Research subjects
Most of my mathematical work has been in number theory and especially in rational points on hyperelliptic curves. Should you have a curve you want to know the rational points on, I'd be happy to take a look at it (I don't promise any results, though!)
ABC-conjecture
In the research for my master's degree, I examined generalisations of the ABC-conjecture (also called the Masser-Oesterlee conjecture) towards more variables. The thesis itself contains a detailed description of Elkies' proof that the ABC-conjecture implies Mordell's conjecture (Faltings' theorem), a generalisation of the construction of n-examples from ABC-examples by Browkin and Brzezinski (giving more ways of doing this) and a list of small, fairly good 4-examples not coming from ABC-examples.
Generalised Fermat equations
In my PhD-thesis, I studied special cases of the diophantine equation x^{r}+y^{s}=z^{t}. There is this remarkable theorem by Darmon and Granville
Theorem (Darmon, Granville): Let r,s,t be positive integers satisfying 1/r+1/s+1/t<1. Then the equation x^{r}+y^{s}=z^{t} has only finitely solutions in pairwise coprime integers x,y,z.
The ABC-conjecture even suggests that all such solutions taken together (i.e., for varying r,s,t) should still form a finite set. There are some highly non-trivial solutions, though. The following list shows all known solutions. The small ones have been known for a long time, while the bigger ones were found by Beukers and Zagier.
1^{r}+2^{3} | = | 3^{2} | (r>6) |
13^{2}+7^{3} | = | 2^{9} | |
2^{7}+17^{3} | = | 71^{2} | |
2^{5}+7^{2} | = | 3^{4} | |
3^{5}+11^{4} | = | 122^{2} | |
17^{7}+76271^{3} | = | 21063928^{2} | |
1414^{3}+2213459^{2} | = | 65^{7} | |
33^{8}+1549034^{2} | = | 15613^{3} | |
43^{8}+96222^{3} | = | 30042907^{2} | |
9262^{3}+15312283^{2} | = | 113^{7} |
Since the ABC-conjecture suggests that the complete list is finite, one may wonder if the known solutions are actually all solutions. This would imply
Conjecture (Tijdeman-Zagier, Beale Prize Problem): Let r,s,t,x,y,z be positive integers with r,s,t>2 and x^{r}+y^{s}=z^{t}. Then x,y,z have a common prime factor.
In my thesis Chabauty methods and covering techniques applied to generalised Fermat equations I show a partial result for this.
Theorem: For (r,s,t)=(2,4,6), (2,6,4), (4,6,2), (2,3,8), (2,8,3), (2,4,5), (2,5,4), all integer solutions to x^{r}+y^{s}=z^{t} with coprime x,y,z can be obtained from the solutions in the list above by inserting minus signs if necessary.
The employed methods first show that such solutions correspond to rational points on algebraic curves of genus > 1. Then the rational points on those curves are obtained by Chabauty methods and covering techniques.
Other Topics
- Chabauty methods. These give a way of determining an upper bound of the number of rational points on a curve of genus > 1. Basically, the problem of determining the rational points on a curve is translated into the problem of determining the intersection of two p-adic analytic varieties. The latter question is often easier to deal with, at least to bound the size of the intersection. The fundamental idea was used by Chabauty in 1941 to partially prove Mordell's conjecture, which was later completely proved by Faltings. In my PhD-thesis, it is shown that these computations are relatively easy to perform if the curve geometrically covers an elliptic curve - i.e. if its jacobian is isogeneous to an abelian variety that over C has some elliptic factors.
- Covering techniques. It may happen in Chabauty-methods that the intersection is not finite. In that case, all is not lost. Covering techniques give a way of covering the rational points of a curve with the rational points of several other curves. Chabauty methods may apply to those curves even if they don't apply to the original one. This idea, first applied by Wetherell to bielliptic curves of genus 2 is presented in a way that is readily applicable to any hyperelliptic curve in my PhD-thesis. Furthermore, it is shown that if the covers are obtained by pulling back along the multiplication-by-2 on the jacobian, the curves cover many elliptic curves (over C at least), so the remark above for Chabauty methods applies.
- Descent. The idea of coverings explained above is a special case of descent. It goes back to Fermat's famous method of infinite descent. Applied to a finite unramified cover D of a curve C, a theorem of Chevalley and Weil guarantees that each rational point on C lifts to one of finitely many twists of D. This is what makes covering techniques work. In the Mordell-Weil theorem, descent is applied to abelian varieties to prove that the group of rational points is finitely generated. Explicit descent allows the computation Selmer groups, which provide upper bounds on the rank of the group of rational points.
- Local-to-Global obstructions. Showing that a diophantine equation does have a solution is usually quite straightforward: You just give the solution. Since the set of rational numbers is enumerable, there is even a procedure that will show that an equation does have a solution in finite time: enumerate all candidates until you find a solution. This procedure obviously will never show that there is no solution. Methods for showing that an equation has no solution are more limited. One way is by showing that the equation already does not have a solution over a bigger (complete) field (e.g., x^{2}+y^{2}=-1 does not have any rational solutions because it does not even have any real ones). Such an equation is said to have a local obstruction to having rational solutions. Some equations have solutions over all completions of Q and yet have no rational solutions. Selmer gave the famous example 3x^{3}+4y^{3}=5z^{3}. Such an equation is said to have a local-to-global obstruction to having rational solutions.
- Mazur's idea on visibility. Local-to-global obstructions play an important role in trying to determine the group of rational points on abelian varieties, such as elliptic curves. Mazur suggested a particularly explicit approach to showing that certain varieties - homogeneous spaces - have a local-to-global obstruction, by realizing them as subvarieties of higher dimensional abelian varieties.