Sujatha Ramdorai
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Graduate Student Supervision
Doctoral Student Supervision
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 be an odd prime and let E be an elliptic curve defined over a quadratic imaginary field where p splits completely. Suppose E has supersingular reduction at the primes above p. The main purpose of this thesis is to study the signed μ-invariants of the dual signed Selmer groups over Zₚ²-extensions of an imaginary quadratic field, as well as the signed μ-invariants of the dual signed Selmer groups over Zₚ-cyclotomic extensions. We give an overview of some of the important results proven for the fine Selmer group and the signed Selmer groups over cyclotomic and Zₚ²-extensions of an imaginary quadratic field. Under appropriate hypotheses, we define and study the fine double-signed residual Selmer groups and extend the results of [35] to Zₚ²-extensions in these settings. We prove that for two residually isomorphic elliptic curves, the vanishing of the signed μ-invariants of one elliptic curve implies the vanishing of the signed μ-invariants of the other. Moreover, we show that the μ-invariant of the classical Selmer groups is bounded by the μ-invariant of the signed Selmer groups. Finally, we show that the Pontryagin duals of the Selmer group and the double-signed Selmer groups have no non-trivial pseudo-null submodules for these extensions, with purely algebraic methods.
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Master's Student Supervision
Theses completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest theses.
The full abstract for this thesis is available in the body of the thesis, and will be available when the embargo expires.
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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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