CM Points on Varieties
In full generality, the André-Oort Conjecture claims the following:
Conjecture (André-Oort):Let S be a Shimura variety, and
Z an irreducible algebraic subvariety of S containing a Zariski-dense
subset of special points. Then Z is a subvariety of Hodge type.
I will not give the general definitions here, as I'm not interested in this level of
generality. Instead, I am rather interested in the case where S=A^n
is just affine n-space, which is a Shimura variety when viewed as a
moduli space of n-tuples of elliptic curves, the tuple (E_1,...,E_n)
corresponding to the point (j(E_1),...,j(E_n)). Then the special points are those
(complex-valued) points for which all of the corresponding elliptic curves
have complex multiplication,
and are called CM points. Lastly, a subvariety Z of A^n is
of Hodge type if it is the product of CM points and modular curves. In our situation
they are also called modular subvarieties. One can also view a modular subvariety
as one which is defined in terms of isogeny relations between the coordinates.
In this case the conjecture is known if n=2 - first proved by Bas Edixhoven,
assuming the Generalised Riemanna Hypothesis (GRH) for imaginary quadratic fields, and
then proved unconditionally by Yves André. More recently, it has been proved
for all n by Edixhoven, but again under the assumption that GRH holds.
My main result was to translate the above results of Edixhoven into characteristic
p: One replaces elliptic curves by rank 2 Drinfeld modules, and the usual
modular curves by Drinfeld modular curves. Then my most general result so far is the
following. Consider A^n as the moduli space of n-tuples of
rank 2 Drinfeld A-modules, where A=F_q[T] is the polynomial
ring over the field of q elements. Then
Theorem. Suppose that q is odd. Let S be the product of
n Drinfeld modular curves, and Z an irreducible algebraic
subvariety of S. Then S contains a Zariski-dense subset of CM
points if and only if S is modular (i.e. the product of CM points and
My thesis can be downloaded
If you have any specific questions, just mail me.
This is not really a serious research interest at the moment, but it is something cute that
I worked on during my undergraduate years. It was useful then because it gave me experience
in working out stuff on my own.
Ducci sequences got started with the following problem, posed and solved in the 1930's.
Suppose you have a 4-tuple of integers (a,b,c,d) and form a new 4-tuple by taking absolute
differences: (|a-b|, |b-c|, |c-d|, |d-a|). Now keep repeating the process, obtaining a
sequence of 4-tuples. Prove that you always obtain the tuple (0,0,0,0) after a finite number
There are many generalizations of this, the earliest appearing in the first paper on the
topic (in which it is mentioned that the problem originated with some Professer E.Ducci, hence
the name), namely that the result is true for n-tuples if and only if n is a power of 2.
Since then quite a bit has been published on this topic, mostly in the Fibonacci Quarterly.
My most recent contribution was to define Ducci sequences over a finite abelian group G as a sequence
of n-tuples u_i=(x_1,...,x_n), x_j in G, generated by the relation
u_(i+1) = (x_1+x_2, x_2+x_3, ..., x_n+x_1). I proved analogues of most of the known Ducci results
in this context. I also have a number of unpublished results, including a proof of a generalized
version of "Richman's Theorem", which was alluded to in some Ducci paper, but apparently never
appeared in print, which relates the periods of Ducci sequences to the multiplicative orders
of certain elements in a finite field. Once I have some more results in this area I might
actually write them up.
I know of at least two other people currently interested in Ducci sequences, and who have
recently contributed to the theory:
Diana Thomas, and
If there are any other Ducci-enthusiasts out there, I'd always appreciate an email.
(My address is flo(a)math.jussieu.fr - with "(a)" meaning "@", of course. I don't
explicitely write it as I'm trying to avoid spam).
F.Breuer, J.M.Robson. Strategy and Complexity of the Game of Squares.
Bull.London.Math.Soc 30 (1998): 274-282.
F.Breuer. A Note on a Paper by Glaser and Schöffl.
The Fibonacci Quarterly 36.5 (1998): 463-466.
F.Breuer. Ducci Sequences Over Abelian Groups.
Comm.Alg 27.12 (1999): 5999-6013.
F.Breuer. Heights of CM Points on Complex Affine Curves.
The Ramanujan Journal 5 (2001): 311-317.
Also appeared as
no269 (November 2000)
F.Breuer. Distinguished Liftings and the André-Oort Conjecture.
Quaestiones Mathematicae 25(3) (2002): 363-380.
F.Breuer. La conjecture d'André-Oort pour le produit de deux
coubres modulaires de Drinfeld. To appear in C.R.Acad.Sci.Paris, Math.
F.Breuer. The André-Oort conjecture for products of Drinfeld
modular curves. Submitted. AODrinfeld1.ps
F.Breuer, A.B. van der Merwe. Ducci Sequences over Finite Fields.
F.Breuer.The André-Oort conjecture and Drinfeld modular curves.
Also available as
Jussieu Prepublication no324 (Avril 2002)
In this paper I prove a characteristic p analogue of the André-Oort conjecture
for the product of two Drinfeld modular curves. A word of warning, though: there are a some mistakes in
this preprint (all of which have been corrected in my thesis, see also my CRAS paper), in particular Lemma 7.13
is false, though its use can be circumvented, and the correct proof is not very different.
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Last modified 6 October 1999.