O-minimality, nonclassical modular functions and diophantine problems
<p>There now exists an abundant collection of conjectures and results, of various complexities, regarding the diophantine properties of Shimura varieties. Two central such statements are the Andre-Oort and Zilber-Pink Conjectures, the first of which is known in many cases, while the second is...
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| Format: | Thesis |
| Language: | English |
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2018
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| _version_ | 1797062925065650176 |
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| author | Spence, H |
| author2 | Pila, J |
| author_facet | Pila, J Spence, H |
| author_sort | Spence, H |
| collection | OXFORD |
| description | <p>There now exists an abundant collection of conjectures and results, of various complexities, regarding the diophantine properties of Shimura varieties. Two central such statements are the Andre-Oort and Zilber-Pink Conjectures, the first of which is known in many cases, while the second is known in very few cases indeed. </p> <p>The motivating result for much of this document is the modular case of the Andre-Oort Conjecture, which is a theorem of Pila. It is most commonly viewed as a statement about the simplest kind of Shimura varieties, namely modular curves. Here, we tend instead to view it as a statement about the properties of the classical modular j-function. It states, given a complex algebraic variety V, that V contains only finitely many maximal special subvarieties, where a special variety is one which arises from the arithmetic behaviour of the j-function in a certain natural way.</p> <p>The central question of this thesis is the following: what happens if in such statements we replace the j-function with some other kind of modular function; one which is less well-behaved in one way or another? Such modular functions are naturally called nonclassical modular functions. This question, as we shall see, can be studied using techniques of o-minimality and point-counting, but some interesting new features arise and must be dealt with.</p> <p>After laying out some of the classical theory, we go on to describe two particular types of nonclassical modular function: <em>almost holomorphic modular functions</em> and <em>quasimodular functions</em> (which arise naturally from the derivatives of the j-function). We go on to prove some results about the diophantine properties of these functions, including several natural Andre-Oort-type theorems, then conclude by discussing some bigger-picture questions (such as the potential for nonclassical variants of, say, Zilber-Pink) and some directions for future research in this area.</p> |
| first_indexed | 2024-03-06T20:52:30Z |
| format | Thesis |
| id | oxford-uuid:38147ede-511d-4c5e-abba-657c2cbfb4f3 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T20:52:30Z |
| publishDate | 2018 |
| record_format | dspace |
| spelling | oxford-uuid:38147ede-511d-4c5e-abba-657c2cbfb4f32022-03-26T13:47:49ZO-minimality, nonclassical modular functions and diophantine problemsThesishttp://purl.org/coar/resource_type/c_db06uuid:38147ede-511d-4c5e-abba-657c2cbfb4f3Number TheoryLogicPure MathematicsEnglishORA Deposit2018Spence, HPila, J<p>There now exists an abundant collection of conjectures and results, of various complexities, regarding the diophantine properties of Shimura varieties. Two central such statements are the Andre-Oort and Zilber-Pink Conjectures, the first of which is known in many cases, while the second is known in very few cases indeed. </p> <p>The motivating result for much of this document is the modular case of the Andre-Oort Conjecture, which is a theorem of Pila. It is most commonly viewed as a statement about the simplest kind of Shimura varieties, namely modular curves. Here, we tend instead to view it as a statement about the properties of the classical modular j-function. It states, given a complex algebraic variety V, that V contains only finitely many maximal special subvarieties, where a special variety is one which arises from the arithmetic behaviour of the j-function in a certain natural way.</p> <p>The central question of this thesis is the following: what happens if in such statements we replace the j-function with some other kind of modular function; one which is less well-behaved in one way or another? Such modular functions are naturally called nonclassical modular functions. This question, as we shall see, can be studied using techniques of o-minimality and point-counting, but some interesting new features arise and must be dealt with.</p> <p>After laying out some of the classical theory, we go on to describe two particular types of nonclassical modular function: <em>almost holomorphic modular functions</em> and <em>quasimodular functions</em> (which arise naturally from the derivatives of the j-function). We go on to prove some results about the diophantine properties of these functions, including several natural Andre-Oort-type theorems, then conclude by discussing some bigger-picture questions (such as the potential for nonclassical variants of, say, Zilber-Pink) and some directions for future research in this area.</p> |
| spellingShingle | Number Theory Logic Pure Mathematics Spence, H O-minimality, nonclassical modular functions and diophantine problems |
| title | O-minimality, nonclassical modular functions and diophantine problems |
| title_full | O-minimality, nonclassical modular functions and diophantine problems |
| title_fullStr | O-minimality, nonclassical modular functions and diophantine problems |
| title_full_unstemmed | O-minimality, nonclassical modular functions and diophantine problems |
| title_short | O-minimality, nonclassical modular functions and diophantine problems |
| title_sort | o minimality nonclassical modular functions and diophantine problems |
| topic | Number Theory Logic Pure Mathematics |
| work_keys_str_mv | AT spenceh ominimalitynonclassicalmodularfunctionsanddiophantineproblems |