Good Supersingular Reduction Research Articles

Iwasawa theory of plus/minus Selmer groups with non-co-free plus/minus local conditions

We often use the plus/minus Selmer groups to study Iwasawa theory for elliptic curves with good supersingular reduction at [Formula: see text]. But, if the plus/minus local conditions are not co-free, it can be difficult to use them effectively. In this paper, we introduce some technical improvements so that we can use the plus/minus Selmer groups even when the plus/minus local conditions are not co-free. In particular, we show that the plus/minus Selmer groups have no proper submodule of finite index and also they satisfy algebraic functional equations even when the said local conditions are not co-free, thus improve our previous results [B. Kim, The algebraic functional equation of an elliptic curve at supersingular primes, Math. Res. Lett. 15(1) (2008) 83–94; The plus/minus Selmer groups for supersingular primes, J. Aust. Math. Soc. 95(2) (2013) 189–200].

Asymptotic formula for Tate–Shafarevich groups of p-supersingular elliptic curves over anticyclotomic extensions

Let p⩾5 be a prime number and E/Q an elliptic curve with good supersingular reduction at p. Under the generalized Heegner hypothesis, we investigate the p-primary subgroups of the Tate–Shafarevich groups of E over number fields contained inside the anticyclotomic Zp-extension of an imaginary quadratic field where p splits.

Conjecture A andμ-invariant for Selmer groups of supersingular elliptic curves

Let p be an odd prime and let E be an elliptic curve defined over a number field F with good reduction at the primes above p. In this survey article, we give an overview of some of the important results proven for the fine Selmer group and the signed Selmer groups over cyclotomic towers as well as the signed Selmer groups over ℤ p 2 -extensions of an imaginary quadratic field where p splits completely. We only discuss the algebraic aspects of these objects through Iwasawa theory. We also attempt to give some of the recent results implying the vanishing of the μ-invariant under the hypothesis of Conjecture A. Moreover, we draw an analogy between the classical Selmer group in the ordinary reduction case and that of the signed Selmer groups of Kobayashi in the supersingular reduction case. We highlight properties of signed Selmer groups, when E has good supersingular reduction, which are completely analogous to the classical Selmer group, when E has good ordinary reduction. In this survey paper we do not present any proofs, however, we have tried to give references of the discussed results for the interested reader.

Rubin's conjecture on local units in the anticyclotomic tower at inert primes

We prove a fundamental conjecture of Rubin on the structure of local units in the anticyclotomic $\mathbb{Z}_p$-extension of the unramified quadratic extension of $\mathbb{Q}_p$ for $p\geq 5$ a prime. Rubin's conjecture underlies Iwasawa theory of the anticyclotomic deformation of a CM elliptic curve over the CM field at primes $p$ of good supersingular reduction, notably the Iwasawa main conjecture in terms of $p$-adic $L$-function. As a consequence, we prove an inequality in the $p$-adic Birch and Swinnerton-Dyer conjecture for Rubin's $p$-adic $L$-function. Rubin's conjecture is also an essential tool in our exploration of the arithmetic of Rubin's $p$-adic $L$-function, which includes a Bertolini--Darmon--Prasanna type formula.

Constructing plus/minus local points over Iwasawa Kummer extensions, and Iwasawa theory for supersingular reduction

Suppose an elliptic curve E/Q has good supersingular reduction at a prime p, and satisfies 1+p−#E(Z/pZ)=0, which is always true if p is supersingular and p>3 by the Hasse inequality. Expanding Kobayashi's plus/minus-local points over the cyclotomic Zp-extension of Qp, where π is any element in Cp with positive p-adic valuation, we construct doubly indexed plus/minus local points over Qp(πpn−m,μpn) (which we will call local Iwasawa Kummer extensions) as n,m vary.Then, we explore various possible plus/minus Selmer groups over Zp-extensions of number fields over which p is ramified, or doubly indexed plus/minus Selmer groups over Iwasawa Kummer extensions, and study or speculate their properties using the above plus/minus local points.Finally, again using the above points, we study the following: Suppose k is a ramified quadratic extension of Qp. We find a bound of [E(mk(μpn)):E+(mk(μpn))+E−(mk(μpn))] as n varies.

