THE LEGACY OF ABEL IN ALGEBRAIC GEOMETRY • Introduction

32 Pages·2010·0.36 MB·English

THE LEGACY OF ABEL IN ALGEBRAIC GEOMETRY
PHILLIP GRIFFITHS
• Introduction
• Origins of Abel’s theorem
• Abel’s theorem and some consequences
• Converses to Abel’s theorem
• Legacies in algebraic geometry — two conjectures
– Webs
– Abel’s DE’s for points on a surface
• Reprise
This paper is based on a talk given at the bicentenary celebration
of the birth of Neils Henrik Abel held in Oslo in June, 2002. The
objectives of the talk were first to recall Abel’s theorem in more or less
its original form, secondly to discuss two of the perhaps less well known
converses to the theorem, and thirdly to present two (from among the
many) interesting issues in modern algebraic geometry that may at
least in part be traced to the work of Abel. Finally, in the reprise I will
suggest that the arithmetic aspects of Abel’s theorem may be a central
topic for the 21st century.
This talk was not intended to be a “documentary” but rather to tell
the story — from my own perspective — of Abel’s marvelous result
and its legacy in algebraic geometry. Another talk at the conference
by Christian Houzel gave a superb historical presentation and analysis
of Abel’s works.
In keeping with the informal expository style of this paper (the only
proof given is one of Abel’s original proofs of his theorem) at the end
are appended a few general references that are intended to serve as a
guide to the literature, and should not be thought of as a bibliography.
1
2 PHILLIP GRIFFITHS
Origins of Abel’s theorem
1.
During the period before and at the time of Abel there was great
interest among mathematicians in integrals of algebraic functions, by
which we mean expressions
(cid:90)
(1.1) y(x)dx
where y(x) is a ‘function’ that satisfies an equation
(1.2) f(x,y(x)) = 0
where f(x,y) ∈ C[x,y] is an irreducible polynomial with complex co-
efficients. Although not formalized until later, it seems to have been
understood that (1.1) becomes well-defined upon choosing a particular
branch of the solutions to (1.2) along a path of integration in the x-
plane that avoids the branch points where there are multiple roots. In
more modern terms, one considers the algebraic curve F◦ in C2 defined
by
f(x,y) = 0 ,
and on F◦ one considers the rational differential ω defined by the re-
striction to F◦ of
ω = ydx .
On the closure F of F◦ in the compactification of C2 given either by
the projective plane P2 or by P1 ×P1 one considers an arc γ avoiding
the singularities of F and the poles of ω, and then (1.1) is defined to
be
(cid:90)
(1.3) ω .
γ
Actually, amongmathematiciansofthetimetheinterestwasinmore
general expressions
(cid:90)
(1.4) r(x,y(x))dx
wherer(x,y)isarationalfunctionofxandy, andy(x)isasabove. The
formal definition of (1.4) is as in (1.3) where now ω is the restriction
to F of the rational differential 1-form r(x,y)dx.
THE LEGACY OF ABEL IN ALGEBRAIC GEOMETRY 3
Of special interest were the hyperelliptic integrals
(cid:90)
p(x)dx
(1.5)
(cid:112)
q(x)
where p(x) and q(x) are polynomials with, say,
q(x) = xn +q xn−1 +···+q
1 n
of degree n and having distinct roots. When n = 1,2 it was well
understood at the time of Abel that these integrals are expressible in
terms of the “elementary” — i.e., trigonometric and logarithmic —
functions. The geometric reason, which was also well understood, is
that any plane curve may be rationally parametrized as expressed by
the picture
(x(t),y(t))
X
t
Plugging the rational functions x(t) and y(t) into (1.5) gives an integral
(cid:90)
r(t)dt ,
where r(t) is a rational function, and this expression may be evaluated
by the partial fraction expansion of r(t).
Therewasparticularinterestinthehyperellipticintegrals(1.5)when
n = 3,4 and important fragments were understood through the works
of Euler, Legendre and others. They go under the general term of
elliptic integrals, for the following reason: Just as the resolution of the
arc length on a circle leads to the trigonometric functions as expressed
by
(cid:90) (cid:90)
(cid:112) dx
(1.6) dx2 +dy2 = √ , x2 +y2 = 1 ,
1−x2
4 PHILLIP GRIFFITHS
there was great interest in the functions that arise in the resolution of
the arc length of all ellipse. Thus, through the substitution
(cid:16)x(cid:17)
t = arcsin
a
the arc length on the ellipse
(cid:90) (cid:112) x2 y2
dx2 +dy2, + = 1 a > b
a2 b2
becomes the elliptic integral
(cid:90) (1−k2x2)dt
(1.7) a , k2 = (a2 −b2)/a2
(cid:112)
(1−t2)(1−k2t2)
in Legendre form.
