Ben Moonen

Linearity properties of Shimura varieties, II

Let A=Ag, 1, n denote the moduli scheme over Z[1/N] of p.p. g-dimensional abelian varieties with a level n structure; its generic fibre can be described as a Shimura variety. We study its ‘Shimura subvarieties’. If x ∈ A is an ordinary moduli point in characteristic p, then we formulate a local ‘linearity property’ in terms of the Serre–Tate group structure on the formal deformation space (= formal completion of A at x). We prove that an irreducible algebraic subvariety of A is a ‘Shimura subvariety’ if, locally at an ordinary point x, it is ‘formally linear’. We show that there is a close connection to a differential-geometrical linearity property in characteristic 0.We apply our results to the study of Oorts conjecture on subvarieties Z ↪ A with a dense collection of CM-points. We give a reformulation of this conjecture, and we prove it in a special case.

Group schemes with additional structures and Weyl group cosets

Let Y be an abelian variety of dimensiongover an algebraically closed fieldkof characteristicp >0. To Y we can associate its p-kernel Y[p], which is a finite commutative k-group scheme of rank p2 gg. In the unpublished manuscript [8], Kraft showed that, fixinggthere are only finitely many such group schemes, up to isomorphism. (As we shall discuss later, Kraft also gave a very nice description of all possible types.) About 20 years later, this result was re-obtained, independently, by Oort. Together with Ekedahl he used it to define and study a stratification of the moduli space akof principally polarized abelian varieties overk.The strata correspond to the pairs (Y, such that the p-kernel is of a fixed isomorphism type. Their results can be found in [11] and [12]; see also related work by van der Geer in [, [17].

Open problems in algebraic geometry

The open problems presented here were collected on the occasion of a workshop on Arithmetic Geometry at the University ofUtrecht, 26{30 June, 2000. This workshop was organized by the editors of the present article, and was made possible by support of: | NWO, the Netherlands Organization for Scientic Research, | KNAW, the Royal Netherlands Academy of Arts and Sciences, | MRI, the Mathematical Research Institute, | the Faculty of Mathematics and Computer Science of the University of Utrecht, and | the Department of Mathematics of the University of Utrecht. We thank these organizations heartily for their support. All problems in this list have been reviewed by at least one referee. In this process many useful suggestions and new references have come up. We thank all referees for their valuable comments.

Modular forms on Schiermonnikoog

Preface Contributors 1. Modular forms Bas Edixhoven, Gerard van der Geer and Ben Moonen 2. On the basis problem for Siegel modular forms with level Siegfried Bocherer, Hidenori Katsurada and Rainer Shulze-Pillot 3. Mock theta functions, weak Maass forms, and applications Kathrin Bringmann 4. Sign changes of coefficients of half integral weight modular forms Jan Hendrik Bruinier and Winfried Kohnen 5. Gauss map on the theta divisor and Greens functions Robin de Jong 6. A control theorem for the images of Galois actions on certain infinite families of modular forms Luis Dieulefait 7. Galois realizations of families of Projective Linear Groups via cusp forms Luis Dieulefait 8. A strong symmetry property of Eisenstein series Bernhard Heim 9. A conjecture on a Shimura type correspondence for Siegel modular forms, and Harders conjecture on congruences Tomoyoshi Ibukiyama 10. Peterssons trace formula and the Hecke eigenvalues of Hilbert modular forms Andrew Knightly and Charles Li 11. Modular shadows and the Levy-Mellin -adic transform Yuri I. Manin and Matilde Marcolli 12. Jacobi forms of critical weight and Weil representations Nils-Peter Skoruppa 13. Tannakian categories attached to abelian varieties Rainer Weissauer 14. Torellis theorem from the topological point of view Rainer Weissauer 15. Existence of Whittaker models related to four dimensional symplectic Galois representations Rainer Weissauer 16. Multiplying modular forms Martin H. Weissman 17. On projective linear groups over finite fields Gabor Wiese.

On the Tate and Mumford–Tate conjectures in codimension 1 for varieties with h2,0 = 1

We prove the Tate conjecture for divisor classes and the Mumford-Tate conjecture for the cohomology in degree 2 for varieties with

Relations between tautological cycles on Jacobians

h^{2,0}=1

Some remarks on modified diagonals

over a finitely generated field of characteristic 0, under a mild assumption on their moduli. As an application of this general result, we prove the Tate and Mumford-Tate conjectures for some classes of algebraic surfaces with

A conjecture on a Shimura type correspondence for Siegel modular forms, and Harder's conjecture on congruences

p_g=1

Algebraic cycles on the relative symmetric powers and on the relative Jacobian of a family of curves. II

.

A letter from the Editors

We study tautological cycle classes on the Jacobian of a curve. We prove a new result about the ring of tautological classes on a general curve that allows, among other things, easy dimension calculations and leads to some general results about the structure of this ring. Further we lift a result of Herbaut and van der Geer-Kouvidakis to the Chow ring (as opposed to its quotient modulo algebraic equivalence) and we give a method to obtain further explicit cycle relations. As an ingredient for this we prove a theorem about how Polishchuks operator D lifts to the tautological subalgebra of CH(J).