# Elliptic Tales

## Curves, Counting, and Number Theory

## Avner Ash, Robert Gross

- 280 pages
- English
- ePUB (mobile friendly)
- Available on iOS & Android

# Elliptic Tales

## Curves, Counting, and Number Theory

## Avner Ash, Robert Gross

## About This Book

A look at one of the most exciting unsolved problems in mathematics today Elliptic Tales describes the latest developments in number theory by looking at one of the most exciting unsolved problems in contemporary mathematicsâthe Birch and Swinnerton-Dyer Conjecture. In this book, Avner Ash and Robert Gross guide readers through the mathematics they need to understand this captivating problem.The key to the conjecture lies in elliptic curves, which may appear simple, but arise from some very deepâand often very mystifyingâmathematical ideas. Using only basic algebra and calculus while presenting numerous eye-opening examples, Ash and Gross make these ideas accessible to general readers, and, in the process, venture to the very frontiers of modern mathematics.

## Frequently asked questions

## Information

**PART I**

## DEGREE

*Chapter 1*

**DEGREE OF A CURVE**

*Road Map*

*degree*is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two different definitions of the degree of an algebraic curve. Our job in the next few chapters will be to show that these two different definitions, suitably interpreted, agree.

*y*

^{2}=

*x*

^{3}â

*x*or

*y*

^{2}=

*x*

^{3}+ 3

*x*as our examples.

#### 1. Greek Mathematics

*algebraic curve*âthat 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 âprimerâ of these results may be found in the

*Elements*(Euclid, 1956). Because 1 and 2 are the lowest degrees, the Greeks were very successful in this part of algebraic geometry. (Of course, they thought only of geometry, not of algebra.)

*doubling the cube*and

*trisecting angles*. Both of these are problems of degree 3, the same degree as the elliptic curves that are the main subject of this book. Doubling the cube requires solving the equation

*x*

^{3}= 2, which is clearly degree 3. Trisecting an angle involves finding the intersection of a circle and a hyperbola, which also turns out to be equivalent to solving an equation of degree 3. See Thomas (1980, pp. 256â261, pp. 352â357, and the footnotes) and Heath (1981, pp. 220â270) for details of these constructions. Squaring the circle is beyond any tool that can construct only algebraic curves; the ultimate reason is that

*Ï*is not the root of any polynomial with integer coefficients.

#### 2. Degree

*x*+

*y*= 0,

*x*

^{2}+

*y*

^{2}= 1, and

*y*

^{2}=

*x*

^{3}â

*x*â 1, respectively. On the other hand, as we will see, the sine curve cannot be described by an algebraic equation.

*y*

^{2}=

*x*

^{3}â

*x*. As we can see in figure 1.2, the graph of this equation has two pieces.

*r*can be described by the equation

*a*and

*b*, and you set

*x*=

*a*,

*y*=

*b*, and

*a*

^{2}+

*b*

^{2}â€

*r*

^{2}.)

*x*,

*y*,

*z*) = (

*t*, 5 â 2

*t*,

*t*),

*t*can be any number.

*y*

^{2}=

*x*

^{3}â

*x*, we see that its solution set includes (0, 0), (1, 0), and (â1, 0), but it is difficult to see what the entire set of solutions is.

*algebraic equations*. That means by definition that both sides of th...