68DANIEL DISEGNIAFTERWORD (FOR THE LAYMAN)
0. — This concluding part is written for all the friends, relatives, and the occasional strangers met at parties who have been wondering what, exactly, I have been
doing all these years. The most frequent questions were: is your thesis going to bejust numbers and formulas, or also words? Did you prove some new theorem orformula? What is your thesis about? Is it useful for something?
The first question needs no answer at this point. The second question has a quickand dirty answer – I proved this formula (that’s Theorem B in the Introduction):
!2
!2
Y
1
1
L0p,W ( fE , 1) = DF 2
1
1hz f , z f iW .
↵}"(})↵}}| p

This, of course, means nothing without context, which brings me to the (longer)
answer to the third question.(42) Before going there, however, let me offer twowords about whether this is new: yes, and no. No, that is, because this work is
a generalisation of the results of another mathematician. Yes, in the sense that inthis situation those results were not known before (were they, no doctorate would
follow), and although perhaps it is not surprising (to the experts) that they are true,
proving them so has required some “new” ideas – which were in turn inspired fromother people’s work. . . But this is not an essay in the philosophy of history ofscientific ideas.
1. — Esoteric as these pages may look like, the questions they deal with are mucholder than all the modern-day musings about gravity, or cells, or molecules. In
fact, the proper name for this subject, Arithmetic, usually elicits reactions betterknown to contemporary artists (“Any child can do that” – arithmetic is identified
with primary school mathematics), so that we are stuck with the (uglyish) nameof “theory of numbers”. The numbers in questions are the integers 0, 1, 2, 3, . . .
and their negatives, as well as the fractions (such as 3/4, 7/5, . . .) – which go under the name of rational numbers. The founding father is usually recognised in
Diophantus of Alexandria (3rd century A.D.), who in a treatise named, indeed,Arithmetica listed many problems and solutions to equations to be solved in integers or rationals. These questions already had a long history by then, starting with
the Pythagorean discovery of the irrationality of the square root of 2 – that is tosay, the non-existence of rational solutions to the quadratic equation x 2 = 2.

(42)

The more practically-minded reader should jump to the end for an answer to the last question.

p-ADIC HEIGHTS OF HEEGNER POINTS ON SHIMURA CURVES69

The general theory of quadratic equations of one variable is known to most highschoolers, and reduced to the Pythagorean question of the squareness of the discriminant(43) in the system of numbers of interest. Similarly, the study of equationsof degree three or four(44) in one variable was reduced to the extraction of third andfourth roots by Italian Renaissance mathematicians. That the same cannot be said
in general for equations of degree five or more was the early-Nineteenth-century
discovery of Abel and independently Galois, who developed a complete theory of
the symmetry of those equations. This was the end of a beautiful story – and the
beginning of a new one, but we will not go there.
2. — Many of the equations studied by Diophantus are rather those in two variables, like x 2 + y 2 = 1 (more generally one could consider systems of several equations in several variables, but two variables are still enough to give us headaches
after almost two millennia). The rational solutions to this equation, like x = 3/5,
y = 4/5, correspond to Pythagorean triples – triples of whole numbers, such as(3, 4, 5), which can be the sides of a right-angled triangle(45) . Of course x 2 + y 2 = 1
is also the equation for a circle: more precisely, this means that the solutions of
this equation in the system of the real numbers (that is, the infinite decimals like
0.7163538902...) form a circle in the x y-plane. That the same can be said of the real
solutions of x 2 + y 2 = 3 should convince the reader that solving equations in thesystem of rational numbers is considerably more complicated than in the system ofreal numbers, once he or she is told that x 2 + y 2 = 3 has no rational solutions atall!(46)
We have thus met the idea of attacking the difficult problem of studying the solutions to an equation in the system of rational numbers by first studying them in simpler
For those struggling to recollect old memories, the discriminant of a x 2 + b x + c is the (in)famous= b 2 4ac.
(44)
Like x 3 + x + 1 = 0, or 2x 4 5x 2 + 7 = 0.
(45)
Because they satisfy the requirements of Pythagorean Theorem on the sum of the squares of thelegs being the square of the hypothenuse: 32 +42 = 52 . There are infinitely many Pythagorean triples,parametrized by (m 2 n 2 , 2mn, m 2 + n 2 ) for any integers m > n.
(46)
This fact is by no means obvious, although not difficult to show: writing x = a/c, y = b /c with acommon denominator c, the problem is equivalent to that of solving(43)

