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In this thesis we investigate the Galois representation attached to an algebraic curve over a local field. In particular, we consider elliptic and hyperelliptic curves with very bad reduction, i.e. which acquire good reduction over a wildly ramified extension. It is a well-known fact that the primes at which an elliptic curve may acquire good reduction over a wildly ramified extension are 2 and 3. For such primes, a classification of the restriction to inertia of the Galois representation is well understood in the literature; in particular the image of inertia is a finite group, which can be either cyclic or non-abelian. However, less is known about the full Galois action. In this work we give an explicit and algorithmic description of the Galois representations which occur in these cases, with a particular focus on the curves with non-abelian inertia image. For a hyperelliptic curve of genus g, the primes of wild reduction are at most 2g +1. In this work we consider the family of hyperelliptic curves which have potentially good reduction at 2g +1, assuming this is a prime, and the largest possible image of inertia. We will see how the result on the Galois representation attached to one such curve is a direct generalisation of the corresponding result for elliptic curves.

To each regular algebraic, conjugate self-dual, cuspidal automorphic representation Π of GL(N) over a CM number field E (or, more generally, to a regular algebraic isobaric sum of conjugate self-dual, cuspidal representations), we can attach a continuous ℓ-adic Galois representation r(Π) of the absolute Galois group of E. The residual Galois representation r̅(Π):GalE̅/E)→GL_N(Fℓ) of π is defined to be the semi-simplification of the reduction of r(Π) (modulo the maximal ideal of Z̅ℓ), with respect to any invariant Z̅ℓ-lattice. The aim of this thesis is to prove a level raising theorem for automorphic representations of GL(2n). More precisely, given a regular algebraic automorphic representation Π of GL(2n) over E, which is either unitary, conjugate self-dual and cuspidal or an isobaric sum Π₁ ⊞Π₂ of two unitary, conjugate self-dual cuspidal representations of GL(n), we want to construct a unitary, conjugate self-dual cuspidal representation Π' of GL(2n) that has the same residual Galois representation as Π and whose component at a finite place w of E is an unramified twist of the Steinberg representation. We prove that this is possible, after replacing Π with its base change along a CM biquadratic extension, under certain assumptions on Π (including a local obstruction at the place w). Our proof uses the results of Kaletha, Minguez, Shin and White on the endoscopic classification of representations of (inner forms of) unitary groups to descend Π to an automorphic representation of a totally definite unitary group G over the maximal totally real subfield of E. We then prove a level raising theorem for the group G; we do this by proving an analogue of "Ihara's lemma" for G, using the strong approximation theorem for the derived subgroup of G.

This thesis is concerned with the Langlands program; namely the global Langlands correspondence, Langlands functoriality, and a conjecture of Gross. In chapter 1, we cover the most important background material needed for this thesis. This includes material on reductive groups and their root data, the definition of automorphic representations and a general overview of the Langlands program, and Gross' conjecture concerning attaching l-adic Galois representations to automorphic representations on certain reductive groups G over ℚ. In chapter 2, we show that odd-dimensional definite unitary groups satisfy the hypotheses of Gross' conjecture and verify the conjecture in this case using known constructions of automorphic l-adic Galois representations. We do this by verifying a specific case of a generalisation of Gross' conjecture; one should still get l-adic Galois representations if one removes one of his hypotheses but with the cost that their image lies in CG(ℚl) as opposed to LG(ℚl). Such Galois representations have been constructed for certain automorphic representations on G, a definite unitary group of arbitrary dimension, and there is a map CG(ℚl) → LG(ℚl) precisely when G is odd-dimensional. In chapter 3, which forms the main part of this thesis, we show that G = Un(B) where B is a rational definite quaternion algebra satisfies the hypotheses of Gross' conjecture. We prove that one can transfer a cuspidal automorphic representation π of G to a π' on Sp2n (a Jacquet-Langlands type transfer) provided it is Steinberg at some finite place. We also prove this when B is indefinite. One can then transfer π′ to an automorphic representaion of GL2n+1 using the work of Arthur. Finally, one can attach l-adic Galois representations to these automorphic representations on GL2n+1, provided we assume π is regular algebraic if B is indefinite, and show that they have orthogonal image.

