometry-of-modular-44o5t0zjdk.pdf

Multiple ζ-Values, Galois Groups, and Geometry of Modular Varieties

Vorliegender Ubersichtsartikel versteht sich als eine Fortsetzung der Zusammenfassung im Jahresbericht der Deutschen Mathematiker-Vereinigung ([1988]). Dort war der erstmals von [1977] beschrittene mehrdimensionale Zugang auf Grund der ungelosten arithmetischen Probleme weitgehend ausgeklammert worden. In der Zwischenzeit konnten bei der Losung dieser Probleme deutliche Fortschritte erzielt werden (sowohl bei der Ausgrenzung der auseren Automorphismen der Galoisgruppe als auch bei der Suche nach rationalen Punkten), so das sich nunmehr dieser Weg fur einen Bericht uber den derzeitigen Kenntnisstand in der konstruktiven Galoisschen Theorie anbietet. Daruber hinaus konnten inzwischen mit den neuen arithmetischen Rationalitatskriterien auch Gruppen als Galoisgruppen uber ℚ nachgewiesen werden, fur die der Nachweis mit den im DMV-Bericht dargestellten algebraischen Rationalitatskriterien nicht gelang.

Der Kenntnisstand in der konstruktiven Galoisschen Theorie

On galois automorphisms of the fundamental group of the projective line minus three points

Galois Groups over ℚ

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Galois Groups over ℚ