# Elliptic Tales: Curves, Counting, and Number Theory

Avner Ash
Robert Gross
Pages: 312
https://www.jstor.org/stable/j.ctt7sx3k

1. Front matter
(pp. i-viii)
(pp. ix-xii)
3. Preface
(pp. xiii-xviii)
4. Acknowledgments
(pp. xix-xxii)
5. Prologue
(pp. 1-10)

Here we will introduce some of the leading characters in our story. To give some form to our tale, we will consider a particular question: Why were Greek mathematicians able to solve certain kinds of algebraic problems and not others? Of course, the Greeks had no concept corresponding exactly to our “algebra,” but we feel free to be anachronistic and to use modern algebraic equations to describe what they were doing. The various concepts and terms we use in the Prologue will be explained in greater detail later in the book.

6. PART I. DEGREE
• Chapter 1 DEGREE OF A CURVE
(pp. 13-25)

In this chapter, we will begin exploring the concept of the degree of analgebraic curve—that is, a curve that can be defined by polynomial equations. We will see that a circle has degree 2. The ancient Greeks also studied lines and planes, which have degree 1. Euclid limited himself to a straightedge and compass, which can create curves only of degrees 1 and 2. A “primer” of these results may be found in theElements(Euclid, 1956). Because 1 and 2 are the lowest degrees, the Greeks were very successful in this part of algebraic geometry. (Of course,...

• Chapter 2 ALGEBRAIC CLOSURES
(pp. 26-41)

The story so far: If we have a polynomialf(x,y) in two variables, we have defined the degreedoffto be the maximum of the degrees of its monomials. If we intersect the graph off= 0 with a parametrized line in the parametert, that is, a line given by the simultaneous equationsx=at+b,y=ct+e, we get a numberNof intersection points. This numberNis equal to the number of solutions of the polynomial equation

$f{(at \, + \, b, \, ct \, + \, e)} \, = \, 0,$

where the lefthand side is a polynomial of degree...

• Chapter 3 THE PROJECTIVE PLANE
(pp. 42-66)

The story so far: If we have a polynomialf(x,y) in two variables of degreed, and if we intersect the graph off= 0 with a parametrized line given by the simultaneous equationsx=at+b,y=ct+e, we get a numberNof intersection points. This numberNis equal to the number of solutions of the polynomial equation

$f(at \, + \, b, \, ct \, + \, e) \, = \, 0,$

wherefis a polynomial of degree at mostd. Problem:Nmay not equald, in which case we experience the frustration of our desire to define geometrically the degree...

• Chapter 4 MULTIPLICITIES AND DEGREE
(pp. 67-81)

If we have a homogeneous polynomialF(x,y,z) in three variables, the degreedofFis the degree of each and every one of the monomials inF(x,y,z). If we homogenize a polynomialF(x,y) of two variables to getF(x,y,z),Fandfwill have the same degree. (Remember the connection:F(x,y, 1) =f(x,y).)

Each solution (x,y,z) to the equationF(x,y,z) = 0, when interpreted as the projective coordinates (x:y:z) of a point in the projective plane, gives us the coordinates of...

• Chapter 5 BÉZOUT’S THEOREM
(pp. 82-92)

Sometimes, we can’t get everything that we want, and sometimes we want something general rather than something specific. For example, let’s suppose that we want to solve the system of equations

$\left\{ {{f(x, \, y) \, = \, 0}\atop {g(x, \, y) \, = \, 0}}$

wheref(x,y) andg(x,y) are polynomials. It could be very hard to find a list of solutions, but perhaps we would settle for knowing the number of solutions. Maybe there is a general theorem telling us how many solutions there are in terms of a simple rule that we can apply tofandg.

The preceding chapters have hinted that this particular problem won’t have...

7. PART II. ELLIPTIC CURVES AND ALGEBRA
• Chapter 6 TRANSITION TO ELLIPTIC CURVES
(pp. 95-99)

In the first part of this book, we discussed at great length equations in two variables of the formf(x,y) = 0, wherefis a polynomial of degreed> 0 with coefficients in a fieldK. Such an equation defines a “plane curve”C. For any fieldK′ that contains the fieldK, we can wonder about the set of all solutions of the equation when the variables are given values fromK′. We call this setC(K′).

We saw that in order to make the degree offequal to the number of intersections of any probing...

• Chapter 7 ABELIAN GROUPS
(pp. 100-115)

When we count mathematical objects such as the number of solutions to an equation, the first fundamental distinction we might make is whether the answer is finite or infinite. If it is finite, we can then ask, “How many?”

What if the number is infinite? The German mathematician Georg Cantor (1845–1918) invented a theory of different sizes of infinity. The smallest infinity is called “countably infinite,” and it occurs when the infinite set can be put into one-to-one correspondence with the set of all counting numbers 1, 2, 3, …. Can more be said when an infinite set is...

