Given a subgroup B of a finitely-generated abelian group A, the saturation B of B is defined to be the largest subgroup of A containing B with finite index. In this thesis we consider a crucial step in the determination of the Mordell-Weil group of an elliptic curve, E(K). Methods such as Descent may produce subgroups H of E(K) with [H:H] > 1. We have determined an algorithm for calculating H given H, and hence for completing the process of finding the Mordell-Weil group. Our method has been implemented in MAGMA with two versions of the programs; one for general number fields K and the other for Q. It builds upon previous work by S. Siksek. Our problem splits into two. First we can use geometry of numbers arguments to establish an upper bound N for the index [H:H]. Second for each remaining prime p < N we seek to prove either that H is p-saturated, i.e. p|[H:H], or to enlarge H by index p. To solve the first problem, 1. We have devised and implemented an algorithm that searches for points on E(K) up to a specified naive height bound. 2. We have devised and implemented an algorithm that calculates the subgroup Egr(K) of points with good reduction at specified valuations. 3. We have implemented joint work with S. Siksek and J. Cremona to calculate an upper bound on the difference of the canonical and naive height of points on an elliptic curve. 4. We have helped to devise and have implemented joint work with S. Siksek and J. Cremona to calculate a lower bound on the canonical heights of non-torsion points on E(K) with K a totally real field. To solve the second problem, 1. As in earlier work by Siksek, we use homomorphisms to prove p-saturation for primes p. We however use the Tate-Lichtenbaum pairing, and we show that, using this pairing, our method will always prove H is p-saturated if that is the case. 2. We show that Siksek's original method will fail for some curves.

We study the L-values of an elliptic curve twisted by an Artin representation. Specifically, we consider the case in which the representation factors through a false Tate curve extension of Q. First, we consider a semistable elliptic curve E; we construct an integral-valued p-adic measure which interpolates the values the L-values of an Artin twist of E, at a family of finite-order character twists. To do this, we exploit the fact that such an L-value may be written as the Rankin convolution of two Hilbert modular forms, when the representation factors through the false Tate curve extension. Recent developments in non-abelian Iwasawa theory predict certain strong congruences between these p-adic L-functions, and we shall establish weakened versions of these congruences. Next, we prove analogous results for an elliptic curve with complex multiplication; we do this using work of Hida and Tilouine on the p-adic interpolation of Hecke L-functions over a CM-field. We go on to investigate the ratio of the automorphic and motivic periods associated to E in this setting. We describe how the p-valuation of this ratio may be explicitly calculated, and use the computer package MAGMA to produce some numerical examples. We end by proving a formula for the growth of this quantity in terms of the Iwasawa invariants associated to the two-variable extension of the CM-field.

In this thesis we discuss scalar field theories, and their applications to gravity. We provide a summary of why there is interest in modifying Einstein’s General Relativity, and discuss why scalar fields make a good candidate for a modification to make. We demonstrate their effects on the dynamics of matter, and discuss the necessity of screening mechanisms in order for these scalar fields to not be ruled out by current observations. We present discussion on two screening mechanisms in particular, the Chameleon and Vainshtein mechanisms. We then present work that aims to study the soft behaviour of scattering amplitudes belonging to single scalar field theories. We generalise current techniques in the literature such that the study of a much wider set of theories is possible. We use this technique to perform a detailed study of a particular family of theories, a so called (1, 2) theory, and demonstrate that the DBI symmetry is the unique way to enhance the soft behaviour of the scattering amplitudes of this family. We also identify the special Galileon as the unique way to maximally enhance the soft behaviour within the (1, 2) class, and verify the validity of recursion techniques to calculate scattering amplitudes. We then move on to studying the Chameleon in more detail. We provide motivation for modifying its high energy behaviour by studying the ‘surfer solution’, and use this to propose the DBI-Chameleon. We demonstrate that this theory avoids the problems the Chameleon suffers in the early Universe and forms a good effective field theory in this regime. Finally we present a UV complete theory describing a massive Galileon, and study its dynamics to verify if it exhibits Vainshtein screening. Theories with Vainshtein screening are usually unable to be UV completed in a Wilsonian way. We present our preliminary findings which suggest screening is possible for at least some parameter values.

