Let $Y$ be an abelian variety over a subfield $k\subset \mathbb{C}$ that is of finite type over $\mathbb{Q}$. We prove that if the Mumford–Tate conjecture for $Y$ is true, then also some refined integral and adelic conjectures due to Serre are true for $Y$. In particular, if a certain Hodge-maximality condition is satisfied, we obtain an adelic open image theorem for the Galois representation on the (full) Tate module of $Y$. We also obtain an (unconditional) adelic open image theorem for K3 surfaces. These results are special cases of a more general statement for the image of a natural adelic representation of the fundamental group of a Shimura variety.

Let C be a family of curves over a non-singular variety S. We study algebraic cycles on the relative symmetric powers C[n] and on the relative Jacobian J. We consider the Chow homology CH*(C[∙]/S) := ⊕n CH*(C[n]/S) as a ring using the Pontryagin product. We prove that CH*(C[∙]/S) is isomorphic to CH*(J/S)[t]〈u〉, the PD-polynomial algebra (variable: u) over the usual polynomial ring (variable: t) over CH*(J/S). We give two such isomorphisms that over a general base are different. Further we give precise results on how CH*(J/S) sits embedded in CH*(C[∙]/S) and we give an explicit geometric description of how the operators and ∂u act. This builds upon the study of certain geometrically defined operators Pi,j (a) that was undertaken by one of us.

Our results give rise to a new grading on CH*(J/S). The associated descending filtration is stable under all operators [N]*, and [N]* acts on as multiplication by Nm. Hence, after − ⊗ ℚ this filtration coincides with the one coming from Beauville's decomposition. The grading we obtain is in general different from Beauville's.

Finally, we give a version of our main result for tautological classes, and we show how our methods give a simple geometric proof of some relations obtained by Herbaut and van der Geer–Kouvidakis, as later refined by one of us.

The present issue of Compositio Mathematica is dedicated to Jozef Steenbrink on the occasion of his sixtieth birthday.

The editors of Compositio Mathematica find it appropriate to communicate some information directly to the readers in the form of this editorial.

Let A=A$_g, 1, n$ 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 Oort‘s conjecture on subvarieties Z [rarrhk ] A with a dense collection of CM-points. We give a reformulation of this conjecture, and we prove it in a special case.