Of special interest were integrals (1.4) that possessed what was
thought to be the very special property of having functional equations
oraddition theorems. Forexample, usingtheobvioussyntheticgeomet-
ric construction of doubling the length of an arc on the circle applied
to the integral (1.6), one recovers the well known formulas for sin2θ
and cos2θ expressed in terms of sinθ and cosθ. More generally one
may derive expressions for sin(θ +θ(cid:48)) etc. which are expressed as ad-
dition theorems for the integral (1.6). In the 18th century the Italian
Count Fagnano discovered a synthetic construction for doubling the
arc length on an ellipse, and when applied to (1.7) this construction
leads to addition theorems for the “elliptic integral” (1.7). As alluded
to above this was thought to be a very special feature, one that was
the subject of intensive study in the late 18th and early 19th centuries.
Abel’s theorem and some consequences
2.
In Abel’s work on integrals of algebraic functions there are two main
general ideas
• abelian sums
• inversion
Together these led Abel to very general forms of
• functional equations
THE LEGACY OF ABEL IN ALGEBRAIC GEOMETRY 5
for the integrals. We will now explain these ideas.
Turning first to what are now called abelian sums, the integrals (1.1)
and more generally (1.4) are highly transcendental functions of the
upper limit of integration and consequently are generally difficult to
study directly.1 Abel’s idea was to consider the sum of integrals to the
variable points of intersection of F = {f(x,y) = 0} with a family of
curves G = {g(x,y,t) = 0} depending rationally on a parameter t.
t
Thus letting
(cid:88)
F ∩G = (x (t),y (t))
t i i
i
be the set of solutions to
(cid:26)
f(x,y) = 0
g(x,y,t) = 0
written additively using the notation of algebraic cycles, the abelian
sum associated to (1.4) is defined to be
(cid:88)(cid:90) xi(t)
(2.1) u(t) = r(x,y(x))dx .
i x0
Below we will amplify on just how this expression is to be understood.
A particularly important example is given by taking the G to be a
t
family of lines as illustrated by the figures
(i) F = {x2 +y2 = 1}
1Theterm“highlytranscendental”needscareininterpretation—cf.thereprise
below. Again Abel, in a paper published in 1826, showed the existence of polyno-
mials R,F such that
(cid:32) √ (cid:33)
(cid:90) F dx P + RQ
√ =ln √
R R− RQ
has solutions for relatively prime polynomials P,Q. Here, R is a polynomial of
degree 2n with distinct roots and F is a polynomial of degree n−1, so that the
integrand is a differential of the 3rd kind. This is an “exceptional” case where the
integral is transcendental but expressible in terms of elementary functions.
6 PHILLIP GRIFFITHS
(ii) F = {y2 = x2 +ax+b}
In both cases we take ω = dx/y and the integrals (1.4) are respectively
(cid:40) (i) (cid:82) √dx
1−x2
(2.2)
(cid:82)
(ii) √ dx
x3+ax+b
Even though the individual terms in the abelian sum are in general
highly transcendental functions, Abel’s theorem expresses the abelian
sum as an elementary function:
Theorem: The abelian sum (2.1) is given by
(cid:88)
(2.3) u(t) = r(t)+ a log(t−t )
λ λ
λ
where r(t) is a rational function of t.
One of the proofs given by Abel is as follows:
Proof: For reasons to appear shortly we define the rational function
q(x,y) = r(x,y)f (x,y) ,
y
so that the integrand in the integrals appearing in the abelian sum is
the restriction to the curve F of
q(x,y)dx
.
f (x,y)
y
Then by calculus
(cid:88) q(x (t),y (t))x(cid:48)(t)
u(cid:48)(t) = i i i .
f (x (t),y (t))
y i i
i
From
(cid:40)
f(x (t),y (t)) = 0
i i
g(x (t),y (t),t) = 0
i i
we have
(cid:18) (cid:19)
g f
x(cid:48)(t) = t y (x (t),y (t))
i f g −f g i i
x y y x
THE LEGACY OF ABEL IN ALGEBRAIC GEOMETRY 7
so that
(cid:88)
(2.4) u(cid:48)(t) = s(x (t),y (t))
i i
i
where s(x,y) is the rational function given by
(cid:18) (cid:19)
qg
t
s(x,y) = (x,y) .
f g −f g
x y y x
(The non-vanishing of the rational function in the denominator is a
consequence of assuming that the curves F and G have no common
t
component). Abel now observes that the right hand side of (2.4) is
a rational function of t — from a complex analysis perspective this is
clear, since u(cid:48)(t) is a single-valued and meromorphic function of t for
t ∈ P1. Integration of the partial fraction expansion of u(cid:48)(t) gives the
result.
In his Paris memoir´e, and also in subsequent writings on the subject
in special cases, Abel gave quite explicit expressions for the right hand
side of (2.4), and therefore for the terms in the formula for u(t) in
his theorem. For example, when the curves G are lines the Lagrange
t
interpolation formula gives the explicit expression for u(cid:48)(t).