Lorentz symmetry is arguably the most fundamental symmetry of physics, at least in its modern conception. On the other hand, some of the issues that plague the currently accepted theory of gravitation could be solved by breaking such symmetry. The theory proposed by Petr Horava in 2009 brings forward exactly this aspect. The theory, dubbed Horava gravity, is a UV complete theory of gravity that is also renormalisable. It represents therefore a good candidate for a quantum theory of gravity. There are some issues though, which typically arise in any theory which explicitely violates Lorentz symmetry. In this thesis we will be concerned with two of these issues, in particular the matter problem and the existence of black holes. The first issue mentioned arises every time we try to couple matter to a Lorentz violating theory of gravity. Indeed, in the matter sector Lorentz symmetry is extremely well constrained, and therefore we need to find a way to avoid the percolation of Lorentz violations to the matter sector through higher order operators. One possible solution based on the separation of scales was proposed in the last few years (Pospelov et al.,2010). While studying the proposed mechanism though, the authors uncovered a naturalness problem in the vector sector of the theory. The solutions they proposed relies on the use of some higher derivative terms that are not normally present in the ``traditional'' Horava theory. It is unclear then what impact this type of terms can have on the whole theory. In our work we precisely addressed this question. We analysed the perturbations around Minkowski of the most generic theory extended to these type of terms, both from the point of view of the stability of the theory and of the renormalisability. What we found is that the theory retains its renormalisability, but some instabilities occur in the scalar sector. More work is hence required in order to understand whether such instabilities could be tamed, or if the mixed derivatives should be abandoned in favour of some alternative solution, not presently available. The second theme we concentrated on is that of the existence of black holes. The definition of black hole in general relativity rely strongly on the causal structure dictated by Lorentz symmetry. As soon as Lorentz symmetry is broken it is therefore unclear whether black holes will still exist. Surprisingly enough black holes have been shown to exist in Lorentz breaking theories, but a rigorous definition was still to be found. In our work we developed the mathematically rigorous definitions for the causal structure of foliated spacetimes and we defined for the first time black holes in such spacetimes. We also uncovered a number of interesting properties of this objects and we developed a local characterisation that allows one to locate horizons without the knowledge of the whole structure of the complete spacetime. Finally we developed the Initial Value Problem for these types of theory in the hope that new simulations of gravitational collapse will be performed using our analysis as a starting point. The thesis is organised as follows. In the first Chapter we give an introduction on gravity and the problems with its renormalization. We also introduce some of the theories that have been proposed to solve this difficulties. In the second Chapter we start discussing Lorentz violations and we provide a proof of the power-counting renormalizability of a toy model of a Lorentz violationg scalar field theory. We also introduce the theories that we will be studying throughout the thesis. In the third Chapter we discuss the mixed derivative extension to Horava gravity and we discuss the consequences of the new terms that occur in the theory. In the fourth and fifth Chapters we introduce the causal structure of spacetimes which violate Lorentz symmetry by means of a preferred foliation, we discuss the notion of black holes and horizons and we formalise some results present in the literature adapting them to our framework. In the sixth Chapter we then discuss the Initial Value Problem for such spacetimes, with some attention to the process of gravitational collapse leading to the formation of black holes. Finally in the last Chapter we draw some conclusions and discuss some ideas for future work.

Dynamically screened scalar field theories form an attractive collection of models that were introduced to drive the late-time expansion of our universe. A common consequence of these theories is a screening mechanism which leads to the suppression of the fifth-forces mediated by the scalar field in sufficiently dense environments. This enables the models within this class of theories to avoid conflict with the stringent results from local tests of gravity, without the need for any fine tuning. The prototypical example of a dynamically screened scalar field is the chameleon model, for which screening arises due to the mass of the scalar responding to the local density. It has been recently demonstrated that atom interferometry is a powerful technique for constraining such scalars, with near future experiments capable of probing a large portion of the model parameter space. The nature of screening however means that closing in on what remains of the chameleon parameter space is going to become increasingly more difficult. This work aims to address this issue by examining the intricacies of how the chameleon field responds to the configuration of an atom interferometry experiment, where it is found that the non-linearities governing the theory can ultimately be harnessed in order to improve the prospects of detection.