On fine Selmer groups and the greatest common divisor of signed and chromatic 𝑝-adic 𝐿-functions

Let E / Q E/\mathbb {Q} be an elliptic curve and p p an odd prime where E E has good supersingular reduction. Let F 1 F_1 denote the characteristic power series of the Pontryagin dual of the fine Selmer group of E E over the cyclotomic Z p \mathbb {Z}_p -extension of Q \mathbb {Q} and let F 2 F_2 denote the greatest common divisor of Pollack’s plus and minus p p -adic L L -functions or Sprung’s sharp and flat p p -adic L L -functions attached to E E , depending on whether a p ( E ) = 0 a_p(E)=0 or a p ( E ) ≠ 0 a_p(E)\ne 0 . We study a link between the divisors of F 1 F_1 and F 2 F_2 in the Iwasawa algebra. This gives new insights into problems posed by Greenberg and Pollack–Kurihara on these elements.

Congruences of algebraic p-adic L-functions and the Main Conjecture of Iwasawa Theory

In this paper, following Perrin-Riou's work, for any eigenform f of weight at least 2 and p-adic slope less than 1, we construct algebraic p-adic L-function Lf(X), and show that for eigenforms f2 of weight 2 and fk of weight k, if f2≡fk(modpN′) and ap(f2)/p≡ap(fk)/p(modpN⋅p) for certain N,N′, then for every Dirichlet character χ of sufficiently large p-power conductor, χ(Lf2(X)) and χ(Lfk(X)) have the same p-adic valuation. Combined with our previous work with Choi on analytic congruences ([2]), this implies the following: Suppose E is an abelian variety over Q associated to f2, ap(E)=0 (always true if E is an elliptic curve over Q and has good supersingular reduction at p>3), and the above conditions for f2 and fk hold true with sufficiently large N,N′ (where the meaning of “sufficiently large” depends only on E). Then, the Main Conjecture of Iwasawa Theory for fk implies the Main Conjecture for E.

On Selmer Groups in the Supersingular Reduction Case

Let $p$ be a fixed odd prime. Let $E$ be an elliptic curve defined over a number field $F$ with good supersingular reduction at all primes above $p$. We study both the classical and plus/minus Selmer groups over the cyclotomic $\mathbb{Z}_p$-extension of $F$. In particular, we give sufficient conditions for these Selmer groups to not contain a non-trivial sub-module of finite index. Furthermore, when $p$ splits completely in $F$, we calculate the Euler characteristics of the plus/minus Selmer groups over the compositum of all $\mathbb{Z}_p$-extensions of $F$ when they are defined.

On the Mordell-Weil ranks of supersingular abelian varieties in cyclotomic extensions

Let F F be a number field unramified at an odd prime p p and let F ∞ F_\infty be the Z p \mathbf {Z}_p -cyclotomic extension of F F . Let A A be an abelian variety defined over F F with good supersingular reduction at all primes of F F above p p . Büyükboduk and the first named author have defined modified Selmer groups associated to A A over F ∞ F_\infty . Assuming that the Pontryagin dual of these Selmer groups is a torsion Z p [ [ Gal ( F ∞ / F ) ] ] \mathbf {Z}_p[[\textrm {Gal}(F_\infty /F)]] -module, we give an explicit sufficient condition for the rank of the Mordell-Weil group A ( F n ) A(F_n) to be bounded as n n varies.

