{"id":{"repo_id":"whiterose","oai_identifier":"oai:etheses.whiterose.ac.uk:1695"},"canonical_url":"https://search.dev.ndltd.org/etd/whiterose/oai:etheses.whiterose.ac.uk:1695","repository":{"repo_id":"whiterose","name":"White Rose University Consortium","base_url":"https://etheses.whiterose.ac.uk/cgi/oai2"},"display":{"title":"Hyperdefinable groups and modularity","abstract":"In this thesis is presented a study of groups of the form G/G^{00}, 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^{00} 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^{00} becomes definable. We obtain that M' is interdefinable with a real closed valued field M_w, and that 1-basedness 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 M-points 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.","abstract_html":"In this thesis is presented a study of groups of the form G/G^{00}, 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^{00} 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^{00} becomes definable. We obtain that M' is interdefinable with a real closed valued field M_w, and that 1-basedness 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 M-points 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.","abstract_has_math":false,"creators":["Penazzi, Davide"],"institution":"University of Leeds","degree_name":"Ph.D","degree_level":"doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Pillay, A."],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-01","date_published":"2011-01","updated_at":"2026-07-24T06:05:10Z","subjects":[],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["uk.bl.ethos.557360"],"render_values":[{"text":"uk.bl.ethos.557360","href":null,"code":true}]}]},"links":{"outbound_url":null,"outbound_label":null,"outbound_source":null},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Pillay, A."]},{"key":"dc:creator","label":"Author","values":["Penazzi, Davide"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-01"]},{"key":"dc:date.issued","label":"Date","values":["2011-01"]},{"key":"dc:publisher.commercial","label":"Dc Publisher Commercial","values":["University of Leeds"]},{"key":"dc:publisher.department","label":"Dc Publisher Department","values":["School of Mathematics (Leeds)"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Leeds"]},{"key":"dc:relation.isreferencedby","label":"Dc Relation Isreferencedby","values":["https://etheses.whiterose.ac.uk/id/eprint/1695/"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["Ph.D"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["uk.bl.ethos.557360"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://etheses.whiterose.ac.uk/id/eprint/1695/1/thesis_maincorr.pdf"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["In this thesis is presented a study of groups of the form G/G^{00}, 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^{00} 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^{00} becomes definable. We obtain that M' is interdefinable with a real closed valued field M_w, and that 1-basedness 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 M-points 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."]},{"key":"dc:format","label":"Dc Format","values":["text"]},{"key":"dc:title","label":"Title","values":["Hyperdefinable groups and modularity"]}]}],"canonical_facts":{"dc:contributor.advisor":["Pillay, A."],"dc:creator":["Penazzi, Davide"],"dc:date":["2011-01"],"dc:date.issued":["2011-01"],"dc:description.abstract":["In this thesis is presented a study of groups of the form G/G^{00}, 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^{00} 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^{00} becomes definable. We obtain that M' is interdefinable with a real closed valued field M_w, and that 1-basedness 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 M-points 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."],"dc:format":["text"],"dc:identifier":["uk.bl.ethos.557360"],"dc:identifier.uri":["https://etheses.whiterose.ac.uk/id/eprint/1695/1/thesis_maincorr.pdf"],"dc:publisher.commercial":["University of Leeds"],"dc:publisher.department":["School of Mathematics (Leeds)"],"dc:publisher.institution":["University of Leeds"],"dc:relation.isreferencedby":["https://etheses.whiterose.ac.uk/id/eprint/1695/"],"dc:title":["Hyperdefinable groups and modularity"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["doctoral"],"dc:type.qualificationname":["Ph.D"]},"updated_at":"2026-07-24T06:05:10Z"}