Special points on products of modular curves

Abstract

We prove the André-Oort conjecture on special points of Shimura varieties for arbitrary products of modular curves, assuming the Generalized Riemann Hypothesis. More explicitly, this means the following. Let n ≥ 0, and let Σ be a subset of C n consisting of points all of whose coordinates are j-invariants of elliptic curves with complex multiplications. Then we prove (under GRH) that the irreducible components of the Zariski closure of Σ are special sub-varieties, i.e., determined by isogeny conditions on coordinates and pairs of coordinates. A weaker variant (Thm. 1.3) is proved unconditionally.

Motivic decomposition of a compactification of a Merkurjev-Suslin variety

“…In the present paper we study certain twisted forms of a smooth hyperplane section of Gr (3,6). These twisted forms are smooth SL 1 (A)-equivariant compactifications of a Merkurjev-Suslin variety corresponding to a central simple algebra A of degree 3.…”

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AbstractWe provide a motivic decomposition of a twisted form of a smooth hyperplane section of Gr(3,6). This variety is a norm variety corresponding to a symbol in K M 3 /3.

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“…Finally, we shall also use the following fact (see [6] Theorem 8.2) called Rost Nilpotence theorem. Let X be a projective homogeneous variety over k. Then for any field extension l/k the kernel of the natural ring homomorphism End(M(X)) → End(M(X l )) consists of nilpotent elements.…”

Hyperbolicity of unitary involutions

“…We assume that there exists a field extension F ′ /F linearly disjoint with an algebraic closure of F such that the motive M F ′ decomposes in a sum of shifts of the motives of Spec F ′ and Spec K ′ , where K ′ is the field K ⊗ F F ′ . Note that the number of F ′ and the number of K ′ appearing in the decomposition do not depend on the choice of F ′ : if F ′′ is another field like that, the Krull-Schmidt principle [1] over the field of fractions of F ′ ⊗ F F ′′ gives the equalities. Here we use an easy version of the Krull-Schimdt principle for motives with finite coefficients of quasi-homogeneous varieties proved also in [10, Corollary 2.2].…”

Motivic decomposition of anisotropic varieties of type F4into generalized Rost motives

“…Finally, we shall also use the following fact (see [CGM,Cor. 8.3]) that follows from Rost Nilpotence Theorem.…”