On the growth of Mordell–Weil ranks in $p$-adic Lie extensions

Let $p$ be an odd prime and $F_{\infty}$ a $p$-adic Lie extension of a number field $F$. Let $A$ be an abelian variety over $F$ which has ordinary reduction at every primes above $p$. Under various assumptions, we establish asymptotic upper bounds for the growth of Mordell-Weil rank of the abelian variety of $A$ in the said $p$-adic Lie extension. Our upper bound can be expressed in terms of invariants coming from the cyclotomic level. Motivated by this formula, we make a conjecture on an asymptotic upper bound of the growth of Mordell-Weil ranks over a $p$-adic Lie extension which is in terms of the Mordell-Weil rank of the abelian variety over the cyclotomic $\mathbb{Z}_p$-extension. Finally, it is then natural to ask whether there is such a conjectural upper bound when the abelian variety has non-ordinary reduction. For this, we can at least modestly formulate an analogue conjectural upper bound for the growth of Mordell-Weil ranks of an elliptic curve with good supersingular reduction at the prime $p$ over a $\mathbb{Z}_p^2$-extension of an imaginary quadratic field.

On the structure of signed Selmer groups

Let F be a number field unramified at an odd prime p and F_infty be the mathbf {Z}_p-cyclotomic extension of F. Generalizing Kobayashi plus/minus Selmer groups for elliptic curves, Büyükboduk and Lei have defined modified Selmer groups, called signed Selmer groups, for certain non-ordinary {{,mathrm{Gal},}}(overline{F}/F)-representations. In particular, their construction applies to abelian varieties defined over F with good supersingular reduction at primes of F dividing p. Assuming that these Selmer groups are cotorsion mathbf {Z}_p[[{{,mathrm{Gal},}}(F_infty /F)]]-modules, we show that they have no proper sub-mathbf {Z}_p[[{{,mathrm{Gal},}}(F_infty /F)]]-module of finite index. We deduce from this a number of arithmetic applications. On studying the Euler–Poincaré characteristic of these Selmer groups, we obtain an explicit formula on the size of the Bloch–Kato Selmer group attached to these representations. Furthermore, for two such representations that are isomorphic modulo p, we compare the Iwasawa-invariants of their signed Selmer groups.

CODIMENSION TWO CYCLES IN IWASAWA THEORY AND ELLIPTIC CURVES WITH SUPERSINGULAR REDUCTION

A result of Bleher, Chinburg, Greenberg, Kakde, Pappas, Sharifi and Taylor has initiated the topic of higher codimension Iwasawa theory. As a generalization of the classical Iwasawa main conjecture, they prove a relationship between analytic objects (a pair of Katz’s$2$-variable$p$-adic$L$-functions) and algebraic objects (two ‘everywhere unramified’ Iwasawa modules) involving codimension two cycles in a$2$-variable Iwasawa algebra. We prove a result by considering the restriction to an imaginary quadratic field$K$(where an odd prime$p$splits) of an elliptic curve$E$, defined over $\mathbb{Q}$, with good supersingular reduction at$p$. On the analytic side, we consider eight pairs of$2$-variable$p$-adic$L$-functions in this setup (four of the$2$-variable$p$-adic$L$-functions have been constructed by Loeffler and a fifth$2$-variable$p$-adic$L$-function is due to Hida). On the algebraic side, we consider modifications of fine Selmer groups over the$\mathbb{Z}_{p}^{2}$-extension of$K$. We also provide numerical evidence, using algorithms of Pollack, towards a pseudonullity conjecture of Coates–Sujatha.

Rank parity for congruent supersingular elliptic curves

A recent paper of Shekhar compares the ranks of elliptic curves E 1 E_1 and E 2 E_2 for which there is an isomorphism E 1 [ p ] ≃ E 2 [ p ] E_1[p] \simeq E_2[p] as G a l ( Q ¯ / Q ) \mathrm {Gal}(\bar {\mathbf {Q}}/\mathbf {Q}) -modules, where p p is a prime of good ordinary reduction for both curves. In this paper we prove an analogous result in the case where p p is a prime of good supersingular reduction.

