| Izvleček: | <p>The purpose of this thesis is to provide model-theoretic structures to study Shimura varieties. Two main problems are studied.</p> <p><b>Transcendence Properties of <em>j</em></b>. This part is aimed at proving a general transcendence property for the <em>j</em> function in the spirit of Schanuel’s conjecture. Such a result was proven for the exponential function by M. Bays, J. Kirby and A. Wilkie, and it relies heavily on the Ax-Schanuel theorem. An analogue of Ax-Schanuel for the j function was proven by J. Pila and J. Tsimerman, and this allowed to adapt the strategy used for the exponential to <em>j</em>.</p> <p>One concrete consequence of the main transcendence result is:</p> <p>Theorem. Let 𝜏 ϵ R be j-generic. Suppose z<sub>1</sub>,...,z<sub>n</sub> ϵ H<sup>+</sup> and g ϵ GL<sub>2</sub>(Q(𝜏 ))<sup>+</sup> are such that z<sub>1</sub>,...,z<sub>n</sub>; gz<sub>1</sub>,...,gz<sub>n</sub> are in different GL<sub>2</sub>(Q)<sup>+</sup>-orbits (pairwise). Then:</p> <p>t.d.(j(z); j'(z); j"(z); j(gz); j'(gz); j"(gz)/𝜏) ≥ 3n:</p> <p>The term <em><em>j</em>-generic</em> is analogous to that of being exponentially transcendental. Using o-minimality, we prove that all but countably many values for 𝜏 can be used.</p> <p>We also analyse a modular version of Schanuel’s conjecture and prove that there are at most countably many essential counterexamples to this conjecture. <p><b>Categoricity of Shimura Varieties.</b> Let p : X<sup>+</sup> → S(C) be a Shimura variety. We associate with it a countable two-sorted first-order language and analyse the complete first-order theory it determines. The question of interest in this part of the thesis of whether or not this theory is categorical. One immediate issue is that the theory cannot determine the size of the fibres of the map between the sorts, so we restrict our analysis to the models with the smallest possible fibres. <p>Categoricity for Shimura curves was established by C. Daw and A. Harris. From their result, it is immediate that some form of arithmetic open image condition is needed for the categoricity of general Shimura varieties. Our goal is to postulate the correct open image condition and prove that it is equivalent to categoricity. This open image condition is not known in general, however, using known cases of the Mumford-Tate conjecture, we obtain unconditional categoricity of two well-known Shimura varieties: A2 and A3, the moduli spaces of principally polarised abelian varieties of dimension 2 and 3, respectively.</p></p></p>
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