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July 6, 2009
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For many applications of the trace formula to problems in the theory of automorphic forms one needs to stabilize the trace formula. Such a stabilization has been achieved by Arthur, building on work of Langlands and Kottwitz, subject to conjectures for orbital integrals known as fundamental lemmas. For the group Sp 4 the standard invariant fundamental lemma has been established by Hales. However, for the stabilization of the full trace formula one needs to prove a fundamental lemma for weighted orbital integrals on Sp 4 . We prove this weighted fundamental lemma and thereby make Arthur's stabilization of the Sp 4 trace formula unconditional.
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Let ℱ( R ) be a set of holomorphic functions on a Reinhardt domain R in a Banach sequence space (as e.g. all holomorphic functions or all m -homogeneous polynomials on the open unit ball of ). We give a systematic study of the sets dom ℱ( R ) of all z ∈ R for which the monomial expansion of every ƒ ∈ ℱ( R ) converges. Our results are based on and improve the former work of Bohr, Dineen, Lempert, Matos and Ryan. In particular, we show that up to any ε > 0 is the unique Banach sequence space X for which the monomial expansion of each holomorphic function ƒ converges at each point of a given Reinhardt domain in X . Our study shows clearly why Hilbert's point of view to develop a theory of infinite dimensional complex analysis based on the concept of monomial expansion, had to be abandoned early in the development of the theory.
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In this paper and its follow-up [Merkulov and Vallette, J. reine angew. Math.], we study the deformation theory of morphisms of properads and props thereby extending Quillen's deformation theory for commutative rings to a non-linear framework. The associated chain complex is endowed with an L ∞ -algebra structure. Its Maurer-Cartan elements correspond to deformed structures, which allows us to give a geometric interpretation of these results. To do so, we endow the category of prop(erad)s with a model category structure. We provide a complete study of models for prop(erad)s. A new effective method to make minimal models explicit, that extends the Koszul duality theory, is introduced and the associated notion is called homotopy Koszul . As a corollary, we obtain the (co)homology theories of (al)gebras over a prop(erad) and of homotopy (al)gebras as well. Their underlying chain complex is endowed with an L ∞ -algebra structure in general and a Lie algebra structure only in the Koszul case. In particular, we make the deformation complex of morphisms from the properad of associative bialgebras explicit. For any minimal model of this properad, the boundary map of this chain complex is shown to be the one defined by Gerstenhaber and Schack. As a corollary, this paper provides a complete proof of the existence of an L ∞ -algebra structure on the Gerstenhaber-Schack bicomplex associated to the deformations of associative bialgebras.
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In [Breuil et Schneider, J. reine angew. Math. 610: 149–180, 2007] is formulated a conjecture on the equivalence of the existence of invariant norms on certain locally algebraic representations of GL d +1 ( L ) and the existence of certain de Rham representations of Gal(/ L ), where L is a finite extension of ℚ p . In this paper, we prove the “easy” direction of the conjecture: the existence of invariant norms implies the existence of admissible filtrations.
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In this paper we find solutions u ε to a certain class of vector-valued parabolic Allen-Cahn equations that as ε → 0 develops as interface a given triod evolving under curve shortening flow.
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In the context of Free Probability Theory, we study two different constructions that provide new examples of factors of type II 1 with prescribed countable fundamental group. First we investigate state-preserving group actions on the almost periodic free Araki-Woods factors satisfying both a condition of mixing and a condition of free malleability in the sense of Popa. Typical examples are given by the free Bogoliubov shifts. Take an ICC w -rigid group G such that ℱ( L ( G )) = {1} (e.g., G = ℤ 2 ⋊ SL(2, ℤ)). For any countable subgroup S ⊂ , we construct an action of G on L (𝔽 ∞ ) such that the associated crossed product L (𝔽 ∞ ) ⋊ G is a type II 1 factor and its fundamental group is S . The second construction is based on a free product. Take ( B ( H ), ψ ) any factor of type I endowed with a faithful normal state and denote by S ⊂ the subgroup generated by the point spectrum of ψ . We show that the centralizer ( L ( G ) ∗ B ( H )) τ * ψ is a type II 1 factor and its fundamental group is S . Our proofs rely on Popa's deformation/rigidity strategy using his intertwining-by-bimodules technique.
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Let G denote a finite group and π : Z → Y a Galois covering of smooth projective curves with Galois group G . For every subgroup H of G there is a canonical action of the corresponding Hecke algebra ℚ[ H \ G / H ] on the Jacobian of the curve X = Z / H . To each rational irreducible representation 𝒲 of G we associate an idempotent in the Hecke algebra, which induces a correspondence of the curve X and thus an abelian subvariety P of the Jacobian JX . We give sufficient conditions on 𝒲, H , and the action of G on Z for P to be a Prym-Tyurin variety. We obtain many new families of Prym-Tyurin varieties of arbitrary exponent in this way.