Signed-Selmer Groups over the ℤ2p-extension of an Imaginary Quadratic Field

AbstractLet E be an elliptic curve over ℚ that has good supersingular reduction at p > 3. We construct what we call the ±/±-Selmer groups of E over the ℤ2p-extension of an imaginary quadratic field K when the prime p splits completely over K/ℚ, and prove that they enjoy a property analogous to Mazur's control theorem.Furthermore, we propose a conjectural connection between the±/±-Selmer groups and Loeffler's two-variable ±/±-p-adic L-functions of elliptic curves.

THE PLUS/MINUS SELMER GROUPS FOR SUPERSINGULAR PRIMES

AbstractSuppose that an elliptic curve $E$ over $ \mathbb{Q} $ has good supersingular reduction at $p$. We prove that Kobayashi’s plus/minus Selmer group of $E$ over a ${ \mathbb{Z} }_{p} $-extension has no proper $\Lambda $-submodule of finite index under some suitable conditions, where $\Lambda $ is the Iwasawa algebra of the Galois group of the ${ \mathbb{Z} }_{p} $-extension. This work is analogous to Greenberg’s result in the ordinary reduction case.

ON THE p-PRIMARY PART OF TATE-SHAFAREVICH GROUP OF ELLIPTIC CURVES OVER ℚ WHEN p IS SUPERSINGULAR

Let E be an elliptic curve over <TEX>$\mathbb{Q}$</TEX> and <TEX>$p$</TEX> be a prime of good supersingular reduction for E. Although the Iwasawa theory of E over the cyclotomic <TEX>${\mathbb{Z}}_p$</TEX>-extension of <TEX>$\mathbb{Q}$</TEX> is well known to be fundamentally different from the case of good ordinary reduction at p, we are able to combine the method of our earlier paper with the theory of Kobayashi [5] and Pollack [8], to give an explicit upper bound for the number of copies of <TEX>${\mathbb{Q}}_p/{\mathbb{Z}}_p$</TEX> occurring in the <TEX>$p$</TEX>-primary part of the Tate-Shafarevich group of E over <TEX>$\mathbb{Q}$</TEX>.

Formal groups of elliptic curves with potential good supersingular reduction

Formal groups of elliptic curves with potential good supersingular reduction

NON-COMMUTATIVE p-ADIC L-FUNCTIONS FOR SUPERSINGULAR PRIMES

Let E/ℚ be an elliptic curve with good supersingular reduction at p with ap(E) = 0. We give a conjecture on the existence of analytic plus and minus p-adic L-functions of E over the ℤp-cyclotomic extension of a finite Galois extension of ℚ where p is unramified. Under some technical conditions, we adopt the method of Bouganis and Venjakob for p-ordinary CM elliptic curves to construct such functions for a particular non-abelian extension.

Signed Selmer groups over p-adic Lie extensions

Let E be an elliptic curve over ℚ with good supersingular reduction at a prime p≥3 and a p =0. We generalise the definition of Kobayashi’s plus/minus Selmer groups over ℚ(μ p ∞ ) to p-adic Lie extensions K ∞ of ℚ containing ℚ(μ p ∞ ), using the theory of (ϕ,Γ)-modules and Berger’s comparison isomorphisms. We show that these Selmer groups can be equally described using Kobayashi’s conditions via the theory of overconvergent power series. Moreover, we show that such an approach gives the usual Selmer groups in the ordinary case.

Formal groups, supersingular abelian varieties and tame ramification

Let us consider an abelian variety defined over Q ℓ with good supersingular reduction. In this paper we give explicit conditions that ensure that the action of the wild inertia group on the ℓ-torsion points of the variety is trivial. Furthermore we give a family of curves of genus 2 such that their Jacobian surfaces have good supersingular reduction and satisfy these conditions. We address this question by means of a detailed study of the formal group law attached to abelian varieties.