Penazzi, Davide (2011) Hyperdefinable groups and modularity. PhD thesis, University of Leeds.

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Abstract
In this thesis is presented a study of groups of the form G/G^{00}, where G is a 1dimensional, definably compact, definably connected, definable group in a saturated real closed field M, with respect to a notion called 1basedness. 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^{00} are only hyperdefinable, i.e., quotients of a definable group by a typedefinable equivalence relation, in M, and therefore we consider a suitable expansion M' in which G/G^{00} becomes definable. We obtain that M' is interdefinable with a real closed valued field M_w, and that 1basedness of G/G^{00} is related to the internality of G/G^{00} to either the residue field or the value group of M_w. In the case when G is the semialgebraic connected component of the Mpoints of an elliptic curve E, there is a relation between the internality of G/G^{00} 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^{00} is interdefinable with a real closed valued field M_w and G/G^{00} is internal to the value group of M_w if and only if E has split multiplicative reduction; G/G^{00} is internal to the residue field of M_w if and only if E has good reduction or nonsplit multiplicative reduction.
Item Type:  Thesis (PhD) 

Academic Units:  The University of Leeds > Faculty of Maths and Physical Sciences (Leeds) > School of Mathematics (Leeds) 
Depositing User:  Ethos Import 
Date Deposited:  07 Oct 2011 08:54 
Last Modified:  07 Mar 2014 11:21 
URI:  http://etheses.whiterose.ac.uk/id/eprint/1695 