## 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 xr+ys=zt. 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 xr+ys=zt 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.

 1r+23 = 32 (r>6) 132+73 = 29 27+173 = 712 25+72 = 34 35+114 = 1222 177+762713 = 210639282 14143+22134592 = 657 338+15490342 = 156133 438+962223 = 300429072 92623+153122832 = 1137

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 xr+ys=zt. 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 xr+ys=zt 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.

### Rational points on algebraic varieties, diophantine equations

While my work up until now has been restricted to rational points on curves of higher genus and problems that can be reduced to such questions, I am very much interested in other cases, especially questions about rational and integer points on higher dimensional varieties. For now, my contributions fall mainly in
• 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.

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• 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.

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• Selmer group computations. The rank of selmer groups can be used to bound the rank of abelian varieties. My contributions in this area are just computational in nature. I have written a program for 2-isogeny descent on elliptic curves over algebraic number fields with class number 1, but this will be extended in the near future to a program for general 2-descent on elliptic curves over arbitrary number fields.