a 2 + b 2 = 3c 2
in integers a, b , c with c not equal to zero. If there is a solution, then the solution with the smallest possible positive c can’t have a, b , c all even, since otherwise the halves of a, b and c would givea smaller solution. Now it is easy to see that the square of an even number (2k)2 = 4k 2 leaves remainder 0 when divided by 4, while the square of an odd number (2k + 1)2 = 4(k 2 + k) + 1 leaves
remainder 1. So looking at the remainders of division by four on either side of our equation, anddenoting “⌘” the relation of equality up to the addition of a multiple of 4, we should have 1 + 0 ⌘ 3
or 0 + 1 ⌘ 3 if only one of a or b is odd (so that c is too); or 1 + 1 ⌘ 0 if both a and b are odd (so thatc is even). This is clearly not the case, so there is no solution.

70

DANIEL DISEGNI

systems of numbers, such as the real numbers and the systems of numbers “up to theaddition of multiples of a fixed integer”, such as the system of numbers up to addition of multiples of 4 used in the last footnote (these are called simply numbers“modulo 4” – note that this is a finite number system, containing only the four elements 0, 1, 2, 3: indeed we have 4 ⌘ 0, and for example 2 + 3 ⌘ 4 + 1 ⌘ 1). This idea
turns out to be very fruitful, and in fact it suffices to give a complete treatment ofequations of degree 2 in two variables, such as x 2 + 3x y 7y 2 + 5x 4 = 0: there
is a solution exactly when there are real solutions(47) and solutions in the numbers“modulo N ” for any integer N ; this can be verified, by a human or a computer, ina finite (and quite short) amount of time, and if there is a solution then there areinfinitely many.
3. — Finally, we can approach the topic of this thesis. For general equations ofdegree three or more, the existence of solutions in the real and finite number systems is not enough to guarantee the existence of a solution in the system of rational
numbers. Even if we know that there is a solution, in general we still don’t knowwhether there are finitely many or infinitely many others. Actually, for equationsof degree at least five, one of the most important recent results in the subject, a theorem of the German mathematician Faltings [12], says that there is always at mosta finite number of solutions. (On the other hand, the study of equations of degree
four can be reduced to the case of degree three.) So in a sense the most important
equations to be studied are those of degree three, called cubic equations, like forexample x 3 + y 3 = 1729.(48) Here are the main questions: is there an efficient way(an algorithm) to detect whether a given cubic equation has a rational solution ornot? If there is a solution, what can be said about how many solutions there are?
The answer to the first question is not known, and this thesis adds (almost) nothing to that. For the second question, it was known to Diophantus that given two
solutions one could construct a third one by a geometric method: picturing the two
given solutions as points P and Q on the cubic curve C in the x y-plane corresponding to the equation, the third one is constructed by intersecting the line through P
and Q with C – since the degree is three, there will be three intersections, that is P ,Q and the new solution. One can perform the same construction starting from justone point P and using the tangent line to that point (this corresponds to the ‘degenerate’ case P = Q); and then iterate the procedure. Then two things can happen:
either one returns to P after a certain number of iterations (in this case P is called
This may not be the case: consider for example x 2 + y 2 = 1. When there are real solutions, they
form a conic in the xy-plane: an ellipse, a parabola or a hyperbola.
(48)
I am choosing this equation because of its curious history: two famous mathematicians, Hardy
and Ramanujan, were once meeting at Ramanujan’s house. Hardy said upon arriving that his taxicab
had a rather unremarkable number, 1729. Ramanujan immediately replied that the number was
remarkable indeed, for being the smallest number expressible in two ways as the sum of two cubes:
1729 = 1000 + 729 = 103 + 93 , and 1729 = 1 + 1728 = 13 + 123 .
(47)

