Notes prepared by
Professor Marcus du Sautoy, University of Oxford, author of Finding Moonshine: The Music of the Primes.

About Marcus du Sautoy

John Tate wins the Abel Prize 2010

If one measured the influence of a mathematician by the number of mathematical ideas that bear their name then John Tate would be a clear winner. Tate cohomology, the Tate duality theorem, Barsotti-Tate groups, the Tate motive, the Tate module, Tate’s algorithm, the Néron-Tate height, Mumford-Tate groups, the Tate isogeny theorem, the Honda-Tate theorem for abelian varieties over finite fields, Serre-Tate deformation theory, Tate-Shafarevich groups, and the Sato-Tate conjectures. Indeed mathematicians have coined a new concept: the Tate index defined as the time it takes to give a talk on number theory before you mention the name Tate. In general this is a very small number.

Even Tate’s PhD was so influential that “Tate’s thesis” has become a byword for the techniques he introduced that revolutionized modern number theory. But it is the new insights that these ideas provide into some of the oldest problems on the mathematical books for which Tate receives the Abel prize.

Tate’s thesis: making sense of the nonsensical

Consider the following equation:

Most people would say that if this equation is at the heart of the award of this year’s Abel prize then surely mathematicians have finally lost it. The famous Indian mathematician Ramanujan wrote to several English mathematicians with his discovery of this strange formula and fully recognized its apparent absurdity as he wrote in the accompanying letter:

But one Cambridge mathematician GH Hardy recognized that this calculation was not nonsense. In fact it represents one of the most startling breakthroughs made by one of the greats of mathematics: Bernhard Riemann. Tate’s thesis provided a new mathematical perspective on this equation that vastly generalised Riemann’s achievement and gave it a context that has allowed mathematicians to extend the ideas to many other areas of number theory.

At their heart all these tools have been developed to try to understand some of the seemingly simplest yet most enigmatic and important numbers in mathematics: the primes.

Prime numbers are the indivisible numbers, numbers like 17 and 23. They are so important to mathematics because all other numbers are built from the primes. If you take a number like 105 then this number is built by multiplying the primes 3, 5 and 7 together: 105=3x5x7. They are like the atoms of arithmetic, the hydrogen and oxygen of the world of mathematics.

The periodic table of chemical elements is probably the most fundamental object in chemistry from which everything else is built. But mathematicians are still missing a periodic table of the primes. The ancient Greeks proved 2000 years ago that there are infinitely many of these numbers. So gone is the chance just to record them in some huge table. Instead we must look for patterns, for some underlying structure or logic to the way the primes are laid out through the universe of numbers.

This challenge goes to the heart of what mathematics is all about. Mathematicians like John Tate are pattern searchers trying to uncover the logic that underpins the world around us. But the challenge of the primes is perhaps the ultimate problem for the pattern searcher. One of the skills of a great mathematician is to offer a new perspective on an old problem. It was the German mathematician Bernhard Riemann who introduced the new mathematics which is at the heart of some of John Tate’s great contributions.

Riemann exploited the idea of something called a zeta function to uncover the secret DNA which controls the behaviour of the primes. A function is like a machine into which you feed a number, the machine calculates away to output another number. Riemann’s zeta function essentially takes in two numbers which can be thought of as defining a coordinates on a map and outputs an answer which is like the height of the landscape above that point on the map. So the zeta function creates a landscape whose contours Riemann discovered hold the secrets to the way the primes are laid out.

The trouble is that the formula for Riemann’s zeta function makes sense only when you input numbers where the first coordinate, the east-west location on the map, is greater than 1. When you put in numbers to the west of this line running through the map the output spirals off to infinity. But Riemann discovered a way to make sense of how to continue this landscape and this is what the formula

represents. The left hand side is telling you that the height in the landscape at a point on the map located at (-1,0) should spiral off to infinity. The right hand side is what Riemann realised the height should be if this landscape is continued. His analysis also revealed a certain symmetry in this landscape which we call a functional equation.

Riemann had to introduce some unnatural looking mathematics to achieve his feat. But it was Tate’s thesis that gave a new perspective that showed why these extra pieces were totally natural. His proof provides such a high degree of clarity to how this machinery works that it has been applied to many other sorts of zeta functions beyond that of Riemann’s. The idea of a zeta function has been used by many number theorists to capture different arithmetic structures. The zeta function transforms arithmetic into a geometric object or landscape whose contours hold the secrets of the arithmetic structure you’re interested in.

The new perspective that Tate introduced in his thesis meant that the same technique could be used on many other sorts of zeta functions which capture different structures. For example, Dirichlet, a contemporary of Riemann, used the idea of a zeta function to tackle the following problem:

Variants of Riemann’s zeta function allowed Dirichlet to capture these special types of primes.

Although mathematicians just study numbers for the love of them, there are now very serious practical reasons why any new knowledge about prime numbers could have a significant impact on the economy because prime numbers are the keys to the codes that keep all the secrets secure as they travel across the internet.

