Zeta Functions - Elliptic Curves: Number Theory and Cryptography - page 440

rank r = 1, with generator P , predicts that

Ω p c p (# E ) h ( P )
# E ( Q ) torsion

L E ( s )=( s − 1) + ··· .

In this case, Ω is the analogue of 4 / √ d and # E ( Q ) torsion plays the role of 2,
which is the number of roots of unity in K . The height h ( P ) gives the size of
P . Similarly, log( η ) gives the size of η .
In general, we can write down a dictionary between elliptic curves and
number fields:

elliptic curves ←→ number fields
points ←→ units
torsion points

←→ roots of unity

Shafarevich-Tate group ←→ ideal class group

This is not an exact dictionary, but it helps to interpret results in one area
in terms of the other. For example, the Dirichlet unit theorem in algebraic
number theory, which describes the group of units in a number field, is the
analogue of the Mordell-Weil theorem, which describes the group of rational
points on an elliptic curve. The finiteness of the ideal class group in algebraic
number theory is the analogue of the conjectured finiteness of the Shafarevich-
Tate group.

Exercises

14.1 Let P 1 be one-dimensional projective space.

(a) Show that the number of points in P 1 ( F q )is q +1.
(b) Let N n =# P 1 ( F q n ). Define the Z -function for P 1 by
Z P 1 ( T )=exp ∞

T n .

N n
n

n =1

Show that

1

Z P 1 ( T )=

qT ) .

(1 − T )(1 −

14.2 Let M = ab

∈ GL 2 ( R ) with det( M ) > 0. Define an action of M

cd

on functions on H by

M )( z )=det( M )( cz + d ) − 2 f ( Mz ) , ( f |

where Mz = az + b

cz + d .