p-ADIC HEIGHTS OF HEEGNER POINTS ON SHIMURA CURVES

71

a torsion point); or one keeps getting new points (i.e. solutions) indefinitely. A
fundamental 1922 theorem of Mordell says the following: all the rational solutions
to the cubic equation of interest can be generated from a finite number of points byperforming the geometric construction just described. The smallest possible such
number of generating points (excluding the torsion points) is called the rank of the
cubic: it is 0 or a positive integer, and it is 0 precisely when the number of solutions
is finite; otherwise, the rank gives as a basic measure of “how infinite” the set ofsolutions is.
4. — What does the number of solutions in other number systems tell us about
the rank of a cubic equation? After extensive computer simulations (a new thingat the time), the mathematicians Birch and Swinnerton-Dyer found in the 1960s a
conjectural answer. Consider for example the equation x 3 + y 3 = 1729; we can lookat its solutions in the systems of numbers modulo p for various prime numbers p(we don’t need to restrict to prime numbers, but they are sufficient, and easier todeal with since one can define the operation of division on the associated number
systems); for example, in the system of numbers modulo 5 we have 23 +13 ⌘ 9 ⌘ 5+
4 ⌘ 4, and 1729 ⌘ 1725+4 ⌘ 4 so that x = 2, y= 1 is a solution in this system. How
many solutions can we expect, in general, in the system of numbers modulo p? We
are looking at one equation in two variables, so roughly speaking we expect to have
one free variable (for example, if every number modulo
p p had a unique cube root,3

then we could take x as the free variable and then y = x 3 1729) – since the freevariables can assume the p values 0, 1, . . . , p 1 we then expect about p solutions.
The actual number of solution will vary around p, and here is the idea: if thisnumber is often larger than p, then this should be because of the existence of rational solutions which “reduce” to solutions in the system of numbers modulo p.
How does one make this idea precise? If N p is the number of solutions modulo p
to a fixed cubic, a good measure of the discrepancy with the expected number ofpoints is its ratio to p, that is N p / p: if there are no rational solution (or only finitely
many of them) this should always be around 1, while if there are infinitely many
then it should often be larger. This information can be packaged into a function ofa variable x (a real number),Np
N2 N3 N5
L(x) =
·
·
··· x
2 3 5
px
where p x is the largest prime number smaller than x. Again, if there are only
finitely many rational solutions, each factor should be about 1 so that we expectthat L(x) does not grow when x grows. On the other hand if there are infinitely
many solutions, Birch and Swinnerton-Dyer observed in their examples that L(x)72

DANIEL DISEGNI

grew likeL(x) ⇠ C log(x) r ,for some real constant C and some non-negative integer r ; and that r was equal
to the rank of the curve.(49) They conjectured that the above relation between thegrowth rate of L(x) and the rank should hold for all cubic equations, and moreover they predicted what the value of C should be (in terms of various quantities
associated to this equation).
Fifty years later, this conjecture is still unproven and likely to remain so for along time – notwithstanding the prize of one million dollars offered by a foundation who ranked it among the seven most important unsolved(50) mathematicalproblems. This thesis, as part of a wider circle of ideas developed in the past three
decades, makes some progress towards it.
5. — The readers who have followed me to this point now know what we aretalking about – yet they may (and should) complain that after several pages they
still have received no explanation as to how any of the words in the title of thiswork relates to the problems I described. In fact, at least two words make sense:
I have tried to explain above how the problem of finding rational solutions to an
equation can be viewed as that of finding points with rational coordinates on a
curve. I will try to explain the other words.
Shimura curves. — Shimura is a Japanese mathematician who studied a certain classof curves, called indeed Shimura curves, which parametrise certain other geometricobjects (let us call those “Shimura objects”). An example should illustrate what thismeans: a circle in the plane can be identified by three numbers, the two coordinatesof its centre (two real numbers) and the length of its radius (a positive real number);
if we consider circles to be equivalent when they can be rigidly moved to coincide,than the radius is the only parameter: that is, the positive real numbers parametrise
the circles up to equivalence.
Shimura curves are less concrete than other curves given by explicit equations,
yet they have a crucial advantage: one can find rational points on them in a naturalway, since they correspond to the “rational Shimura objects” – in the above example, circles with rational radius can be thought of as “rational circles” (of course,
in the example it is trivial to find rational points on the – rather straight – “curve”
constituted by the positive real numbers, but by now my reader will be convinced
To get a sense at why this conjecture could not have been made before the advent of computers,
notice that log(x), the inverse function of the exponential function, grows very slowlly. So if in the
study of the growth of L(x) one wants to see when it surpasses, say, the value of 9, and if r = 1, then
one needs to find the number of solutions to his equation modulo primes up to 109 = 1, 000, 000, 000.
(50)
One of the seven has been solved since being included in the list – but the million dollars offered
for it was refused by the winner.
(49)