We shall now give applications of Abel’s theorem to the two integrals
in(2.2). BotharebasedonthesecondofAbel’sideasmentionedabove,
namely to invert the integral (1.4) by defining the coordinates x(u),
y(u) on the curve F as single-valued functions of the variable u by
setting
(cid:90) x(u),y(u))
(2.5) u = ω
(x0,y0)
where ω is the restriction to the curve F of r(x,y)dx. For example, for
the integral (i) in (2.2) we obviously have
(cid:90) (sinu,cosu)
u = ω .
(0,1)
8 PHILLIP GRIFFITHS
The right hand side in (2.3) may be evaluated using the Lagrange
interpolation formula and this leads to the relation
(cid:90) x1 dx (cid:90) x2 dx (cid:90) x1y2+x2y1 dx
√ + √ = √
1−x2 1−x2 1−x2
0 0 0
which we recognize as the addition formula for the sin function.
Beforeturningtothesecondintegralin(2.2), weremarkthatalready
in his Paris memoir´e Abel singled out a “remarkable” class of abelian
integrals(1.4), now called integrals of the 1st kind, bytheconditionthat
the right hand side of (2.3) reduce to a constant — this is evidently
equivalent to the abelian integral (1.4) being locally a bounded func-
tion of the upper limit of integration. Abel explicitly determined the
integrals of the 1st kind for a large number of examples. For instance
for the hyperelliptic curves
y2 = p(x)
where p(x) is a polynomial of degree n + 1 with distinct roots, Abel
showed that the integrals of the 1st kind are
(cid:40)
g(x)dx
ω =
y
degg(x) (cid:53) (cid:2)n(cid:3) .
2
In particular, assuming that the cubic x3+ax+b has distinct roots, the
expression (ii) in (2.2) is an integral of the 1st kind. Abel’s theorem for
the family of lines meeting the cubic may be expressed by the relation
(2.6) u +u +u = c
1 2 3
where c is a constant and
(cid:90) (x(u),y(u)) dx
(2.7) u =
y
(x0,y0)
with u = u plugged into (2.7) for i = 1,2,3 in (2.6). Differentiation of
i
(2.7) gives
x(cid:48)(u)
1 =
y(u)
so that
(2.8) y(cid:48)(u) = x(y) .
THE LEGACY OF ABEL IN ALGEBRAIC GEOMETRY 9
Choosing (x ,y ) appropriately (specifically the flex [0,0,1] on the in-
0 0
tersection of F with the line at infinity in P2) we will have
(cid:40)
c = 0
x(−u) = x(u)
and (2.6) becomes the famous addition theorem for the elliptic integral
(2.9) x(u +u ) = R(x(u ),x(cid:48)(u ),x(u ),x(cid:48)(u ))
1 2 1 1 2 2
where R is a rational function that expresses the x-coordinates of the
third point of intersection of a line with F as a rational function of the
coordinates of the other two points.
Of course, x(u) is the well-known Weierstrass p-function and the
above discussion gives the functional equation (2.9) and differential
equation
x(cid:48)(u)2 = x(u)3 +ax(u)+b
satisfied by the p-function. We give two remarks amplifying this dis-
cussion.
The first is that in order to define the integral
(cid:90)
dx
(2.10) √
x3 +ax+b
one cuts the x-plane, including the point at infinity, along slits con-
necting two of the roots of x3 +ax+b and connecting the third root
to x = ∞
∞
δ(cid:13)
1
δ(cid:13)
2
√
Then x3 +ax+b is single-valued on the slit plane, and one may
envision the algebraic curve F as a 2-sheeted covering of the x-plane
where crossing a slit takes one to the “other sheet” — i.e., F is the
√
Riemann surface associated to the algebraic function x3 +ax+b.
The topological picture of F is the familiar torus
10 PHILLIP GRIFFITHS
δ(cid:13)
2
δ(cid:13)
1
The integral (2.10) is then interpreted as an integral along a path on
the Riemann surface. The choice of path is only well-defined up to
linear combinations of δ and δ . In particular, from (2.7) we infer that
1 2
(cid:40)
x(u+λ ) = x(λ )
i i
(2.11)
y(u+λ ) = y(λ )
i i
where
(cid:73)
dx
λ =
i
y
δi
are the periods of dx/y. Letting Λ be the lattice in the complex plane
generated by λ and λ , we have the familiar parametrizaton
1 2
C/Λ −→ F
↓ ∪|
u −→ (x(u),x(cid:48)(u))
of the cubic curve by the p-function and its derivative.
In his Paris memoir´e, Abel gave in generality the essential analytic
properties of elliptic functions, defined as those functions that arise by
inversion of the integral of the first kind on curves having one such
integral.
Remark that the dimension of the space of integrals of the first kind
is one definition of the genus of the algebraic curve F (or arithmetic
genus, in case F is singular). Following Abel’s pioneering work, the
extension of the above story to curves of arbitrary genus was carried
out by Jacobi, Riemann and other 19th-century mathematicians.
A second remark is that the functions x(u),y(u) in (2.7) may be
defined locally with (2.8) holding, and the functional equation (2.9) is
valid where defined. But then this functional equation may be used to
extend x(u) and y(u) to entire meromorphic functions — e.g., if x(u) is