Arithmetic properties of signed Selmer groups at non-ordinary primes

Abstract

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).

Comparing anticyclotomic Selmer groups of positive co-ranks for congruent modular forms

“…Indeed, in extending this work to supersingular elliptic curves, Kim needed to use the modified ±-Selmer groups of Kobayashi [Kob03], since the normal Selmer group associated to a p-supersingular elliptic curve over Q ∞ is never cotorsion. Similar results were obtained for p-non-ordinary modular forms by the authors [HL18].…”

p r -Selmer companion modular forms

“…12) and(4.13), by the definition of π r Bloch-Kato Selmer group, the remaining j = k − 2 case of Theorem 4.10 in the (p-good) case is obtained. This completes the proof of Theorem 4.10 when assumptions (1), (2) and (p-ord) are satisfied.…”

“…Let f 1 and f 2 be two weight k normalized cuspforms which are congruent mod π r . If f 1 and f 2 are p-ordinary, then we compare π r -Bloch-Kato Selmer local condition at p with the π r -Greenberg Selmer local condition at p. On the other hand, if f 1 and f 2 are non-ordinary at p, then we compare π r -Bloch-Kato Selmer local conditions at p with the π r -signed Selmer local conditions at p ( [12]). The theory of signed Selmer groups for non-ordinary modular forms are developed using works of Lei-Loeffler-Zerbes ( [16,17]).…”

“…Now we take † = i and define signed i Selmer group for i ∈ {1, 2} (cf. [12,16]). To define this, it is necessary to assume that h is "good" at p and also non-ordinary at p i.e.…”

“…+ q 9 − 2q 10 + 2q 12 + O(q 13 ), g(q) = q − q 2 − 3q 3 + q 4 − 3q 5 + 3q 6 − q 7 − q 8 + 6q 9 + 3q 10 − 3q 12 + O(q 13 ) By computing the minimal discriminant of 1246B and 1246C and using [4, Proposition 2.12 (c)] we can show that ∀ ∈ S = {2, 7, 89}, cond (ρ f ) = = cond (ρ g ). Also using [19] we get that ρ f and ρ g are irreducible and equivalent.…”