Hyperdefinable groups and modularity

• Main Author: Penazzi, Davide
• Other Authors: Pillay, A.
• Published: University of Leeds 2011
• Subjects: 510
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.557360

In this thesis is presented a study of groups of the form G/G , where G is a 1-dimensional, definably compact, definably connected, definable group in a saturated real closed field M, with respect to a notion called 1-basedness. In particular G will be one of the following: 1. ([-1,1),+ mod 2) 2. ([1/b,b),*mod b 2 3. (SO_2(M)*) and truncations 4. (E(M) 0,+) and truncations, where E is an elliptic curve over M, where a truncation of a linearly or circularly ordered group (G,*) is a group whose underlying set is an interval [a,b) containing the identity of G, and whose operation is *mod(b*a {-1}). Such groups G/G are only hyperdefinable, i.e., quotients of a definable group by a type-definable equivalence relation, in M, and therefore we consider a suitable expansion M' in which G/G becomes definable. We obtain that M' is interdefinable with a real closed valued field M_w, and that 1-basedness of G/G is related to the internality of G/G to either the residue field or the value group of M_w. In the case when G is the semialgebraic connected component of the M-points of an elliptic curve E, there is a relation between the internality of G/G to the residue field or the value group of M_w and the notion of algebraic geometric reduction. Among our results is the following: If G = E(M) 0, the expansion of M by a predicate for G is interdefinable with a real closed valued field M_w and G/G is internal to the value group of M_w if and only if E has split multiplicative reduction; G/G is internal to the residue field of M_w if and only if E has good reduction or nonsplit multiplicative reduction.