Volume 2006, Issue 594

Zagier proved that the traces of singular moduli, i.e., the sums of the values of the classical j -invariant over quadratic irrationalities, are the Fourier coefficients of a modular form of weight 3/2 with poles at the cusps. Using the theta correspondence, we generalize this result to traces of CM values of (weakly holomorphic) modular functions on modular curves of arbitrary genus. We also study the theta lift for the weight 0 Eisenstein series for SL 2 (ℤ) and realize a certain generating series of arithmetic intersection numbers as the derivative of Zagier's Eisenstein series of weight 3/2. This recovers a result of Kudla, Rapoport and Yang.

Let K be a local field of mixed characteristic not absolutely ramified. Fontaine-Laaille theory (see [ J. M. Fontaine et G. Laffaille , Construction de représentations p -adiques, Ann. Sci. ENS 15 (1982), 547–608. ]) gives a description of the torsion crystalline ℤ p representations of the absolute Galois group of K ( p denotes the characteristic of the residual field). Improving the former works, Breuil introduced new modules and obtained an integer and torsion theory for the semi-stable representations (see [ C. Breuil , Construction de représentations p -adiques semi-stables, Ann. Sci. ENS 31 (1997), 281–327]). In this paper, we follow Breuil's works and adapt them to the case where the local field K can be absolutely ramified. However, we would have a limitation on the index of absolute ramification.

We prove a formula for the Chow group of zero cycles on a quasiprojective threefold X over a field of characteristic zero with Cohen-Macaulay isolated singularities, in terms of an inverse limit of relative Chow groups of a desingularization X ˜ relative to multiples of the exceptional divisor. As an application, we give a necessary and sufficient condition for the Chow group of 0-cycles on the affine cone over a smooth projective and arithmetically Cohen-Macaulay surface to vanish. This partially answers a conjecture of Srinivas in affirmative.

Let X be a smooth complex projective variety of dimension greater than or equal to 2, L an ample line bundle and k ≫ 0 an integer. We prove that a very generic global section of the twisted tangent sheaf gives rise to a foliation of X without any proper algebraic invariant subvarieties of nonzero dimension. As a corollary we obtain a dynamical characterization of ampleness for line bundles over smooth projective surfaces.

We prove compactness of solutions to some fourth order equations with exponential nonlinearities on four manifolds. The proof is based on a refined bubbling analysis, for which the main estimates are given in integral form. Our result is used in a subsequent paper to find critical points (via minimax arguments) of some geometric functional, which give rise to conformal metrics of constant Q -curvature. As a byproduct of our method, we also obtain compactness of such metrics.

Let M be a complex projective manifold of dimension n + 1 and ƒ a meromorphic function on M obtained by a generic pencil of hyperplane sections of M . The n -th cohomology vector bundle of ƒ 0 = ƒ | M −ℛ , where ℛ is the set of indeterminacy points of ƒ, is defined on the set of regular values of ƒ 0 and we have the usual Gauss-Manin connection on it. Following Brieskorn's methods in [ Brieskorn, E ., Die Monodromie der isolierten Singularitäten von Hyperflächen, Manuscr. Math. 2 (1970), 103–161.], we extend the n -th cohomology vector bundle of ƒ 0 and the associated Gauss-Manin connection to ℙ 1 by means of differential forms. The new connection turns out to be meromorphic on the critical values of ƒ 0 . We prove that the meromorphic global sections of the vector bundle with poles of arbitrary order at ∞ ∈ ℙ 1 is isomorphic to the Brieskorn module of ƒ in a natural way, and so the Brieskorn module in this case is a free ℂ[ t ]-module of rank β n , where ℂ[ t ] is the ring of polynomials in t and β n is the dimension of n -th cohomology group of a regular fiber of ƒ 0 .

We show that there exist genus one curves of every index over the rational numbers, answering affirmatively a question of Lang and Tate. The proof is “elementary” in the sense that it does not assume the finiteness of any Shafarevich-Tate group. On the other hand, using Kolyvagin's construction of a rational elliptic curve whose Mordell-Weil and Shafarevich-Tate groups are both trivial, we show that there are infinitely many genus one curves of every index over every number field.

We introduce a class of C *-algebras which can be viewed as a generalization of the classical Cuntz-Krieger algebras. Our approach is based on a flexible “generators and relations”-concept. The main result is a canonical uniqueness theorem stating that the C *-algebras of this class are uniquely determined by their generators and relations. We can show that rank one Cuntz-Krieger algebras with infinitely large transition matrices fall in this class, and this provides an alternative proof of a result of Exel and Laca. Further we analyze a subclass of rank two Cuntz-Krieger algebras inspired by shifts of finite type in dimension two, with an infinite set of generators and relations.