We study analytic aspects of the Dedekind zeta function of a Galois extension. Specifically, we are interested in its mean values. In the first part of this thesis we give a formula for the second moment of the Dedekind zeta function of a quadratic field times an arbitrary Dirichlet polynomial of length $T^{1/11-\epsilon}$. In the second part, we derive a hybrid Euler-Hadamard product for the Dedekind zeta function of an arbitrary number field. We rigorously calculate the $2k$th moment of the Euler product part as well as conjecture the $2k$th moment of the Hadamard product using random matrix theory. In both instances we are restricted to Galois extensions. We then conjecture that the $2k$th moment of the Dedekind zeta function of a Galois extension is given by the product of the two. By using our results from the first part of this thesis we are able to prove both conjectures in the case $k=1$ for quadratic fields. We also re-derive our conjecture for the $2k$th moment of quadratic Dedekind zeta functions by using a modification of the moments recipe. Finally, we apply our methods to general non-primitive $L$-functions and gain a conjecture regarding their moments. Our main idea is that, to leading order, the moment of a product of distinct $L$-functions should be the product of the individual moments of the constituent $L$-functions.

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In the first part of this thesis various problems in diophantine approximation are considered, which generalize well known theorems of Dirichlet and Kronecker. A brief survey is presented in the first chapter, including a discussion on the scope of elementary methods. It is demonstrated here that stronger results are possible by elementary means than have previously been obtained. In the subsequent chapters non-elementary methods are used. Results are proved for fractional parts of quadratic forms in several variables which improve upon previous work. New theorems are demonstrated for the distribution modulo one of 'almost all" additive forms in many variables, including the particularly interesting case of a linear form in positive variables. In chapter four new bounds are given for exponential sums over primes, which greatly improve upon the work of I.M. Vinogradov. Some applications to diophantine approximation problems involving primes are given in chapters 4 and 5, the latter chapter also improving upon previous work on the problem of a linear form in three prime variables. In the second section, topics in multiplicative number theory are discussed. It is shown that almost-primes are very well distributed in almost all very short intervals, improving upon previous work by a considerable factor. Sieve methods are then employed to tackle three other problems. New results are in this way obtained for primes in short intervals, for the distribution of the square roots of primes (modulo one), and for the distribution of [alpha] modulo one for irrational [alpha]. This last chapter contains a new method for tackling sums over primes which has other applications.

After introducing basic notation and results in chapter one, we begin studying the model theory of the Peano axioms, P, proper in the second chapter where we give a proof of Rabin's theorem :- that P is not axiomatizable by any consistent set of [sigma]n sentences for any n[epsilon][omega], and also answer a question of Gaifman raised in Another problem, from the same article, is partially answered in chapter three, where we show every countable non-standard model, M, 'of P has an elementary equivalent end extension solving a Diophantine equation with coefficients in M, that was not solvable in M. In chapter four we investigate substructures of countable non-standard models of P, and show that every such model M, contains 2 substructures all isomorphic to M. Other related results are also proved. Chapter five contains theorems on indescernibles and omitting certain types in models of P. Chapter six is concerned with the following problem the set), of elementary substructures of M, is lattice ordered by inclusion. Which lattices are of the form for some We show that the pentagon lattice is of this form (answering a question suggested in [7] p. 280)and produce a class of non-modular lattices all of whose members are not of the form for any M = N, the standard model of P. Elementary co-final extensions of models of P are also investigated in this chapter. Finally, chapter seven concludes the thesis by posing some open problems suggested by the preceding text.

The theory of overconvergent modular symbols, developed by Rob Pollack and Glenn Stevens, gives a beautiful and effective construction of the p-adic L-function of a modular form. In this thesis, we develop the theory of overconvergent modular symbols over a completely general number field and use it to construct p-adic L-functions for automorphic forms for GL2. In particular, we prove control theorems that say that the natural specialisation map from overconvergent to classical modular symbols is an isomorphism on the small slope subspaces, hence attaching a unique overconvergent modular symbol to a small slope cuspidal automorphic eigenform .Φ. From this overconvergent symbol we then obtain a p-adic distribution that interpolates certain critical L-values of .Φ. The text is comprised of two largely independent parts. In the first, we develop the theory in concrete detail over imaginary quadratic fields, and in the process present a constructive definition of the p-adic L-function in this setting. In the second, which was joint work with Daniel Barrera Salazar (Université de Montréal), we provide an analogous theory over general number fields, though not in the same explicit detail.

