The notion of the truncated Euler characteristic for Iwasawa modules is a generalization of the the usual Euler characteristic to the case when the cohomology groups are not finite. Let $p$ be an odd prime, $E_1$ and $E_2$ be elliptic curves over a number field $F$ with semistable reduction at all primes $v|p$ such that the $\operatorname{Gal}(\bar{F}/F)$-modules $E_1[p]$ and $E_2[p]$ are irreducible and isomorphic. We compare the Iwasawa invariants of certain imprimitive multisigned Selmer groups of $E_1$ and $E_2$. Leveraging these results, congruence relations for the truncated Euler characteristics associated to these Selmer groups over certain $\mathbb{Z}_p^m$-extensions of $F$ are studied. Our results extend earlier congruence relations for elliptic curves over $\mathbb{Q}$ with good ordinary reduction at $p$.
In this paper we study when two congruent $l$-adic Galois representations have congruent Selmer groups. We obtain results for representations from cyclotomic characters, Hecke characters and adjoints of modular forms.
We study the functional equation for the multi-signed Selmer groups for non-ordinary motives whose Hodge-Tate weights are $0$ and $1$, defined by Büyükboduk and the first named author. This generalizes simultaneously Greenberg's result for ordinary motives and Kim's result for supersingular elliptic curves.
Let $f$ be an elliptic modular form and $p$ an odd prime that is coprime to the level of $f$. We study the link between divisors of the characteristic ideal of the $p$-primary fine Selmer group of $f$ over the cyclotomic $\mathbb{Z}_p$ extension of $\mathbb{Q}$ and the greatest common divisor of signed Selmer groups attached to $f$ defined using the theory of Wach modules. One of the key ingredients of our proof is a generalization of a result of Wingberg on the structure of fine Selmer groups of abelian varieties with supersingular reduction at $p$ to the context of modular forms.
Let $p$ be an odd prime number, and $E$ an elliptic curve defined over a number field with good reduction at every prime of $F$ above $p$. In this short note, we compute the Euler characteristics of the signed Selmer groups of $E$ over the cyclotomic $\Zp$-extension. The novelty of our result is that we allow the elliptic curve to have mixed reduction types for primes above $p$ and that we allow mixed signs in the definition of the 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 sh…
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 $\mathbf{Z}_p$-cyclotomic extensions. First, we provide a definition of residual and non-primitive 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 si…
Let $E$ be an elliptic curve defined over a number field with good reduction at all primes above a fixed odd prime $p$, where at least one of which is a supersingular prime of $E$. In this paper, we will establish the algebraic functional equation for the mixed signed Selmer groups of an elliptic curve with good reduction at every prime above $p$ over a multiple $\mathbb{Z}_p$-extension.
Let $p$ be an odd prime. We attach appropriate signed Selmer groups to an elliptic curve $E$, where $E$ is assumed to have semistable reduction at all primes above $p$. We then compare the Iwasawa $λ$-invariants of these signed Selmer groups for two congruent elliptic curves over the cyclotomic $\mathbb{Z}_p$-extension in the spirit of Greenberg-Vatsal and B. D. Kim. As an application of our comparsion formula, we show that if the $p$-parity conjecture is true for one of the congruent elliptic curves, then it is also true for the other elliptic curve. In the midst of proving this latter result…
Let $p$ be an odd prime number, and let $E$ be an elliptic curve defined over a number field $F'$ such that $E$ has semistable reduction at every prime of $F'$ above $p$ and is supersingular at at least one prime above $p$. Under appropriate hypotheses, we compute the Akashi series of the signed Selmer groups of $E$ over a $\mathbb{Z}_p^d$-extension over a finite extension $F$ of $F'$. As a by-product, we also compute the Euler characteristics of these Selmer groups.