Here’s a little challenge: 126619 is not a prime number but built out of two smaller primes multiplied together. Can you find the primes which built this 6 digit number. It’s like having a molecule of salt and trying to work out what are the atoms, the sodium and chlorine that built this molecule. In the case of 126619 if you manage to find the primes 127 and 997 did the job then you’ve in fact just cracked a modern day internet code. Because cracking a number into its prime building blocks is precisely the same as cracking a code. However Internet companies are using slightly larger numbers with 200 digits or more to protect credit cards.

The tools that Tate and others have introduced to understand prime numbers could ultimately have very serious real world applications in showing us ways to crack these codes. But if these prime number codes are ever cracked then there are new codes hiding in the wings that depend on another bit of mathematics upon which Tate has had a big impact.

Elliptic curves: finding patterns in numbers

Ever since we’ve been playing with numbers mathematicians have been exploring relationships between those numbers. For example if you take the numbers 3,4 and 5 then the square of the first two can be added together to give the square of the third:

This is an example of a solution to Pythagoras’s famous equation:

which expresses the relationship between the lengths of three sides of a right angled triangle. Finding a solution where the lengths are all whole numbers was very useful for the ancient Egyptians because a rope marked off with these three lengths could be used to ensure that buildings were built with true right angles, essential if you are going to build a structure like the pyramids.

But is 3, 4 and 5 the only choice of whole numbers that has this relationship? The Egyptians weren’t the only ancient culture interested in solving this famous equation. A tablet in the Babylonian collection in Columbia University called Plimpton 322 lists fifteen other triples of whole numbers that satisfy this equation including the following example:

The ancient Greeks realised that there are infinitely many solutions with whole numbers and Diophantus in the third century AD wrote down a formula to generate all of them in a systematic fashion. This success led to mathematicians considering a whole range of different equations in the search for whole number solutions.

For example Archimedes challenged mathematicians in Alexandria to solve the following equations about counting cows:

Of course cows cannot be cut in half so the solution the mathematician is after is to find whole numbers which satisfy the conditions laid down by Archimedes. The mathematicians of Alexandria failed to find the number of cows which with hindsight was not surprising. A general solution was discovered for the first time in 1880 and the smallest number of cows that will satisfy Archimedes’ conditions is of the order of 7.76 x 10²⁰⁶⁵⁴⁴. There are only estimated to be 10⁸⁰ atoms in the observable universe!

It wasn’t until the dawn of the computer that the number of cows was finally written out in all its gory details. At its heart is the challenge to find whole number solutions (x,y) to equations of the following shape:

where n is a number. The eighteenth century mathematician Euler named these Pell equations after the seventeenth century English mathematician John Pell. As with many attributions in mathematics, other names are probably more worthy of being attached to these equations. For example the Indian mathematician Brahmagupta studied them extensively in the 7^th century AD. They are particularly interesting because whole number solutions can be used to make fractions x/y that give good approximations to the square root of n.

The machinery developed by John Tate has been instrumental in getting to grips with a slight variation of these equations. Changing the quadratic in x to a cubic equation creates a class of equations called elliptic curves. Despite such a small change the challenge of solving these equations has stumped generations of mathematicians. Modern number theory has been concerned with the question of finding whole number solutions or even fractions that satisfy equations like the following:

There are a multitude of different elliptic curves depending on the equation used but they all essentially look like Y²=X³+aX+b where each curve corresponds to choosing different values for a and b.

One of the central questions concerning these sorts of equations is to find values of X and Y where both X and Y are whole numbers or fractions which satisfy the equation. For example setting X=3 and Y=5 in the equation Y²=X³-2 works because 5²=25=27-2=3³-2. Are there more solutions? For this equation it turns out there are infinitely many pairs of whole number or fractions that satisfy this equation. But change the equation to Y²=X³-43X+166 and suddenly there are only seven pairs of numbers that fit: (X,Y)=(0,0), (3,8), (3,-8), (-5,16), (-5,-16),(11,32),(11,-32).

One of the central questions about these equations is the following: How can you tell which equations have infinitely many rational solutions and which only finitely many?

Tate has been responsible for creating sophisticated machinery that has helped us to investigate the mysteries of these elliptic curves. For example the Tate-Shaferevich group gives us a measure of how many solutions found in different sorts of numbers called p-adic numbers can be used to create solutions which are ordinary fractions.

The amazing thing is that the machinery that Riemann developed to understand primes can also be used to understand solutions to these equations. In particular certain landscapes defined by L-functions, variants of the zeta function for understanding primes, have points in them hiding treasure which explains the solutions of these elliptic curves. One of the most important conjectures in mathematics, the Birch-Swinnerton-Dyer Conjecture, speculates that there is one point in these landscapes whose height will tell you whether an elliptic curve has infinitely many points where both coordinates are fractions? Tate’s ideas have been central to our attempts to uncover that treasure.