This thesis studies Grothendieck-Witt spectra of quadric hypersurfaces. In particular, we compute Witt groups of quadrics. Besides, by calculating Grothendieck-Witt groups of a deleted quadric over an algebraically closed field of characteristic different from 2, we improve Dugger and Isaksen’s condition (some powers of 2 dividing some binomial coefficients) on an old problem Hurwitz concerning the existence of sums-of-squares formulas.

This thesis focuses on three decision problems in the mapping class groups of surfaces, namely the reducibility, pseudo-Anosov and conjugacy problems. For a fixed surface, we use ideal triangulations to model both its mapping class group and space of measured laminations. This allows us to state these problems combinatorially. We give new solutions to each of these problems that, unlike the existing solutions which are based on the Bestvina–Handel algorithm, run in polynomial time when given a suitable certificate. This allows us to show that in fact each of these problems lies in the complexity class NP∩ co-NP instead of just EXPTIME. At the heart of each of our solutions is the maximal splitting sequence of a projectively invariant measured lamination, as described by Agol. The complexity of this sequence bounds the difficulty of determining many of the properties of such a lamination, including whether it is filling. In Chapter 4 we give explicit polynomial upper bounds on the periodic and preperiodic lengths of such a sequence. This allows us to construct the running time bounds needed to show that these problems lie in NP∩co-NP. We finish with a discussion of an implementation of these algorithms as part of the Python package flipper. We include several examples of properties of mapping classes that can be computed using it.

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In this thesis we prove several results in analytic number theory. 1. We show that there exist 3-digit palindromic primes in base b for a set of b having density 1 and that if b is sufficiently large then there is a $3$-digit palindrome in base b having precisely two prime factors. 2. We prove various estimates for averages of sums of Kloosterman fractions over primes. The first of these improves previous results of Fouvry-Shparlinski and Baker. 3. By using the q-analogue of van der Corput's method to estimate short Kloosterman sums we study the divisor function in an arithmetic progression to modulus q. We show that the expected asymptotic formula holds for a larger range of q than was previously known, provided that q has a certain factorisation. 4. Let ‖x‖ denote the distance from x to the nearest integer. We show that for any irrational α and any ϴ< 8/23 there are infinitely many n which are the product of two primes for which ‖nalpha‖ ≤ n ^-ϴ. 5. By establishing an improved level of distribution we study almost-primes of the form f(p,n) where f is an irreducible binary form over Z. 6. We show that for an irreducible cubic f ? Z[x] and a full norm form $mathbf N$ for a number field $K/Q$, satisfying certain hypotheses, the variety $$f(t)=mathbf N(x_1,ldots,x_k) e 0$$ satisfies the Hasse principle. Our proof uses sieve methods.

Pour une représentation automorphe cuspidale de GL(2,F) avec F un corps de nombres totalement réel, tel que est de type (k, r) et satisfait une condition de pente non critique, l’on construit une distribution p-adique sur le groupe de Galois de l’extension abélienne maximale de F non ramifiée en dehors de p et 1. On démontre que la distribution obtenue est admissible et interpole les valeurs critiques de la fonction L complexe de la représentation automorphe. Cette construction est basée sur l’étude de la cohomologie de la variété modulaire de Hilbert à coefficients surconvergents. / For each cohomological cuspidal automorphic representation for GL(2,F) where F is a totally real number field, such that is of type (k, r) tand satisfies the condition of non critical slope we construct a p-adic distribution on the Galois group of the maximal abelian extension of F unramified outside p and 1. We prove that the distribution is admissible and interpolates the critical values of L-function of the automorphic representation. This construction is based on the study of the overconvergent cohomology of Hilbert modular varieties.