Homework Assignment 6

(Math 252: Modular Abelian Varieties)

William A. Stein

Oct. 22 (Due: Oct. 29)

1. Formation of Néron models commutes with products: Let A and B be abelian varieties over Q. Show that if A and B are the Néron models of A and B, respectively, then the Néron model of A ×_{Q} B is A ×_{Z} B. [Hint: Use the universal property of products–see Hartshorne’s book, Section 3, page 87. You can just work formally, if you are not familiar with schemes. Also, I chose the base to be Q for concreteness; the statement is true in general.]

2. The Néron models functor:

(a) Suppose φ : A → B is a homomorphism of abelian varieties. Prove that there is a homomorphism A → B of Néron models whose generic fiber is φ.

(b) Suppose 0 → A → B → C → 0 is an exact sequence of abelian varieties. Prove that there is a complex of Néron models 0 → A → B → C → 0 whose generic fiber is 0 → A → B → C → 0. [Hint: By “complex” I mean that the sequence of Néron models need not be exact. The only condition is that the composition of A → B with B → C is the 0 map.]

3. Component groups over F_{p} and \overline{F}_{p}:

Give an example of an abelian variety A and a prime p such that

Φ_{A,p}(\overline{F}_{p}) ≠ Φ_{A,p}(F_{p}).

[Hint: Your A should be an elliptic curve. You may use Corollary 15.2.1, which is on page 359 of Silverman’s book Arithmetic of Elliptic Curves, which we now recall. Let K be a finite extension of Q_{p} with valuation v normalized so that v(π) = 1, where π is a generator of the maximal ideal of the ring of integers of K, and let k be the residue class field of K. Corollary 15.2.1. Let E be an elliptic curve over K. If E has split multiplicative reduction, then Φ_{E,v}(k) is cyclic of order − ord_{v}(j(E)). In all other cases, Φ_{E,v}(k) has order at most 4. Now choose an E with nonsplit multiplicative reduction over Q_{p}, which attains split multiplicative reduction over an unramified extension of Q_{p}.]