• Chapter 8 NONSINGULAR CUBIC EQUATIONS
(pp. 116-134)

How to turn a nonsingular cubic curve into an abelian group is not that hard to describe geometrically. The hard part, which we will not perform fully, is verifying that all of the axioms described in chapter 7 are satisfied.

By now, we have arranged our definitions so that any line intersects the curve defined by a cubic equation in 3 points, provided that we use the definition of “intersect” that we have constructed. Our goal in this chapter is to make use of some consequences of that fact.

We begin with a nonsingular cubic curveE, defined over a...

• Chapter 9 SINGULAR CUBICS
(pp. 135-151)

Let$\overline {C}$be a projective cubic curve defined by the homogeneous equationG(X,Y,Z) = 0. In this section, we will be working with the part of$\overline {C}$which lies in the finite plane, so in this chapter we return to using the notation$\overline {C}$to stand for the projective curve, andCto represent the part of$\overline {C}$, that lies in the finite plane. We will suppose that the coefficients of the equation and the allowable values ofX,Y, andZare elements of a fieldK. We will further suppose that the characteristic ofK...

• Chapter 10 ELLIPTIC CURVES OVER Q
(pp. 152-158)

In chapter 8, we took an equation of the formy² =x³ +Ax+B, and discussed how to turn the set of solutions to this equation, along with$\cal{O}$, into a group. In that chapter, our examples mostly used coefficients in finite fields. That finiteness allowed us to specify the group law completely by listing all of the elements of the group. In this chapter, we turn instead to the case where the coefficients and unknowns are taken to be rational numbers. The first important theorem says that the group of rational solutions is finitely generated....

8. PART III. ELLIPTIC CURVES AND ANALYSIS
• Chapter 11 BUILDING FUNCTIONS
(pp. 161-180)

The number theory discussed in this book concerns counting the number of solutions to systems of equations of various kinds. For example, in Part I, we counted the number of solutions to a system of two polynomial equations. We discussed Bézout’s Theorem, which tells us how many solutions to expect when we count properly.

In other contexts, as we will see shortly, we indulge in an infinite sequence of counts. For example, a problem may depend on a parameter that takes on the values 0, 1, 2, …. The result of our counting is then an infinite sequence of counts....

• Chapter 12 ANALYTIC CONTINUATION
(pp. 181-198)

Think of a polynomial in one variable—we will usef(x) = 2x² − 4x+ 1 as an example—and make a table as follows (see table 12.1): List a few entries for consecutive integer values ofx. Underneath the first row, containing the values off(x) for those consecutive integersx, list the row of successive differences. For example, under the −1 in the first row, we list −2 because that is the difference of −1 and the following value 1. Now repeat this process again on the second row, and then the third row, and so on....

• Chapter 13 L-FUNCTIONS
(pp. 199-214)

The concept and notation for anL-function seem to go back to Lejeune Dirichlet, in his famous paper from 1837, “Beweis eines Satzes über die arithmetische Progression.” In this paper, he proved that if you start with two positive integersaandbthat share no common prime factor, then the sequence

$a, a \, + \, b, a \, + 2b, a \, + \, 3b, a \, + 4b, \dots$

contains infinitely many prime numbers. We won’t describe the proof here; it uses a function that Dirichlet denoted byLand which was later calledL(χ,s). We will describe this function and some of its more general cousins at the end of this chapter.

There are...

• Chapter 14 SURPRISING PROPERTIES OF L-FUNCTIONS
(pp. 215-224)

TheL-functions defined in the preceding chapter are not obviously functions of the complex variables, although we told you they were. For example, letEbe an elliptic curve defined over Q. To state the BSD Conjecture, it is essential to viewL(E,s) as an actual function, not just a formal series, so that we can analytically continue it at least to the point where we can evaluate it ats= 1. As you will see in chapter 15,s= 1 is where all the action takes place.

We will see in this section that the...

• Chapter 15 THE CONJECTURE OF BIRCH AND SWINNERTON-DYER
(pp. 225-244)

In this chapter, we will look at cubic equationsy² =x³ +Ax+BwhereAandBare integers, and we will ask: How many rational solutions are there to this equation? Answering this question is the goal we’ve been aiming at all along.

Let’s fixAandB. Then we are looking for rational numbersxandysolving that equation:y² =x³ +Ax+B. At first, that doesn’t seem so hard. All we need to solve it is to find a rational numberxandysolving that equation:y² =x³...

• EPILOGUE
(pp. 245-248)

We could have explained the Birch–Swinnerton-Dyer Conjecture in many fewer pages than we have used in this book. We wanted to move at a leisurely pace and give a bigger and more detailed picture. The BSD Conjecture has its natural context within the larger scope of modern algebraic geometry and number theory. Our story exemplifies how number theoretic problems that are very elementary to state can easily give rise to new and important mathematical ideas. Some of the methods we’ve explained were perhaps not obvious but after you get used them, they seem to be natural. As some philosopher...

9. Bibliography
(pp. 249-250)
10. Index
(pp. 251-253)