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132 6. Proof of Mordell’s Finite Generation Theorem

Exercises

1. Let f (x) be a monic polynomial of degree n over k, and let ( f ) denote the ideal generated
by f in k[x]. Show that R f = k[x]/( f ) is an algebra of dimension n over k generated by
one element. If f (x) factors into distinct linear factors, then show that R f is isomorphic to
n
k as algebras over k. When f (x) factors with repeated linear factors, describe a structure
theorem for R f as a direct sum of indecomposible algebras.
2
2. Let y = f (x) = (x −e 1 )(x −e 2 )(x −e 3 ) with distinct e i , and let E be the elliptic curve
over k defined by this equation. Show that we can define, using the notation of Exercise
∗ 2
1, a homomorphism g : E(k) → R /(R ) by the relation for 2P = 0,
∗
f f
∗ 2
g(P) = x(P) − e mod(R ) , where e ≡ x mod( f ).
f
Define g(P) for 2P = 0insuchawaythat g is a group morphism. Show that ker(g) ⊂
∗ 2
∗ 2
∗
2E(k),and im(g) ⊂ R ∗ /(R ) , where R ∗ consists of all a ∈ R with N(a) ⊂ (k ) .
f,1 f f,1 f
Finally relate these results with (4.3).
§5. Quasilinear and Quasiquadratic Maps
This is a preliminary section to our discussion of heights on projective space and on
elliptic curves. From heights we derive the norm function on the group of rational
points on an elliptic curve.

−1
(5.1) Definition. For a set X a function h : X → R is proper provided h ([−c,+c])
is finite for all c ≥ 0.
In general a map between two locally compact spaces is proper if and only if the
inverse image of compact subsets is compact. Thus a function h : X → R is proper
if and only if h is a proper map when X has the discrete topology and R the usual
topology.

(5.2) Definition. Two functions h, h : X → R are equivalent, denoted h ∼ h ,

provided h − h is bounded, that is, there exists a > 0 such that |h(x) − h (x)|≤ a
for all x ∈ X.
(5.3) Remark. Equivalence of real valued functions on a set is an equivalence re-
lation. If two functions are equivalent, and if one function is proper, then the other
function is proper.

Using this equivalence relation, we formulate quasilinearity, quasibilinearity, and
quasiquadratic and then relate them to the usual algebraic concepts.

(5.4) Definition. Let A be an abelian group.
(1) A function u : A → R is quasilinear provided u(x + y) and u(x) + u(y)
are equivalent functions A × A → R, i.e., u(x, y) − u(x + y) − u(x) − u(y) is
bounded on A × A.