1. Hensel’s lemma 2. Metric definition of p-adic integers Zp and p-adic rationals Qp 3. Elementary/clumsy definitions of adeles A and ideles J 4. Uniqueness of objects characterized by mapping properties 5. Existence of limits 6. Zp and Ẑ as limits 7. Qp and A as colimits 8. Abelian solenoids (R×Qp)/Z[1/p] and A/Q 9. Non-abelian solenoids and SL2(Q)\SL2(A) Although we will also give the more ty...

Canonical forms are developed for several sets of matrices that are normal with respect to an indefinite inner product induced by a nonsingular Hermitian, symmetric, or skewsymmetric matrix. The most general result covers the case of polynomially normal matrices, i.e., matrices whose adjoint with respect to the indefinite inner product is a polynomial of the original matrix. From this result, c...

In this article we prove a Grothendieck trace formula for L-functions of not necessarily commutative adic sheaves.

1.1. Motivation. Let E be an elliptic curve over a p-adic integer ring R, and assume that E has supersingular reduction. Consider the 2-dimensional Fp-vector space of characteristic-0 geometric p-torsion points in the associated 1-parameter formal group Ê over R. It makes sense to ask if, in this vector space, there is a line whose points x are nearer to the origin than are all other points (wi...