Characteristic numbers, Jiang subgroup and non-positive curvature

Curvature, Cones, and Characteristic Numbers

We study Einstein metrics on smooth compact 4-manifolds with an edge-cone singularity of specified cone angle along an embedded 2-manifold. To do so, we first derive modified versions of the GaussBonnet and signature theorems for arbitrary Riemannian 4-manifolds with edge-cone singularities, and then show that these yield non-trivial obstructions in the Einstein case. We then use these integral...

Root Numbers and Ranks in Positive Characteristic

For a global fieldK and an elliptic curve Eη overK(T ), Silverman’s specialization theorem implies rank(Eη(K(T ))) ≤ rank(Et(K)) for all but finitely many t ∈ P(K). If this inequality is strict for all but finitely many t, the elliptic curve Eη is said to have elevated rank. All known examples of elevated rank for K = Q rest on the parity conjecture for elliptic curves over Q, and the examples ...

Curvature and Characteristic Numbers of Hyper-kähler Manifolds

Characteristic numbers of compact hyper-Kähler manifolds are expressed in graphtheoretical form, considering them as a special case of the curvature invariants introduced by L. Rozansky and E. Witten. The appropriate graphs are generated by “wheels,” and the recently proved Wheeling theorem is used to give a formula for the L 2-norm of the curvature of an irreducible hyper-Kähler manifold in te...

Congruence Subgroup Growth of Arithmetic Groups in Positive Characteristic

We prove a new uniform bound for subgroup growth of a Chevalley group G over the local ring F[[t]] and also over local pro-p rings of higher Krull dimension. This is applied to the determination of congruence subgroup growth of arithmetic groups over global fields of positive characteristic. In particular, we show that the subgroup growth of SLn(Fp[t]) (n ≥ 3) is of type nlog n . This was one o...

of non-positive Gaussian curvature