Volume 2011, Issue 661

We complete the proof of the equivariant Tamagawa number conjecture for Tate motives over absolutely abelian fields by proving a refined cyclotomic main conjecture at the prime 2.

In this paper we construct explicitly through Eisenstein series, a square integrable residual automorphic representation of the special orthogonal group SO 2 n . We show that this representation comes from an elliptic Arthur parameter and it appears in the space L 2 (SO 2 n ()\SO 2 n ( )) with multiplicity one. Next, we consider parameters whose Hecke matrices, at the unramified places, have eigenvalues bigger in absolute value than those of the parameter constructed before. The main result is that these parameters cannot be cuspidal. We establish bounds for the eigenvalues of Hecke operators, as consequences of Arthur's conjectures for SO 2 n . Next, we calculate the character and the twisted characters for the representations that we considered and we prove an identity of traces. Finally, we find the composition of the global and local Arthur's packets associated to our parameter . All the results in this paper are true if we replace by any number field F .

On a locally Noetherian scheme X over a field of positive characteristic p , we study the category of coherent X -modules M equipped with a p e -linear map, i.e. an additive map C : X X satisfying rC ( m ) C ( r p e m ) for all m M , r X . The notion of nilpotence, meaning that some power of the map C is zero, is used to rigidify this category. The resulting quotient category, called Cartier crystals, satisfies some strong finiteness conditions. The main result in this paper states that, if the Frobenius morphism on X is a finite map, i.e. if X is F -finite, then all Cartier crystals have finite length. We further show how this and related results can be used to recover and generalize other finiteness results of HartshorneSpeiser Ann. Math. 105: 4579, 1977, Lyubeznik J. reine angew. Math. 491: 65130, 1997, Sharp Trans. Amer. Math. Soc. 359: 42374258, 2007, EnescuHochster Alg. Num. Th. 2: 721754, 2008, and Hochster Contemp. Math. 448: 119127, 2007 about the structure of modules with a left action of the Frobenius. For example, we show that over any regular F -finite scheme X Lyubeznik's F -finite modules J. reine angew. Math. 491: 65130, 1997 have finite length.

In this paper we study the Fuchsian RiemannHilbert (inverse monodromy) problem corresponding to Frobenius structures on Hurwitz spaces. We find a solution to this RiemannHilbert problem in terms of integrals of certain meromorphic differentials over a basis of an appropriate relative homology space over a Riemann surface, study the corresponding monodromy group and compute the monodromy matrices explicitly for various special cases.

Let ( t ) ( t ) have degree d 2. For a given rational number x 0 , define x n 1 ( x n ) for each n 0. If this sequence is not eventually periodic, and if does not lie in one of two explicitly determined affine conjugacy classes of rational functions, then x n 1 x n has a primitive prime factor in its numerator for all sufficiently large n . The same result holds for the exceptional maps provided that one looks for primitive prime factors in the denominator of x n 1 x n . Hence the result for each rational function of degree at least 2 implies (a new proof) that there are infinitely many primes. The question of primitive prime factors of x n x n is also discussed for uniformly bounded.

Using the theory of hyperkhler manifolds, we generalize the notion of Enriques surfaces to higher dimensions, explore their properties, and construct several examples using group actions on Hilbert schemes of points or moduli spaces of stable sheaves.

We show the rank (i.e. minimal size of a generating set) of lattices cannot grow faster than the volume.