Number Theory: An Introduction to Mathematics

3 Cubic Curves 549

A remarkable application of the Rogers–Ramanujan identities to the hard hexagon
model of statistical mechanics was found by Baxter (1981). Many other models in sta-
tistical mechanics have been exactly solved with the aid of theta functions. A unifying
principle is provided by the vast theory of infinite-dimensional Lie algebras which has
been developed over the past 25 years.
The numberp(n)of partitions ofnincreases rapidly withn. It was first shown by
Hardy and Ramanujan (1918) that

p(n)∼eπ

√

2 n/ (^3) / 4 n√3asn→∞.
They further obtained an asymptotic series for p(n), which was modified by
Rademacher (1937) into a convergent series, from which it is even possible to cal-
culatep(n)exactly. A key role in the difficult proof is played by the behaviour under
transformations of the modular group ofDedekind’s eta function
η(τ)=q^1 /^12

∏

k≥ 1

( 1 −q^2 k),

whereq=eπiτandτ∈H (the upper half-plane).
The paper of Hardy and Ramanujan contained the first use of the ‘circle method’,
which was subsequently applied by Hardy and Littlewood to a variety of problems in
analytic number theory.

3 CubicCurves...............................................

We d e fi n e a naffine plane curveover a fieldKto be a polynomialf(X,Y)in two
indeterminates with coefficients fromK, but we regard two polynomialsf(X,Y)and
f∗(X,Y)as defining the same affine curve iff∗=λffor some nonzeroλ∈K.The
degreeof the curve is defined without ambiguity to be the degree of the polynomialf.
If

f(X,Y)=aX+bY+c

is a polynomial of degree 1, the curve is said to be anaffine line.If

f(X,Y)=aX^2 +bXY+cY^2 +lX+mY+n

is a polynomial of degree 2, the curve is said to be anaffine conic.Iff(X,Y)is a
polynomial of degree 3, the curve is said to be anaffine cubic. It is the cubic case in
which we will be most interested.
LetCbe an affine plane curve over the fieldK, defined by the polynomialf(X,Y).
We s a y t h a t(x,y)∈K^2 is apointor, more precisely, aK -pointof the affine curveC
iff(x,y)=0. TheK-point (x,y)issaidtobenon-singularif there exista,b∈K,
not both zero, such that

f(x+X,y+Y)=aX+bY+···,