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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!)
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
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.
Since the ABC-conjecture suggests that the complete list is finite,
one may wonder if the known solutions are actually all solutions. This
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.
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
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.
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.
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