Let $E$ be an elliptic curve over $\mathbb{Q}$ with good supersingular reduction at a prime $p\geq 3$ and $a_p=0$. We generalise the definition of Kobayashi's plus/minus Selmer groups over $\mathbb{Q}(μ_{p^\infty})$ to $p$-adic Lie extensions $K_\infty$ of $\mathbb{Q}$ containing $\mathbb{Q}(μ_{p^\infty})$, using the theory of $(ϕ,Γ)$-modules and Berger's comparison isomorphisms. We show that these Selmer groups can be equally described using the "jumping conditions" of Kobayashi via the theory of overconvergent power series. Moreover, we show that such an approach gives the usual Selmer group…
Let $p$ be an odd prime number, and let $E$ be an elliptic curve defined over a number field which has good reduction at every prime above $p$. Under suitable assumptions, we prove that the $η$-eigenspace and the $\barη$-eigenspace of mixed signed Selmer group of the elliptic curve are pseudo-isomorphic. As a corollary, we show that the $η$-eigenspace is trivial if and only if the $\barη$-eigenspace is trivial. Our results can be thought as a reflection principle which relate an Iwasawa module in a given eigenspace with another Iwasawa module in a "reflected" eigenspace.
Let $E/\mathbb{Q}$ be an elliptic curve and $p$ an odd prime where $E$ has good supersingular reduction. Let $F_1$ denote the characteristic power series of the Pontryagin dual of the fine Selmer group of $E$ over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ and let $F_2$ denote the greatest common divisor of Pollack's plus and minus $p$-adic $L$-functions or Sprung's sharp and flat $p$-adic $L$-functions attached to $E$, depending on whether $a_p(E)=0$ or $a_p(E)\ne0$. We study a link between the divisors of $F_1$ and $F_2$ in the Iwasawa algebra. This gives new insights into probl…
In this paper, we study the Selmer groups of two congruent Galois representations over an admissible $p$-adic Lie extension. We will show that under appropriate congruence condition, if the dual Selmer group of one satisfies the $\M_H(G)$-property, so will the other. In the event that the $\M_H(G)$-property holds, and assuming certain further hypothesis on the decomposition of primes in the $p$-adic Lie extension, we compare the ranks of the $π$-free quotient of the two dual Selmer groups. We then apply our results to compare the characteristic elements attached to the Selmer groups. We also s…
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.
For an odd prime $p$ and a supersingular elliptic curve over a number field, this article introduces a fine signed residual Selmer group, under certain hypotheses on the base field. This group depends purely on the residual representation at $p$, yet captures information about the Iwasawa theoretic invariants of the signed $p^\infty$-Selmer group that arise in supersingular Iwasawa theory. Working in this residual setting provides a natural framework for studying congruences modulo $p$ in Iwasawa theory.
The plus and minus norm groups are constructed by Kobayashi as subgroups of the formal group of an elliptic curve with supersingular reduction, and they play an important role in Kobayashi's definition of the signed Selmer groups. In this paper, we study the cohomology of these plus and minus norm groups. In particular, we show that these norm groups are cohomologically trivial. As an application of our analysis, we establish certain (quasi-)projectivity properties of the non-primitive mixed signed Selmer groups of an elliptic curve with good reduction at all primes above $p$. We then build on…
The study of $n$-Selmer group of elliptic curve over number field in recent past has led to the discovery of some deep results in the arithmetic of elliptic curves. Given two elliptic curves $E_1$ and $E_2$ over a number field $K$, Mazur-Rubin\cite{mr} have defined them to be {\it $n$-Selmer companion} if for every quadratic twist $χ$ of $K$, the $n$-Selmer groups of $E_1^χ$ and $E_2^χ$ over $K$ are isomorphic. Given a prime $p$, they have given sufficient conditions for two elliptic curves to be $p^r$-Selmer companion in terms of mod-$p^r$ congruences between the curves. We discuss an analogu…
Let $p$ be an odd prime and let $E$ be an elliptic curve defined over a number field $F$ with good reduction at 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 $\mathbb{Z}_p^2$-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 unde…
Let $K$ be an imaginary quadratic field where $p$ splits. We study signed Selmer groups for non-ordinary modular forms over the anticyclotomic $\mathbf{Z}_p$-extension of $K$, showing that their $μ$-invariants vanish. This generalizes and gives a new proof of a recent result of Matar on the vanishing of the $μ$-invariants of plus and minus signed Selmer groups for elliptic curves.
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.
