Geometry of Numbers Lectures by Akshay Venkatesh, Notes by Tony Feng

Reductive Groups over Fields

These are lecture notes that Tony Feng live-TEXed from a course given by Brian Conrad at Stanford in“winter” 2015, which Feng and Conrad edited afterwards. Two substitute lectures were delivered (by Akshay Venkatesh and Zhiwei Yun) when Conrad was out of town. This is a sequel to a previous course on the general structure of linear algebraic groups; some loose ends from that course (e.g., Cheva...

Beilinson’s conjectures II

The Analytic Class Number Formula and L-functions

Artin Approximation and Proper Base Change

and an abelian sheaf F of torsion abelian groups on X, the natural base change map gRf∗F ∼ −→ Rf ′ ∗(g F) is an isomorphism. By limit arguments and considerations with geometric stalks as discussed last time, we can arrange that F is a Z/nZ-sheaf for some n > 0 and it suffices to treat the case where S = Spec (A) for a strictly henselian local ring and S′ = Spec (k′) for a separably closed fiel...

Isogeny invariance of the BSD conjecture

• Ω = ∫ E(R) |ω|, and ω is the global section of Ω 1 N(E)/Z corresponding to a choice of basis of the Z-line Cot0(N(E)) for the Néron model N(E) of E over Z, • cp is the number of connected components of N(E)Fp that are geometrically connected over Fp, or equivalently (by Lang’s theorem applied to N(E)0Fptorsors) have a rational point, so cp coincides with the number of Fp-points of the finite ...