Let K be a real quadratic number field. Let p be a prime which is inert in K. We denote the completion of K at the place p by Kp. Let ƒ > 1 be a positive integer coprime to p. In this thesis we give a p-adic construction of special elements u(r, ??) ∈ Kxp for special pairs (r, ??) ∈ (ℤ/ƒℤ)x x Hp where Hp = ℙ¹(ℂp) ℙ¹(ℚp) is the so called p-adic upper half plane. These pairs (r, ??) can be thought of as an analogue of classical Heegner points on modular curves. The special elements u(r, ??) are conjectured to be global p-units in the narrow ray class field of K of conductor ƒ. The construction of these elements that we propose is a generalization of a previous construction obtained in [DD06]. The method consists in doing p-adic integration of certain ℤ-valued measures on ??=ℤpxℤp pℤpxpℤp . The construction of those measures relies on the existence of a family of Eisenstein series (twisted by additive characters) of varying weight. Their moments are used to define those measures. We also construct p-adic zeta functions for which we prove an analogue of the so called Kronecker's limit formula. More precisely we relate the first derivative at s = 0 of a certain p-adic zeta function with -logₚ NKp/Qp u(r, ??). Finally we also provide some evidence both theoretical and numerical for the algebraicity of u(r, ??). Namely we relate a certain norm of our p-adic invariant with Gauss sums of the cyclotomic field Q (zetaf, zetap). The norm here is taken via a conjectural Shimura reciprocity law. We also have included some numerical examples at the end of section 18.

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 2000. / Includes bibliographical references (p. 82).

No description available.

The purpose of this thesis is to construct nontrivial elements in the Bloch--Kato Selmer group of the symmetric cube of the Galois representation attached to a cuspidal holomorphic eigenform 𝐹 of level 1. The existence of such elements is predicted by the Bloch--Kato conjecture. This construction is carried out under certain standard conjectures related to Langlands functoriality. The broad method used to construct these elements is the one pioneered by Skinner and Urban in [SU06a] and [SU06b]. The construction has three steps, corresponding to the three chapters of this thesis. The first step is to use parabolic induction to construct a functorial lift of 𝐹 to an automorphic representation π of the exceptional group G₂ and then locate every instance of this functorial lift in the cohomology of G₂. In Eisenstein cohomology, this is done using the decomposition of Franke--Schwermer [FS98]. In cuspidal cohomology, this is done assuming Arthur's conjectures in order to classify certain CAP representations of G₂ which are nearly equivalent to π, and also using the work of Adams--Johnson [AJ87] to describe the Archimedean components of these CAP representations. This step works for 𝐹 of any level, even weight 𝑘 ≥ 4, and trivial nebentypus, as long as the symmetric cube 𝐿-function of 𝐹 vanishes at its central value. This last hypothesis is necessary because only then will the Bloch--Kato conjecture predict the existence of nontrivial elements in the symmetric cube Bloch--Kato Selmer group. Here this hypothesis is used in the case of Eisenstein cohomology to show the holomorphicity of certain Eisenstein series via the Langlands--Shahidi method, and in the case of cuspidal cohomology it is used to ensure that relevant discrete representations classified by Arthur's conjecture are cuspidal and not residual. The second step is to use the knowledge obtained in the first step to 𝓅-adically deform a certain critical 𝓅-stabilization 𝜎π of π in a generically cuspidal family of automorphic representations of G₂. This is done using the machinery of Urban's eigenvariety [Urb11]. This machinery operates on the multiplicities of automorphic representations in certain cohomology groups; in particular, it can relate the location of π in cohomology to the location of 𝜎π in an overconvergent analogue of cohomology and, under favorable circumstances, use this information to 𝓅-adically deform 𝜎π in a generically cuspidal family. We show that these circumstances are indeed favorable when the sign of the symmetric functional equation for 𝐹 is -1 either under certain conditions on the slope of 𝜎π, or in general when 𝐹 has level 1. The third and final step is to, under the assumption of a global Langlands correspondence for cohomological automorphic representations of G₂, carry over to the Galois side the generically cuspidal family of automorphic representations obtained in the second step to obtain a family of Galois representations which factors through G₂ and which specializes to the Galois representation attached to π. We then show this family is generically irreducible and make a Ribet-style construction of a particular lattice in this family. Specializing this lattice at the point corresponding to π gives a three step reducible Galois representation into GL₇, which we show must factor through, not only G₂, but a certain parabolic subgroup of G₂. Using this, we are able to construct the desired element of the symmetric cube Bloch--Kato Selmer group as an extension appearing in this reducible representation. The fact that this representation factors through the aforementioned parabolic subgroup of G₂ puts restrictions on the extension we obtain and guarantees that it lands in the symmetric cube Selmer group and not the Selmer group of 𝐹 itself. This step uses that 𝐹 is level 1 to control ramification at places different from 𝓅, and to ensure that 𝐹 is not CM so as to guarantee that the Galois representation attached to π has three irreducible pieces instead of four.

Thesis advisor: Dubi Kelmer / We prove a second moment formula for incomplete Eisenstein series on the homogeneous space Γ\G with G the orientation preserving isometry group of the real (n + 1)-dimensional hyperbolic space and Γ⊂ G a non-uniform lattice. This result generalizes the classical Rogers' second moment formula for Siegel transform on the space of unimodular lattices. We give two applications of this moment formula. In Chapter 5 we prove a logarithm law for unipotent flows making cusp excursions in a non-compact finite-volume hyperbolic manifold. In Chapter 6 we study the counting problem counting the number of orbits of Γ-translates in an increasing family of generalized sectors in the light cone, and prove a power saving estimate for the error term for a generic Γ-translate with the exponent determined by the largest exceptional pole of corresponding Eisenstein series. When Γ is taken to be the lattice of integral points, we give applications to the primitive lattice points counting problem on the light cone for a generic unimodular lattice coming from SO₀(n+1,1)(ℤ\SO₀(n+1,1). / Thesis (PhD) — Boston College, 2018. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

Thesis (doctoral)-- Rheinische Friedrich-Wilhelm-Universität zu Bonn, 1989. / Includes bibliographical references (p. 99-102).

Thesis (M.S.)--Brigham Young University. Dept of Mathematics, 2006. / Includes bibliographical references (p. 94-96).

We study the analytic properties of the Eisenstein Series of $frac {1}{2}$-integral weight associated with the Hecke congruence subgroup $Gamma_0(4)$. Using these properties we obtain asymptotics for sums of certain Dirichlet $L$-series. We also obtain a formula reducing the study of Selberg's Eigenvalue Conjecture to the study of the nonvanishing of the Eisenstein Series $E(z,s)$ for Hecke congruence subgroups $Gamma_0(N)$ at $s=frac {1+i}{2}$.

No description available.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第22231号 / 理博第4545号 / 新制||理||1653(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 池田 保, 教授 雪江 明彦, 教授 並河 良典 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM