Meng Fai Lim

Notes on the fine Selmer groups

In this paper, we study the fine Selmer groups attached to a Galois module defined over a commutative complete Noetherian ring with finite residue field of characteristic p. Namely, we are interested in its properties upon taking residual representation and within field extensions. In particular, we will show that the variation of the fine Selmer group in a cyclotomic

Poitou–Tate duality over extensions of global fields

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Fine Selmer groups of congruent Galois representations

-extension is intimately related to the variation of the class groups in the cyclotomic tower. We also discuss some examples of pseudo-nullity of fine Selmer groups.

Akashi series, characteristic elements and congruence of Galois representations

Abstract In this paper, we are interested in the Poitou–Tate duality in Galois cohomology. We will formulate and prove a theorem for a nice class of modules (with a continuous Galois action) over a pro-p ring. The theorem will comprise of the Tate local duality, Poitou–Tate duality and the Poitou–Tateʼs exact sequence.

On completely faithful Selmer groups of elliptic curves and Hida deformations

Abstract In this paper, we study the fine Selmer groups of two congruent Galois representations over an admissible p-adic Lie extension. We show that under appropriate congruence conditions, if the dual fine Selmer group of one is pseudo-null, so is the other. Our results also compare the π-primary submodules of the two dual fine Selmer groups. We then apply our results to compare the structure of Galois group of the maximal abelian unramified pro-p extension of an admissible p-adic Lie extension and the structure of the dual fine Selmer group over the said admissible p-adic Lie extension.

On the pseudo-nullity of the dual fine Selmer groups

In this paper, we compare the Akashi series of the Pontryagin dual of the Selmer groups of two Galois representations over a strongly admissible p-adic Lie extension. Namely, we show that whenever the two Galois representations in question are congruent to each other, the Akashi series of one is a unit if and only if the Akashi series of the other is also a unit. We will also obtain similar results for the Euler characteristics of the Selmer groups and the characteristic elements attached to the Selmer groups.

THE GROWTH OF THE SELMER GROUP OF AN ELLIPTIC CURVE WITH SPLIT MULTIPLICATIVE REDUCTION

Abstract In this paper, we study completely faithful torsion Z p 〚 G 〛 -modules with applications to the study of Selmer groups. Namely, if G is a nonabelian group belonging to certain classes of polycyclic pro-p group, we establish the abundance of faithful torsion Z p 〚 G 〛 -modules, i.e., non-trivial torsion modules whose global annihilator ideal is zero. We then show that such Z p 〚 G 〛 -modules occur naturally in arithmetic, namely in the form of Selmer groups of elliptic curves and Selmer groups of Hida deformations. It is interesting to note that faithful Selmer groups of Hida deformations do not seem to have appeared in literature before. We will also show that faithful Selmer groups have various arithmetic properties. Namely, we show that faithfulness is an isogeny invariant, and we will prove “control theorem” results on the faithfulness of Selmer groups over a general strongly admissible p-adic Lie extension.

Comparing the

In this paper, we will study the pseudo-nullity of the dual fine Selmer group and its related question. We investigate certain situations, where one can deduce the pseudo-nullity of the dual fine Selmer group of a general Galois module over an admissible p-adic Lie extension F∞ from the knowledge of the pseudo-nullity of the Galois group of the maximal abelian unramified pro-p extension of F∞ at which every prime of F∞ above p splits completely. In particular, this gives us a way to construct examples of the pseudo-nullity of the dual fine Selmer group of a Galois module that is unramified outside p. We will illustrate our results with many examples.

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Let p be a fixed prime. Let E be an elliptic curve that has split multiplicative reduction at some prime with the corresponding Tamagawa factor being divisible by p. We study the variation of the p-ranks of the Selmer groups of E in pro-p algebraic extensions.