Gunther Cornelissen works on deformation of group actions on curves, non-archimedean uniformization of curves, automorphic forms for function fields of curves and links between undecidability and arithmetic geometry.
Arithmetic geometry in positive characteristic
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Let us provisionally call an algebraic curve the solution set to a polynomial equation in two variables. How many symmetries can it have? How can one deform the curve (vary the coefficients) and still preserve the symmetries? Over the complex numbers, one can give an upper bound only depending on a topological invariant of the curve, the so-called genus g (if g>1, then the bound is linear in g). Over fields of positive characteristic (e.g., finite fields), the bound is no longer linear, and curves can have very many symmetries. The curve (xp-x)(yp-y)=1, for example, has genus g=(p-1)2 and 2p(p-1)=2g1/2(g1/2+1)2 automorphisms (given by interchanging x and y , by adding an element from the finite field F with p elements to x or y and by multiplying x and y by two non-zero elements from F that are inverse to each other). One can deform the coefficients as follows: (xp-x)(yp-y)=c for some non-zero constant c and still have a curve, not isomorphic to the original one, but with the same automorphism group. Observe that forc=0, this curve degenerates into a ``chess board'' picture x(x-1)...(x-p+1)y(y-1)...(y-p+1)=0. See the picture on the left. |
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Joint research
with Fumiharu Kato (Kyoto University) and Ariane Mézard (Paris Orsay)
has focused on the general problem: given an algebraic curve over a field
of positive characteristic, can one bound its number of automorphisms in
terms of geometrically relevant information (such as ``ordinarity''), determine
its formal deformation space, and determine parametrized families of curves
with the same automorphisms.
Such an algebraic curve over a finite field has a function field K. Studying K is in many ways similar to studying number fields. In the 1970, a new level of sophistication came to the field when Drinfeld introduced elliptic modules. These objects lead to a function field analogue of modular forms, modular curves, the theorem of Shimura-Taniyama-Weil, etc. In this context, we are studying Eisenstein series (location of the zeros, relations), group actions on modular curves and its use in realizing Galois groups over function fields and more recently spaces of automorphic forms related to the zeta-function. On the left: poster
of the workshop `Automorphisms of Curves
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Non-archimedean
uniformization, rigid analytic geometry
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Just
like Riemann surfaces of genus
>1 can be uniformized by the complex
upper half plane, i.e., written as a quotient G\H, where Gis
some group of Moebius transformations, one can write certain algebraic
curves over non-archimedean valued fields as similar quotients and study
their properties by using group theory. Instead of the complex upper half
plane, one gets a combinatorial object, the so-called Bruhat-Tits tree.
Joint work with Fumiharu Kato (Kyoto) has focused on the theory of symmetries
of curves from this point of view. At the very basis are such concepts
as ``p-adic platonic solids''. The theory (esp. when combined with
the algebraic theory of symmetries from above) turns out to be a mathematical
analogue of the holography principle in quantum gravitation.
On the left: a 2-adic icosahedron by F.Kato, which also appeared on the cover of the AMS Notices. |
Diophantine sets and Hilbert's tenth problem
Hilbert's tenth problem asks whether there is an algorithm to decide whether or not a diophantine equation (polynomials equation with integer coefficients, such as Fermat's equation) has a solution in integers or not. It was shown in the nineteen seventies that this is impossible by Davis, Matijasevich, Putnam and Robinson.
Joint research with Karim Zahidi (Antwerp) focuses on the variant of this problem where one asks for the existence of a solution in rational numbers of not. The answer is unknown, but we have proven that ways of proving a negative answer also lead to a negative answer to a conjecture by Bary Mazur about the topological nature of the set of rational solutions to diophantine equation, and we have found intimate links between a more general decision problem for rational numbers and the arithmetic of elliptic curves, in particular, elliptic divisibility sequences.
-> go to a lecture about Hilbert's
tenth problem by Gunther Cornelissen, view Hilbert's orginal problem and
listen to a short radio speech by Hilbert at the
American
Institute for Mathematics.
Applications of our research
On the left: poster of the workshop "Arithmetic Geometry and High Energy
Physics (Lorentz Center, Leiden) (2005)", organised by Gunther
Cornelissen, Matilde Marcolli and Andrew Waldron.