This thesis concerns the arithmetic statistics of even Galois representations. The first chapter shows how level-raising leads to the study of the splitting of primes in number fields that depend on the prime. In addition, the parity of Galois representations is placed in the context of reciprocity in the Langlands program. The second chapter concerns an even, reducible residual Galois representation in even characteristic. By thickening the image with cohomology classes, all lifts of the representation are ensured to be irreducible. The global reciprocity law of Galois cohomology is applied to lift the representation to mod 8, and smooth quotients of the local deformation rings at the primes where the representation is ramified are found. By using the generic smoothness of the local deformation rings at trivial primes and the Wiles-Greenberg formula, a balanced global setting is created, in the sense that the Selmer group and the dual Selmer group have the same rank. The Selmer group is computed explicitly and shown to have rank three. Finally, the distribution over primes of the ranks of Selmer groups in a family of even representations obtained by allowing ramification at auxiliary primes is studied. The infinitude of primes for which the Selmer rank increases by one is proved, and the density of such primes is shown to be 1/192.

Let E be a non-CM elliptic curve defined over Q. Fix an algebraic closure Q of Q. We get a Galois representation ρE : Gal(Q/Q) →GL2( ˆZ) associated to E by choosing a compatible bases for the N-torsion subgroups of E(Q). Associated to an open subgroup G of GL2( ˆZ) satisfying −I ∈G and det(G) = ˆZ×, we have the modular curve (XG ,πG ) over Q which loosely parametrises elliptic curves E such that the image of ρE is conjugate to a subgroup of Gt. In this article we give a complete classification of all such genus 0 modular curves that have a rational point. This classification is given in finitely many families. Moreover, each such modular curve can be explicitly computed.

Let $p$ be an odd prime and $q$ a power of $p$. By the celebrated theorem of Khare and Wintenberger (previously Serre's conjecture), an absolutely irreducible odd $2$-dimensional Galois representation $\bar{\rho}:\operatorname{G}_{\mathbb{Q}}\rightarrow \operatorname{GL}_2(\mathbb{F}_q)$ (satisfying favorable conditions) lifts to a characteristic zero Galois representation associated to a Hecke eigencuspform. Hamblen and Ramakrishna prove the analog of (the weak form of) Serre's conjecture for residually reducible $2$-dimensional Galois representations. A higher dimensional generalization of their result is proved in chapter three. Let $\bar{\rho}:\operatorname{G}_{\mathbb{Q}}\rightarrow \operatorname{GSp}_{2n}(\mathbb{F}_q)$ be a reducible and indecomposable Galois representation which is unramified outside a finite set of primes $S$ and whose image lies in a Borel subgroup. It is shown that if $\bar{\rho}$ satisfies some additional conditions, it lifts to characteristic zero Galois representation which is geometric in the sense of Fontaine-Mazur. \par In chapter four, we examine the problem of lifting a two dimensional Galois representation $\bar{\rho}:\operatorname{G}_{\mathbb{Q},S}\rightarrow \operatorname{GL}_2(\mathbb{F}_q)$ to a cuspidal Hida Family which is isomorphic to the Iwasawa algebra $\Lambda$ via the weight-space map. This was achieved for an odd, ordinary and absolutely irreducible $\bar{\rho}$ by Ramakrishna for a suitable choice of auxiliary local deformation conditions. We show that if $\bar{\rho}$ is reducible and indecomposable, one may indeed lift $\bar{\rho}$ to a Hida family $\mathbb{T}$ such that the image of the weight space map contains a congruence class of weights in $\operatorname{Spec} \Lambda$ modulo $p^2$. This Hida family is in some sense close to $\operatorname{Spec} \Lambda$, more precisely, we show that it represents a deformation functor which is arranged to have a hull isomorphic to $\operatorname{Spec} \Lambda$ (this isomorphism is not via the weight-space map). In chapter five, we apply results of Hamblen-Ramakrishna to construct some special Galois extensions. For every prime $p\geq 5$ for which a certain condition on the class group $\operatorname{Cl}(\mathbb{Q}(\mu_p))$ is satisfied, we construct a $p$-adic analytic Galois extension of the infinite cyclotomic extension $\mathbb{Q}(\mu_{p^{\infty}})$ with some special ramification properties. In greater detail, this extension is unramified at primes above $p$ and tamely ramified above finitely many rational primes and its Galois group over $\mathbb{Q}(\mu_{p^{\infty}})$ is isomorphic to a finite index subgroup of $\operatorname{SL}_2(\mathbb{Z}_p)$ which contains the principal congruence subgroup. For the primes $107,139,271$ and $379$ such extensions were first constructed by Ohtani and Blondeau. The strategy for producing these special extensions at an abundant number of primes is through lifting two-dimensional reducible Galois representations which are diagonal when restricted to $p$.

