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Dissertations completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest dissertations.
Let p ≥ 5 be a prime and E/ℚ be an elliptic curve of conductor N that is ordinary at p. Let K/ℚ be an imaginary quadratic field. This thesis is concerned with the dual Selmer group of E over the anticyclotomic extension of K, especially when p is split in K and K satisfies the Heegner hypothesis for E: every prime ℓ dividing N is split in K/ℚ.A fundamental result in this setting is the non-existence of non-zero finite submodules. Using purely algebraic methods, we extend this result to new cases under verifiable hypotheses concerning the Heegner point of E over K. Under similar hypotheses, we establish the vanishing of the μ-invariant using simpler techniques than the literature.As an application, we study the variation of λ-invariants for p-residually isomorphic elliptic curves. This type of congruence question is also explored on the analytic side. Namely, we look at the Bertolini-Darmon-Prasanna (BDP) p-adic L-functions for p-congruent modular forms.
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Classical norm principles examine the behavior of quadratic forms and central simple algebras under the extensions of the base field. Merkurjev reformulated them as a property of algebraic groups, and Gille reformulated them as a property of torsors under algebraic groups using Galois cohomology. After recalling the classical norm principles, we will show that Merkurjev’s formulation of the norm principle for reductive linear algebraic groups is equivalent to Gille’s formulation for torsors under semisimple groups. We will then prove that it is sufficient to show the norm principle for simply connected groups. Among classical groups, the only case for which the norm principle is open, are groups of type Dₙ. Absolutely simple, simply connected, classical groups of type Dₙ are spinor groups of central simple algebras with orthogonal involution. We will reduce the norm principle for the spinor groups to the case that the field extension has degree 2. We then focus on the spinor groups of skew-hermitian forms defined over quaternion algebras and will reduce the question in this case to the case that the skew-hermitian form is anisotropic. Let K be a complete discretely valued field with residue field k with char(k) ≠ 2. Suppose that the norm principle holds for spinor groups Spin(h) for every nonsingular skew-hermitian form h defined over every quaternion algebra with canonical involution defined over finite separable extensions of k. Then we will show that it holds for spinor groups Spin(H) for every nonsingular skew-hermitian form H defined over every quaternion algebra with canonical involution defined over K.
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Theses completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest theses.
In this thesis, we establish congruences for values of Dedekind Zeta functions attached to a specific family of totally real fields. Our main theoremgeneralizes [7, Proposition 2.5]. The proof relies on Iwasawa’s construction ofp – adic L– functions and an application of Local Class Field Theory. As a consequence, we derive a criterion for the p – indivisibility of generalized Bernoullinumbers Bn,χ associated with Dirichlet characters χ of p – power order, thetriviality of p-torsion in certain even K-groups of specific totally real fields, andcongruence modulo p between Euler characteristic of certain arithmetic groups.Our findings demonstrate the applicability of similar methods to establishcongruences for Dirichlet L– values at negative odd integers, provided that thecorresponding Dirichlet characters satisfy specific congruence criteria moduloa prime p. Our results generalize and offer an alternate approach to somecongruences demonstrated in .
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In the first part, we introduce theory of p-adic analysis for one variable p-adic functions and then use them to construct Kubota-Leopoldt p-adic L-functions.In the second part, we give a description of the Iwasawa modules attached to p-adic Galois representations of the absolute Galois group of K in terms of the theory of (φ,Γ)-modules of Fontaine. When the representation is de Rham when K be finite extension of Qp. This gives a natural construction of the exponential map of Perrin-Riou which is used in the construction and the study of p-adic L-functions.In the third part, we give formulas for Bloch-Kato’s exponential map and its dual for an alsolutely crystalline p-adic representation V . As a corollary of these computation, we can give a improved description of Perrin-Riou’s exponential map, which interpolates Bloch-Kato’s exponentials for the twists of V. Finally we use this map to reconstruct Kubota-Leopoldt p-adic L-functions.
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