When the conjecture was first made by the two British mathematicians Bryan Birch and Peter Swinnerton-Dyer it was not known how to continue the landscape to even define the height at the critical point they had identified. As Tate once commented “This remarkable conjecture relates the behavior of an L-function at a point where it is not at present known to be defined to the order of a group (the Tate-Shaferevich group) which is not known to be finite!'” But thanks to the proof of Fermat’s Last Theorem and techniques similar to those of Tate’s thesis we can now extend the landscape and reach the point where x marks the spot. But we still are battling to survey the contours of this point and reveal the treasure that we hope is buried there.

You might say who cares? But increasingly the mathematics of elliptic curves is being used by mobile phones, smart cards as well as air traffic control systems to protect our secrets. In these new codes using elliptic curves, your credit card number or message is converted by clever maths onto a point on this curve. To encrypt the message, the mathematics moves the point around to another point using the geometry underlying these curves. To undo this geometric procedure requires cracking some mathematics that currently we can’t do.

For thousands of years we’ve been wrestling with questions about numbers that are as old as mathematics itself. The numbers 1,2,3… are the simplest yet most intractable concepts of mathematics. Just as the telescope allowed astronomers to see new worlds, Tate’s mathematics has provided tools and insights which have allowed the mathematicians of this generation to see further into the universe of numbers than ever before. He truly deserves the title of the Galileo of number theory.

Notes prepared by Marcus du Sautoy, Simonyi Professor for the Public Understanding of Science and Professor of Mathematics at the University of Oxford.

For more details on the story of zeta functions to understand primes consult The Music of the Primes (Harper Perennial).

email: dusautoy@maths.ox.ac.uk

tel: 0044-7958-049484

Some more mathematical details of finding points on elliptic curves

The Y²=X³-2 equation defines a curve that can be drawn on a piece of graph paper. Put in a value of X and calculate the equation X³-2; then take its square root to get the corresponding value of Y. For example, put X=3 then X³-2= 27-2=25. To get Y take the square root of 25 (since Y2=X³-2) so Y is 5 or -5 (since minus times minus is plus there are always two square roots). Notice that the graph you get is symmetrical about the horizontal axis because all the square roots have a mirror root which is negative.

We found a very nice point on this elliptic curve because when you put X equal to a whole number (X=3), the corresponding Y was also a whole number (Y=5 or -5). Are there any other points like this? Let’s try putting X=2. First, calculate X³-2=8-2=6, which means that in this case Y=√6 or -√6. In the first example 25 had a whole number square root but the square root of 6 is not so tidy. In fact the Ancient Greeks proved that there is no fraction, let alone whole number, that when you square it the result is 6. √6 written as a decimal number races off to infinity with no pattern at all:

One of the biggest open problems of mathematics relates to finding the points on this curve where both the X and the Y are fractions. Most of the time they aren’t because when you put in X, the Y will not be a fraction since most numbers don’t have a nice square root. We were lucky to find that X=3, Y=5 was a nice point on the curve, but are there any more?

The Ancient Greeks came up with a beautiful piece of geometry which showed how to get more whole points (X,Y) where both X and Y are fractions once you’ve found one. Draw a line which just touches the first point you’ve found (it mustn’t go through it but just be at the correct angle to glance the curve). We call this the tangent to the curve. By extending this line one finds that it will cut the curve at a new point. The exciting discovery is that the coordinates of this new point will also both be fractions.

For example, if we take the point (X,Y)=(3,5) on the elliptic curve Y²=X³-2, draw in the line touching this point, it will intersect at a new point (X,Y)=(129/100, 383/1000) where both coordinates are fractions. With this new point we could repeat the procedure and get another point where both X and Y are fractions: (2340922881/45427600,93955726337279/306182024000).

Without this bit of geometry it would be very tough to discover that feeding in the fraction X=2340922881/45427600 will give you a Y that is also a fraction.

In this example you can keep repeating this bit of geometry and get infinitely many pairs of fractions (X,Y) that are points on this curve. For a general elliptic curve Y²=X³+aX+b, if you’ve got one point (X₁,Y₁) on the curve where both X₁ and Y₁ are fractions then setting

X₂= ((3X₁²+a)²-8X₁Y₁²)/4Y₁²
Y₂=(X₁⁶+5aX₁⁴+20bX₁³-5a²X₁²-4abX₁-a³-8b²)/ 8Y₁³

will give you another point on the curve where both X₂ and Y₂ are fractions.

For our curve Y²=X³-2 this generates infinitely many points on the curve where both X and Y are fractions, but there are other curves where it’s impossible to get infinitely many points. For example, take the curve defined by the following equation

On this curve it turns out that there are only a finite number of points where both X and Y are fractions:

In fact they all have whole number coordinates. Trying to use the geometry trick or algebra to get more points with fractions just delivers one of these seven points again.

It is machinery like that of the ancient Greeks that Tate has been so adept at creating that has helped us to investigate the mysteries of these elliptic curves.

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