Let $A$ be a square-free abelian variety defined over a number field $K$. Let $S$ be a density one set of prime ideals $\p$ of $\mathcal{O}_K$. A famous theorem of Faltings says that the Frobenius polynomials $P_{A,\p}(x)$ for $\p\in S$ determine $A$ up to isogeny. We show that the prime factors of $|A(\FF_\p)|=P_{A,\p}(1)$ for $\p\in S$ also determine $A$ up to isogeny over an explicit finite extension of $K$. The proof relies on understanding the $\ell$-adic monodromy groups which come from the $\ell$-adic Galois representations of $A$, and the absolute Weyl group action on their weights. We also show that there exists an explicit integer $e\geq 1$ such that after replacing $K$ by a suitable finite extension, the Frobenius polynomials of $A$ at $\p$ must equal to the $e$-th power of a separable polynomial for a density one set of prime ideals $\p\subseteq\mathcal{O}_K$.

Let $E_{/\mathbf{Q}}$ be an elliptic curve. The Sato--Tate conjecture, now a theorem, tells us that the angles $\theta_p =\cos^{-1}\left(\frac{a_p}{2\sqrt p}\right)$ are equidistributed in $[0,\pi]$ with respect to the measure $\frac{2}{\pi}\sin^2\theta\, d\theta$ if $E$ is non-CM (resp.~$\frac{1}{2\pi} d \theta + \frac 1 2 \delta_{\pi/2}$ if $E$ is CM). In the non-CM case, Akiyama and Tanigawa conjecture that the discrepancy \[ D_N = \sup_{x\in [0,\pi]} \left| \frac{1}{\pi(N)} \sum_{p\leqN} 1_{[0,x]}(\theta_p) - \int_0^x \frac{2}{\pi}\sin^2\theta\, d\theta\right| \] asymptotically decays like $N^{-\frac 1 2+\epsilon}$, as is suggested by computational evidence and certain reasonable heuristics on the Kolmogorov--Smirnov statistic. This conjecture implies the Riemann hypothesis for all $L$-functions associated with $E$. It is natural to assume that the converse (``generalized Riemann hypothesis implies discrepancy estimate'') holds, as is suggested by analogy with Artin $L$-functions. We construct, for compact real tori, ``fake Satake parameters'' yielding $L$-functions which satisfy the generalized Riemann hypothesis, but for which the discrepancy decays like $N^{-\epsilon}$ for any fixed $\epsilon>0$. This provides evidence that for CM abelian varieties, the converse to ``Akiyama--Tanigawa conjecture implies generalized Riemann hypothesis'' does not follow in a straightforward way from the standard analytic methods. We also show that there are Galois representations $\rho\colon Gal(\overline{\mathbf{Q}} /\mathbf{Q}) \to GL_2(\mathbf{Z}_l)$, ramified at an arbitrarily thin (but still infinite) set of primes, whose Satake parameters can be made to converge at any specified rate to any fixed measure $\mu$ on $[0,\pi]$ for which $\cos_\ast\mu$ is absolutely continuous with bounded derivative.

