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Consider central L-values of even weight elliptic or Hilbert modular forms f twisted by ideal class characters χ of an imaginary quadratic extension K. Fixing χ , and assuming K is inert at each prime dividing the level, one knows simple exact formulas for averages over newforms f of squarefree levels satisfying a parity condition on the number of prime factors. These averages stabilize when the level is large with respect to K (the "stable range"). In weight 2, we obtain exact formulas for a simultaneous average over both f and χ . We allow for non-squarefree levels with any number of prime factors, and ramification or splitting of K above the level. Under elementary conditions on the level, these double averages are "stable" in all ranges. Two consequences are generalizations of the aforementioned stable (single) averages and effective results on nonvanishing of central L-values. [ABSTRACT FROM AUTHOR]

In the first part of this paper, we give a global description of simply connected maximal Lorentzian surfaces whose group of isometries is of dimension 1 (i.e. with a complete Killing field), in terms of a 1-dimensional generally non-Hausdorff manifold (the space of Killing orbits) and a smooth function defined there. In the second part, we study the completeness of such surfaces and prove in particular that under the hypothesis of bounded curvature, completeness is equivalent to null completeness. We also give completeness criterions involving the topological structure of the space of Killing orbits, or both the topology of this space and the geometry of the surface. [ABSTRACT FROM AUTHOR]

We use gauge theoretic and algebraic methods to examine sufficient conditions for smooth points on the moduli space of flat connections on a compact manifold and on the character variety of a finitely generated and presented group. We give a complete proof of the slice theorem for the action of the group of gauge transformations on the space of flat connections. Consequently, the slice is smooth if the second cohomology of the manifold with coefficients in the semisimple part of the adjoint bundle vanishes. On the other hand, we find that the smoothness of the slice for the character variety of a finitely generated and presented group depends not only on the second group cohomology but also on the relation module of the presentation. However, when there is a single relator or if there is no relation among the relators in the presentation, our condition reduces to the minimality of the second group cohomology. This is also verified using Fox calculus. Finally, we compare the conditions of smoothness in the two approaches. [ABSTRACT FROM AUTHOR]