p-ADIC HEIGHTS OF HEEGNER POINTS ON SHIMURA CURVES

73

that for other curves this is not so). Shimura and others made the striking conjecture that every cubic curve is related to some Shimua curve, in the precise sensethat one can find a transformation (a function) from a Shimura curve to the cubic
curve, which sends rational points to rational points. (Here is an example to illustrate the concept: the parabola y = x 2 is related to the real line with coordinate t
by the function which sends t to the point (x = t , y = t 2 ) on the parabola – andif t is rational then so are x and y.) The proof of this conjecture by Andrew Wiles
in 1993 was the essential ingredient towards his proof of the famous Fermat’s Last
Theorem, which had been waiting for one for four centuries.
Heegner points. — Heegner was a German high school teacher and amateur mathematician, who in 1952 had the brilliant idea of producing rational points on cubic
curves from the natural rational points on the related Shimura curve. This is to datethe only systematic way of finding solution to cubic equations.(51) One can raise thequestion of whether the so-called Heegner points thus obtained are torsion points
or not, and here is the answer:(52) suppose that L(x) grows like C log(x) r ; then the
point is torsion exactly when r = 1 (as an exercise, the reader can try to extract
what this implies for the conjecture of Birch and Swinnerton-Dyer). Notice that,since r is supposed to equal the rank, it is no surprise that the Heegner point is
torsion when r = 0, as in this case every solution is torsion. On the other hand,when r is at least two, we should expect many (non-torsion) rational points, yet the
only known method for finding them fails, and we are at a loss.
p-adic heights. — A measure of the complexity of a rational number written as a
reduced fraction m/n is the maximum of the number of digits of m and n. One
can refine this notion to study the complexity of a point on a cubic curve – andthe resulting measure of complexity of the point is called the height of the point.
It is a non-negative real number, which is zero precisely when the point is torsion.
A study of the height of Heegner points is what has led to the result mentioned
just above on the “growth versus rank” part of the Birch and Swinnerton-Dyer
conjecture.

Here is an example of its power – relegated to a footnote in order to avoid scaring any readerswith the size of the numbers involved. The equation 1063y 2 = x 3 x has rank one, and the simplest
non-torsion solution has the x-coordinate q 2 /1063, where
11091863741829769675047021635712281767382339667434645
q=.
317342657544772180735207977320900012522807936777887(This was found as a Heegner point by Noam Elkies. The reader may make a guess on how long it
would take to find it with a naive search.)(52)
Due to Gross and Zagier [13]: my more attentive followers may recognize in the latter the nameof my laurea thesis advisor.
(51)

74

DANIEL DISEGNIThe p-adic height is another measure of the complexity of a point, which is not areal number but rather a p-adic number, that is an element of a certain infinite system of numbers obtained by combining together the systems of numbers modulo
p, p 2 , p 3 , . . . for a fixed prime number p. Now reading the title should give someideas as to what this thesis studies – which is useful for understanding the constant
C in the growth of L(x). (This constant is important: it is related to the problem
of deciding on the very existence of any points on some other related cubic curves.)
6. — Finally, a word of explanation on the formula I started this section with (andwhich is indeed my main result): the Heegner point being studied is z f (here f is
the name of the corresponding cubic curve on which it lies), and hz f , z f iW is its padic height; on the left-hand side, the quantity L0p,W ( fE , 1) is related to the function
L(x) introduced above.
As mentioned before, the results of this work are not new in the study of rational
solutions to cubic equations.(53) The novelty here is that they are proved for acertain class of systems of equations in any number of variables which share someof the features of cubic equations of one variable (some of these systems are related
to the study of one-variable equations of degree higher than three), and over systemsof numbers which are more general than the system of rational numbers.
7. — I have thus far escaped the question of what all this is useful for. Jacobi’s
noble answer “pour l’honneur de l’esprit humain” may not satisfy every palate inan era of tight budgets. It is certainly unclear that the world is a better place after
these pages. Yet the same could be said of the vast majority of basic research.(54) Siparva licet, nothing could have sounded more abstruse than antimatter when Diracsuggested its existence in 1928; yet today it is used daily in PET(55) scans in every
modern hospital. More to the point, no one from Diophantus to Heegner could
foresee the advent of the internet; yet the cryptographic methods that protect ourelectronic transactions and communications are using all the arithmetic knowledge
developed in the past two millennia – and cubic curves have a significant part in it.
Whether the ideas of this work will prove useful for improving on those methods,
for nothing at all, or for something else which has not been invented yet, is as
unclear as it is unpredictable: we will have to wait and see.
DANIEL DISEGNI • E-mail : disegni@math.columbia.edu

They are due to the French mathematician Bernadette Perrin-Riou [30].
Not to mention many other more marketable cultural artifacts: opinion pieces, TV ads, mortgagebacked securities. . .
(55)
Positron Emission Tomography (positrons are the antiparticles of electrons).
(53)(54)