Theorem 15.1. Let E/Fq be an elliptic curve over a finite field, and let πE be the Frobenius endomorphism of E. Then E is supersingular if and only if trπE ≡ 0 mod p. Proof. Let q = pn and let π be the p-power Frobenius map π(x, y) = (xp, yp) (note that π is an isogeny, but not necessarily an endomorphism, since E need not be defined over Fp). We have π̂π = [p], where [p] denotes the multiplicat...

Any such neighborhood is called an expansive neighborhood of the identity in X. An automorphism τ of X is said to be expansive if the cyclic group generated by τ acts expansively on X. It is easy to check that when X is compact and ρ is an endomorphism action of a semigroup Γ, these two notions of expansiveness coincide. The notion of expansiveness plays an important role in the study of dynami...

We prove that any stable, additive, combinatorial model category M has a canonical model enrichment over Sp(sAb) (symmetric spectra based on simplicial abelian groups). So to any object X ∈ M one can attach an endomorphism ring object in this category, denoted hEndad(X). Since the homotopy theory of ring objects in Sp(sAb) is equivalent to the homotopy theory of differential graded algebras, on...

Let f be a continuous ring endomorphism of ZpJxK/Zp of degree p. We prove that if f acts on the tangent space at 0 by a uniformizer and commutes with an automorphism of infinite order, then it is necessarily an endomorphism of a formal group over Zp. The proof relies on finding a stable embedding of ZpJxK in Fontaine’s crystalline period ring with the property that f appears in the monoid of en...

Let X be a projective bundle. We prove that X admits an endomorphism of degree > 1 and commuting with the projection to the base, if and only if X trivializes after a finite covering. When X is the projectivization of a vector bundle E of rank 2, we prove that it has an endomorphism of degree > 1 on a general fiber only if E splits after a finite base change. It is clear that, for a complex pro...