This dissertation focus on automorphy lifting theorems and related questions. There are two primary components.The first deals with residually dihedral Galois representations. Namely, fix an odd prime $p$, and consider a continuous geometric representation $\rho: G_F \to \GL_n(\mathcal{O})$, where $F$ is either a totally real field if $n=2$, or a CM field if $n>2$, and $\mathcal{O}$ is the integer ring of a finite extension of $\mathbb{Q}_p$. The goal is to prove the automorphy of representations whose residual representation $\bar{\rho}$ has the property that the restriction to $G_{F(\zeta_p)}$ is reducible, where $\zeta_p$ denotes a primitive $p$-th root of unity. This means the classical Taylor-Wiles hypothesis fails and classical patching techniques do not suffice to prove the automorphy of $\rho$. Building off the work of Thorne, we prove an automorphy theorem in the $n=2$ case and apply the result to elliptic curves. The case $n>2$ is examined briefly as well.The second component deals with the generic unobstructedness of compatible systems of adjoint representations. Namely, given a compatible system of representations, one can consider the adjoints of the residual representations and determine whether the second Galois cohomology group with the adjoints as coefficients vanishes for infinitely many primes. Such a question relates to classical problems such as Leopoldt's conjecture. While theorems are hard to prove, we discuss heuristics and provide computational evidence.

Let $F$ be a totally real number field, $k$ a finite field of characteristic $p$ and $\ol{\rho}: \tr{Gal}(\ol F/F) \ra GL_n(k)$ a continuous Galois representation. Under some technical hypotheses on $\ol{\rho}$ we extend the method of Khare and Ramakrishna of constructing ramified characteristic zero lifts of $\rh$ from the setting of $GL_2$ to $GL_n$. As an application of this method we prove the existence of closed points $x \in \tr{Spec}(R^{\tr{ord}}[1/p])$, on a certain nearly ordinary deformation ring $R^{\tr{ord}}$, such that Mazur's dimension conjecture is true locally at $x$. In the process we obtain examples of ordinary Galois representations $\rho: G_{F, S\cup Q} \ra GL_n(\mc O)$, where $\mc O$ is a finite extension of $\Z_p$, such that its adjoint Selmer group $H^1_{\mc L} (G_{F, S\cup Q}, ad^0\, \rho \otimes \Q_p/\Z_p)$ is finite. We then determine the exact size of these Selmer groups in terms of the ramification index of the primes in the auxiliary set.

In this Dissertation we consider stripping primes from the level of genus 2 cuspidal Siegel eigenforms. Specifically, given an eigenform of level Nlr which satisfies certain mild conditions, where l is a prime not dividing N, we construct an eigenform of level N which is congruent to our original form. To obtain our results, we use explicit constructions of Eisenstein series and theta functions to adapt ideas from a level stripping result on elliptic modular forms. Furthermore, we give applications of this result to Galois representations and provide evidence for an analog of Serre's conjecture in the genus 2 case.

Let $F$ be a totally real number field, $p$ a rational prime, and $\chi$ a finite order totally odd abelian character of Gal$(\overline{F}/F)$ such that $\chi(\mathfrak{p})=1$ for some $\mathfrak{p}|p$. Motivated by a conjecture of Stark, Gross conjectured a relation between the derivative of the $p$-adic $L$-function associated to $\chi$ at its exceptional zero and the $\mathfrak{p}$-adic logarithm of a $p$-unit in the $\chi$ component of $\overline{F}^\times$. In a recent work, Dasgupta, Darmon, and Pollack have proven this conjecture assuming two conditions: that Leopoldt's conjecture holds for $F$ and $p$, and that if there is only one prime of $F$ lying above $p$, a certain relation holds between the $\mathscr{L}$-invariants of $\chi$ and $\chi^{-1}$. The main result of this work removes both of these conditions, thus giving an unconditional proof of the conjecture. We also describe some applications towards simplifying the Iwasawa Main Conjecture at the weight one and Leopoldt zeroes, as well as some partial results on Gross's conjecture in the higher rank setting.

