Points on algebraic curves over function fields, primes in arithmetic progressions : beyond Bombieri-Pila and Bombieri-Vinogradov theorems

Abstract : E.Bombieri and J.Pila introduced a method to bound the number of integral points in a small given box (under some conditions). In algebraic part we generalise this method to the case of function fields of genus $0$ in ove variable. Then we apply the result to count the number of elliptic curves falling in the same isomorphic class with coefficients lying in a small box.Once we are done the natural question is how to improve this bound for some particular families of curves. We study the case of elliptic curves and use the fact that the necessary part of Birch Swinnerton-Dyer conjecture holds over function fields. We also use the properties of height functions and results about sphere packing.In analytic part we give an explicit version of Bombieri-Vinogradov theorem. This theorem is an important result that concerns the error term in Dirichlet's theorem in arithmetic progressions averaged over moduli $q$ up to $Q$. We improve the existent result of such type given in cite{Akbary2015}. We reduce the logarithmic power by using the large sieve inequality and Vaughan identity.
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Alisa Sedunova. Points on algebraic curves over function fields, primes in arithmetic progressions : beyond Bombieri-Pila and Bombieri-Vinogradov theorems. Number Theory [math.NT]. Université Paris-Saclay, 2017. English. ⟨NNT : 2017SACLS178⟩. ⟨tel-01585244⟩

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