Final Report Summary - EMRCC (Effective methods in rigid and crystalline cohomology)
* Please, notice that this summary may be published
The purpose of the project was to develop and implement new methods for performing certain computations in number theory. Such computations are motivated from two directions. First, there are many unsolved problems within number theory and new
light can be thrown upon them by doing explicit computations. Second, over the last
few decades number theory has shown itself to be applicable, most notably to cryptography, and to apply number theory one needs good algorithms. Let me illustrate the former by describing the most spectacular outcome of the project, the discovery of new formulae for “points on elliptic curves”.
Elliptic curves are arguably the most beautiful structures in all of mathematics. They are defined by equations which are just complicated enough to endow them with immensely rich arithmetic properties: any simpler and the arithmetic would become somewhat dull, and surprisingly the arithmetic of more complicated curves is also less interesting. Concretely an “elliptic curve” is defined by a particular type of equation, and by “arithmetic” we mean the solutions of this equation in whole numbers or rational fractions. A simple example is the equation Y2 + Y = X3 + X2 with the solution P = (X,Y) = (0,0). This solution looks rather boring, but what makes elliptic curves so special is that there is a classical geometric method for “multiplying” any solution by a whole number to give further solutions. There is a simple recipe for this, and following this one finds, for example, the solution 7 × P = (21,-99). It turns out that P is not boring at all, since by multiplying it by one whole number after another one generates an infinite sequence of whole number or fractional solutions which are all different. We call P a solution of “infinite order”.
A classical problem in number theory is to construct in an explicit manner whole number or fractional solutions, what one calls points on the elliptic curve. A breakthrough on this was achieved by Kurt Heegner in the 1950s in his solution of a famous problem due to Gauss. Heegner gave an explicit construction of such points, and P = (0,0) on the curve above is an example. In the 1990s Henri Darmon found a different construction, giving a formula for a new type of point of infinite order on an elliptic curve. These two constructions though stood alone though as the fruits of many centuries (arguably millennia) of research on elliptic curves.
Working with Henri Darmon and Victor Rotger, together we developed an entirely new approach and in the closing months of the project uncovered a treasure trove of such formulae, giving points of infinite order on elliptic curves in many new situations. Our work was highly experimental, and used original methods for computing with important objects in number theory called "modular forms" which the PI had developed earlier on in the project. Our discoveries were a delightful conclusion to a stimulating five years of research.
The purpose of the project was to develop and implement new methods for performing certain computations in number theory. Such computations are motivated from two directions. First, there are many unsolved problems within number theory and new
light can be thrown upon them by doing explicit computations. Second, over the last
few decades number theory has shown itself to be applicable, most notably to cryptography, and to apply number theory one needs good algorithms. Let me illustrate the former by describing the most spectacular outcome of the project, the discovery of new formulae for “points on elliptic curves”.
Elliptic curves are arguably the most beautiful structures in all of mathematics. They are defined by equations which are just complicated enough to endow them with immensely rich arithmetic properties: any simpler and the arithmetic would become somewhat dull, and surprisingly the arithmetic of more complicated curves is also less interesting. Concretely an “elliptic curve” is defined by a particular type of equation, and by “arithmetic” we mean the solutions of this equation in whole numbers or rational fractions. A simple example is the equation Y2 + Y = X3 + X2 with the solution P = (X,Y) = (0,0). This solution looks rather boring, but what makes elliptic curves so special is that there is a classical geometric method for “multiplying” any solution by a whole number to give further solutions. There is a simple recipe for this, and following this one finds, for example, the solution 7 × P = (21,-99). It turns out that P is not boring at all, since by multiplying it by one whole number after another one generates an infinite sequence of whole number or fractional solutions which are all different. We call P a solution of “infinite order”.
A classical problem in number theory is to construct in an explicit manner whole number or fractional solutions, what one calls points on the elliptic curve. A breakthrough on this was achieved by Kurt Heegner in the 1950s in his solution of a famous problem due to Gauss. Heegner gave an explicit construction of such points, and P = (0,0) on the curve above is an example. In the 1990s Henri Darmon found a different construction, giving a formula for a new type of point of infinite order on an elliptic curve. These two constructions though stood alone though as the fruits of many centuries (arguably millennia) of research on elliptic curves.
Working with Henri Darmon and Victor Rotger, together we developed an entirely new approach and in the closing months of the project uncovered a treasure trove of such formulae, giving points of infinite order on elliptic curves in many new situations. Our work was highly experimental, and used original methods for computing with important objects in number theory called "modular forms" which the PI had developed earlier on in the project. Our discoveries were a delightful conclusion to a stimulating five years of research.