In this article, we set up a strategy to prove one divisibility towards the main Iwasawa conjecture for the Selmer groups attached to the twisted adjoint modular Galois representations associated to Hida families. This conjecture asserts the equality of the p-adic L-function interpoling the critical values of the symmetric square of the modular forms in these families and the characteristic ideal of the associated Selmer group. The idea is to introduce a third characteristic ideal containing informations on the congruences between cuspidal Siegel modular forms of genus 2 and the Klingen type E…
Let $f$ and $g$, of weights $k'>k\geq 2$, be normalised newforms for $Γ_0(N)$, for square-free $N>1$, such that, for each Atkin-Lehner involution, the eigenvalues of $f$ and $g$ are equal. Let $λ\mid\ell$ be a large prime divisor of the algebraic part of the near-central critical value $L(f\otimes g,\frac{k+k'-2}{2})$. Under certain hypotheses, we prove that $λ$ is the modulus of a congruence between the Hecke eigenvalues of a genus-two Yoshida lift of (Jacquet-Langlands correspondents of) $f$ and $g$ (vector-valued in general), and a non-endoscopic genus-two cusp form. In pursuit of this we a…
The aim of this article is to give some numerical data related to the order of the Selmer groups in twist families of elliptic curves. To do this we assume the Birch and Swinnerton-Dyer conjecture is true and we use a celebrated theorem of Waldspurger to get a fast algorithm to compute $% L_{E}(1)$. Having an extensive amount of data we compare the distribution of the order of the Selmer groups by functions of type $α\frac{(\log \log (X))^{1+\varepsilon}}{\log (X)}$ with $\varepsilon $ small. We discuss how the "best choice" of $α$ is depending on the conductor of the chosen elliptic curves an…
The elliptic curve $E_k \colon y^2 = x^3 + k$ admits a natural 3-isogeny $ϕ_k \colon E_k \to E_{-27k}$. We compute the average size of the $ϕ_k$-Selmer group as $k$ varies over the integers. Unlike previous results of Bhargava and Shankar on $n$-Selmer groups of elliptic curves, we show that this average can be very sensitive to congruence conditions on $k$; this sensitivity can be precisely controlled by the Tamagawa numbers of $E_k$ and $E_{-27k}$. As consequences, we prove that the average rank of the curves $E_k$, $k\in\mathbb Z$, is less than 1.21 and over $23\%$ (resp. $41\%$) of the cur…
For a totally real field $F$, a finite extension $\mathbf{F}$ of $\mathbf{F}_p$ and a Galois character $χ: G_F \to \mathbf{F}^{\times}$ unramified away from a finite set of places $Σ\supset \{\mathfrak{p} \mid p\}$ consider the Bloch-Kato Selmer group $H:=H^1_Σ(F, χ^{-1})$. In an earlier paper of the authors it was proved that the number $d$ of isomorphism classes of (non-semisimple, reducible) residual representations $\overlineρ$ giving rise to lines in $H$ which are modular by some $ρ_f$ (also unramified outside $Σ$) satisfies $d \geq n:= \dim_{\mathbf{F}} H$. This was proved under the assu…
Let f be a newform of weight 2k-2 and level 1. In this paper we provide evidence for the Bloch-Kato conjecture for modular forms. We demonstrate an implication that under suitable hypothesis if a prime divides the algebraic part of L(k,f), then the prime divides the order of the Selmer group associated to f. We demonstrate this by establishing a congruence between the Saito-Kurokawa lift of f and a cuspidal Siegel eigenform that is not a Saito-Kurokawa lift. We then examine what this congruence says in terms of Galois representations to produce a non-trivial p-torsion element in the Selmer gro…
The notion of the truncated Euler characteristic for Iwasawa modules is an extension of the notion of the usual Euler characteristic to the case when the homology groups are not finite. This article explores congruence relations between the truncated Euler characteristics for dual Selmer groups of elliptic curves with isomorphic residual representations, over admissible $p$-adic Lie extensions. Our results extend earlier congruence results from the case of elliptic curves with rank zero to the case of higher rank elliptic curves.