In this thesis, we will study mock-modular forms of weight one and their Fourier coefficients. In particular, we will concentrate on the mock-modular forms whose shadows are dihedral newforms arising from ray class group characters of imaginary quadratic fields. We will show that certain linear combinations of their Fourier coefficients are logarithms of CM values of the modular j-function. We will also make a conjecture about the algebraicity of the individual Fourier coefficients.

The purpose of this thesis is to outline the Hecke stability method (HSM), a novel method for the computation of modular forms.The HSM relies on the following idea: A finite-dimensional space of ratios of modular forms that is stable under the action of a Hecke operator should consist of modular forms (i.e., without poles). This principle is correct over the complex numbers, but more care is required over finite fields due to complications arising near the supersingular points on modular curves. Formalizing this main idea as a theorem comprises most of our theoretical work.Though it can be utilized in a variety of settings, the main application of the Hecke stability method is the computation of weight 1 modular forms. These spaces cannot be computed using the algorithms (e.g., modular symbols algorithms) that are typically employed to compute modular forms of higher weight.Furthermore, to provide a complete picture of the weight 1 modular forms of level N, we must account for certain sporadic discrepancies between the space of classical forms and the space of mod p modular forms. Ultimately, our approach is motivated by the effect this "ethereality" phenomenon may have on the statistics of number fields via the theory of modular Galois representations.

We prove modularity of some two dimensional 2-adic Galois representations over a totally real field that are nearly ordinary at all places above 2 and that are residually dihedral. We do this by employing the strategy of Skinner and Wiles using Hida families together with the 2-adic patching method of Khare and Wintenberger. As an application we deduce modularity of some elliptic curves over totally real fields that have good ordinary or multiplicative reduction at places above 2.

In this work, we investigate congruences between modular cuspforms. Specifically, we start with a given cuspform and count the number of cuspforms congruent to it as we vary the weight or level. This counting problem is equivalent to understanding the ranks of certain completed Hecke rings. Using the deep modularity results of Wiles, et al., we investigate these Hecke rings by studying the deformation theory of the residual representation corresponding to our given cuspform. This leads us to consider certain Selmer groups attached to this residual representation. In this setting, we can apply standard theorems from local and global Galois cohomology to achieve our results.

Let k be a positive integer divisible by 4, ℓ > k an odd prime, and f a normalized elliptic cuspidal eigenform of weight k - 1, level 4 and non-trivial character. This thesis provides evidence for the Bloch-Kato conjecture in the following way. Let L(Symm2 f, s) denote the symmetric square L-function of f and rf the residual (mod ℓ) Galois representation attached to f. We prove that if ordℓ(L alg(Symm2 f, k)) > 0 then ℓ divides the size of the Selmer group H1f&parl0;K,r f&vbm0;GK&parl0;-1&parr0;&parr0; , where K = Q( -1 ). Our method uses an idea of Ribet [Rib76] to introduce an intermediate step showing that ℓ-divisibility of Lalg(Symm 2 f, k) implies the existence of a congruence between modular forms on a group containing GL2 as a Levi subgroup of a maximal parabolic. Here we use the unitary group U(2,2) and following Gritsenko [Gri90] we lift f to a CAP modular Hecke eigenform Ff on U(2,2). We then construct a different (non-CAP) Hecke eigenform on U(2, 2) with Fourier coefficients congruent to those of Ff. Using this congruence we prove that ℓ divides the size of the Selmer group.