We extend many results on Selmer groups for elliptic curves and modular forms to the non-ordinary setting. More precisely, we study the signed Selmer groups defined using the machinery of Wach modules over Zpcyclotomic extensions. First, we provide a definition of residual and nonprimitive Selmer groups at non-ordinary primes. This allows us to extend techniques developed by Greenberg (for p-ordinary elliptic curves) and Kim (p-supersingular elliptic curves) to show that if two p-non-ordinary modular forms are congruent to each other, then the Iwasawa invariants of their signed Selmer groups are related in an explicit manner. Our results have several applications. First of all, this allows us to relate the parity of the analytic ranks of such modular forms generalizing a recent result of the first-named author for p-supersingular elliptic curves. Second, we can prove a Kida-type formula for the signed Selmer groups generalizing results of Pollack and Weston.2010 Mathematics Subject Classification. 11R18, 11F11, 11R23 (primary); 11F85 (secondary).

Recently, there has been much interest in studying the torsion subgroups of elliptic curves baseextended to infinite extensions of Q. In this paper, given a finite group G, we study what happens with the torsion of an elliptic curve E over Q when changing base to the compositum of all number fields with Galois group G. We do this by studying a group theoretic condition called generalized G-type, which is a necessary condition for a number field with Galois group H to be contained in that compositum. In general, group theory allows one to reduce the original problem to the question of finding rational points on finitely many modular curves. To illustrate this method, we completely determine which torsion structures occur for elliptic curves defined over Q and base-changed to the compositum of all fields whose Galois group is A4.More results of this nature can be found in [14,19], where the torsion structures that occur when K is allowed to be a quadratic extension of Q are completely classified. Since then much work has been put into classifying the torsion structures of elliptic curves defined over cubic extensions of Q with results recently announced by Etropolski, Morrow and Zureick-Brown, and independently by the second author. The case where [K : Q] = 4 is still completely wide open.Another approach is to start with an elliptic curve defined over Q and consider what torsion subgroups occur when it is base-extended to a field K/Q. The cases when E is defined over Q and K is a number field of degree d with d = 2, 3, 4, 5, 7 or d is not divisible by 2,3,5,7 have been completely settled in [3, 11-13, 26]. There have been a number of papers that consider the question of what torsion structures occur over a fixed infinite extension of Q. For example, in [9, 10, 20] all the possible torsion structures that occur when E/Q is base-extended to the compositum of all quadratic extensions of Q are classified. More recently,in [7] all of the torsion structures that occur when E/Q is base-extended to the compositum of all degree 3 extensions of Q are classified. It is worth noting at this point that when working over an infinite extension of Q, we no longer have the Mordell-Weil Theorem to ensure that the torsion subgroup remains finite. Because of this we have to make sure that we choose an infinite extension of Q carefully.In this paper, we provide a general framework for studying the torsion subgroups of rational elliptic curves over certain infinite extensions of Q. Before proceeding we remind the reader of the following definition.

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Abstract. A recent paper of Shekhar compares the ranks of elliptic curves E 1 and E 2 for which there is an isomorphismas Gal(Q/Q)-modules, where p is a prime of good ordinary reduction for both curves. In this paper we prove an analogous result in the case where p is a prime of good supersingular reduction.

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