We studied the physical meaning of tidal force singularities in non-relativistic spacetimes. Typical examples of such spacetimes include Lifshitz spacetimes, Schr\"{o}dinger spacetimes and hyperscaling violation spacetimes. First I will discuss the extension of singularity-free hyperscaling violation geometry. To understand the physical meaning of singularity in the deep non-relativistic IR bulk, I will calculate string scattering amplitudes to find a field theory interpretation of bulk singularity. Since geometric quantities like singularities or horizons are not physical observables in higher spin theory, we will discuss whether it is possible to resolve such singularities in non-relativistic spacetimes from higher spin theory context. We will show singularity resolution cannot be performed in $2+1$ dimensional higher spin theory. Finally, we will give an explicit construction of Schr\"{o}dinger spacetime solutions in $3+1$ dimensional higher spin theory.

We study problems related to the metric of a Riemannian manifold with a particular focus on certain cohomogeneity one metrics. In Chapter 2 we study a set of cohomogeneity one Einstein metrics found by A. Dancer and M. Wang. We express these in terms of elementary functions and nd explicit sectional curvature formulae which are then used to determine sectional curvature asymptotics of the metrics. In Chapter 3 we construct a non-standard parametrix for the heat kernel on a product manifold with multiply warped Riemannian metric. The special feature of this parametrix is that it separates the contribution of the warping functions and the heat data on the factors; this cannot be achieved via the standard approach. In Chapter 4 we determine explicit formulae for the resolvent symbols associated with the Laplace Beltrami operator over a closed Riemannian manifold and apply these to motivate an alternative method for computing heat trace coecients. This method is entirely based on local computations and to illustrate this we recover geometric formulae for the heat coecients. Furthermore one can derive topological identities via this approach; to demonstrate this application we nd explicit formulae for the resolvent symbols associated with Laplace operators on a Riemann surface and recover the Riemann-Roch formula. In the nal chapter we report on an area of current research: we introduce a class of symbols for pseudodi erential operators on simple warped products which is closed under composition. We then extend the canonical trace to this setting, using a cut - o integral, and nd an explicit formula for the extension in terms of integrals over the factor.

Modular degree is an interesting invariant of elliptic curves. It is computed by variety of methods. After computer calculations, Watkins conjectured that given E/IQ of rank R, 2R I deg(<I», where <I> : Xo(N) ---+ E is the optimal map (up to isomorphism of E) and deg(<I» is the modular degree of E. In fact he observed that 2R+K divides the degree of the modular degree and 2K depends on #W, where W is the group of Atkin-Lehner involutions, #W=2w(N), N is the conductor of the elliptic curve and w(N) counts the number of distinct prime factors of N. The goal of this thesis is to study this conjecture. We have proved that 2R+K I deg( <I»would follow from an isomorphism of complete intersection of a universal deformation ring and a Hecke ring, where 2K = #W', the cardinality of a certain subgroup of the group of Atkin-Lehner involutions. I attempt to verify 2R+K I deg(<I» for certain Ellipitic Curves E, when N is 250 by using a computer algebra package Magma. I have verified when N is squarefree. Tables are in Section 5.1.

The possible model theoretic compactifications of a cover of the muliplicative group of a field are discussed. The members of a large class of such compactifications are shown to be covers of toric varieties. The structure of the members of this class is shown to be Analytic Zariski on a dense open subset. Some equivalences between model theoretic notions and those of toric geometry are established. An elimination of imaginaries result is proved for the theory of the covers. The notion of universality for a class is introduced. Some connections with mirror symmetry are discussed.

No description available.

A singularity is said to be weakly-exceptional if it has a unique purely log terminal blow up. In dimension 2, V. Shokurov proved that weakly exceptional quotient singularities are exactly those of types Dn, E6, E7, E8. This thesis classifies the weakly exceptional quotient singularities in dimensions 3, 4 and 5, and proves that in any prime dimension, all but finitely many irreducible groups give rise to weakly exceptional singularities. It goes on to provide an algorithm that produces such a classification in any given prime dimension.

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.

On étudie l'existence de métrique à courbure scalaire hermitienne constante sur des variétés presque-Kähler obtenues par lissage d'orbifolds Kähler à courbure scalaire riemannienne constante et à singularités A1. On démontre que si un tel orbifold n'a pas de champs de vecteurs holomorphes (non triviaux) alors un lissage presque Kähler (Mє, ωє) admet une structure presque-Kähler à courbure scalaire hermitienne constante. De plus, on démontre que pour є > O assez petit, les (Mє, ωє) sont toutes symplectiquement équivalentes à une variété symplectique fixée (M , ω) qui possède un cycle évanescent admettant un représentant Hamiltonien stationnaire pour la structure presque complexe associée. / We study the existence of metrics of constant Hermitian scalar curvature on almost-Kähler manifolds obtained as smoothings of a constant scalar curvature Kähler orbifold, with A1 singularities. More precisely, given such an orbifold that does not admit nontrivial holomorphie vector fields, we show that an almost-Kähler smoothing (Mє, ωє) admits an almost-Kähler structure (Jє, gє) of constant Hermitian curvature. Moreover, we show that for є > O small enough, the (Mє, ωє) are all symplectically equivalent to a fixed symplectic manifold (M , ω) in which there is a surface S homologous to a 2-sphere, such that [S] is a vanishing cycle that admits a representant that is Hamiltonian stationary for gє .

A consequence of the Birch and Swinnerton-Dyer conjecture is that the parity of the rank of abelian varieties is expected to be given by their global root numbers. This is known as the parity conjecture. Assuming the finiteness of the Shafarevich-Tate groups, the parity conjecture is equivalent to the p-parity conjecture for all prime p, that is the p∞ Selmer rank is expected to be given by the global root number. In this thesis we study the parity of the 2∞ Selmer rank of Jacobians of hyperelliptic curves of genus 2 defined over number fields. This forces us to assume the existence of a Richelot isogeny (the analogue of a 2-isogeny for elliptic curves) to provide an expression for the parity of their 2∞ Selmer rank as a sum of local factors Λv modulo 2. Based on a joint work with T. and V. Dokchitser and A. Morgan on arithmetic of hyperelliptic curves over local fields, this makes the parity of the 2∞ Selmer rank of such semistable Jacobians computable in practice. By introducing a set of polynomial invariants in the roots of the defining polynomials of the underlying curves of a specific family of Jacobians, we provide an expression for the local discrepancy existing between the local factors Λv and the local root numbers, and prove the 2-parity conjecture in this case. One outcome of this result it that, using the theory of regulator constants, one can lift the assumption on the existence of an isogeny and prove the parity conjecture for a class of semistable Jacobians of genus 2 curves assuming finiteness of their Shafarevich-Tate group.

In this document we formulate and discuss conjecture 1.2.1, giving an upper bound for the number of lines on K3 surfaces occurring as complete intersections of three quadrics in P5. In the case that these quadrics contain in their span a quadric of rank 4, we construct a pair of elliptic fibrations, each of which realises the lines on the surface as either sections or line components within the singular fibres, and the general fibre is realised as an intersection of two quadrics in P3. The possibilities for singular fibres are limited by the Euler number of the surface, while the rank of the group of sections is bounded by the rank of its Picard group. In the cases where this rank is low, these bounds are enough to prove the stated conjecture in the torsion-free case by utilising the height-pairing. In the remaining cases, if a surface has more lines than the stated conjecture, we discuss how these techniques can be used to construct necessary conditions on the configurations of the lines on the surface, along with an example of how this could work in practice.