In section 2, we show a $p$-parity result in a $D_{2p^{n}}$-extension of number fields $L/K$ ($p\geq 5$) for the twist $1\oplus η\oplus τ$: W(E/K,1\oplus η\oplus τ)=(-1)^{< 1\oplusη\oplus τ, X_{p}(E/L)>}, where $E$ is an elliptic curve over $K,$ $η$ and $τ$ are respectively the quadratic character and an irreductible representation of degree 2 of $Gal(L/K)=D_{2p^{n}},$ and $X_{p}(E/L)$ is the $p$-Selmer group. The main novelty is that we use a congruence result between $% ε_{0}$-factors (due to Deligne) for the determination of local root numbers in bad cases (places of additive reduction abov…
Let $k > 3$ be an integer and $p$ a prime with $p > 2k-2$. Let $f$ be a newform of weight $2k-2$ and level 1 so that $f$ is ordinary at $p$ and $\barρ_{f}$ is irreducible. Under some additional hypotheses we prove that $ord_{p}(L_{alg}(k,f)) \leq ord_{p}(# S)$ where $S$ is the Pontryagin dual of the Selmer group associated to $ρ_{f} \otimes ε^{1-k}$ with $ε$ the $p$-adic cyclotomic character. We accomplish this by first constructing a congruence between the Saito-Kurokawa lift of $f$ and a non-CAP Siegel cusp form. Once this congruence is established, we use Galois representations to obtain th…
Congruences of $p$-adic $L$-functions of modular forms at non-ordinary primes
Institute of Mathematics University of the Philippines Diliman（菲律宾大学达曼数学研究所）；Department of Mathematics and Statistics University of Ottawa（渥太华大学数学与统计系）
Let $E/\mathbb{Q}$ be an elliptic curve and let $K$ be an imaginary quadratic field. Under a certain Heegner hypothesis, Kolyvagin constructed cohomology classes for $E$ using $K$-CM points and conjectured they did not all vanish. Conditional on this conjecture, he described the Selmer rank of $E$ using his system of classes. We extend work of Wei Zhang to prove new cases of Kolyvagin's conjecture by considering congruences of modular forms modulo large powers of $p $. Additionally, we prove an analogous result, and give a description of the Selmer rank, in a complementary "definite" case (usi…
Let $E$ be an elliptic curve defined over a number field $K$ where $p$ splits completely. Suppose that $E$ has good reduction at all primes above $p$. Generalizing previous works of Kobayashi and Sprung, we define multiply signed Selmer groups over the cyclotomic $\mathbb{Z}_p$-extension of a finite extension $F$ of $K$ where $p$ is unramified. Under the hypothesis that the Pontryagin duals of these Selmer groups are torsion over the corresponding Iwasawa algebra, we show that the Mordell-Weil ranks of $E$ over a subextension of the cyclotomic $\mathbb{Z}_p$-extension are bounded. Furthermore,…
We formulate integral Iwasawa main conjectures for suitable twists of a newform $f$ that is non-ordinary at $p$, over the cyclotomic $\mathbb{Z}_p$-extension, the anticyclotomic $\mathbb{Z}_p$-extensions (in both the definite and the indefinite cases) as well as the $\mathbb{Z}_p^2$-extension of an imaginary quadratic field $K$ where $p$ splits. In order to do so, we define Kobayashi-Sprung-style signed Coleman maps, which we use to introduce doubly-signed Selmer groups. In the same spirit, we construct signed (integral) Beilinson-Flach elements (out of the collection of unbounded Beilinson-Fl…
This paper aims at studying the Iwasawa $λ$-invariant of the $p$-primary Selmer group. We study the growth behaviour of $p$-primary Selmer groups in $p$-power degree extensions over non-cyclotomic $\mathbb{Z}_p$-extensions of a number field. We prove a generalization of Kida's formula in such a case. Unlike the cyclotomic $\mathbb{Z}_p$-extension, where all primes are finitely decomposed; in the $\mathbb{Z}_p$-extensions we consider, primes may be infinitely decomposed. In the second part of the paper, we study the relationship for Iwasawa invariants with respect to congruences, obtaining refi…
Let k be a positive integer divisible by 4, l>k a prime, and f an elliptic cuspidal eigenform of weight k-1, level 4, and non-trivial character. Let ρ_f be the l-adic Galois representation attached to f. In this paper we provide evidence for the Bloch-Kato conjecture for a twist of the adjoint motif of ρ_f in the following way. Let L(f,s) denote the symmetric square L-function of f. We prove that (under certain conditions) the l-adic valuation of the algebraic part of L(f, k) is no greater than the l-adic valuation of the order of S, where S is (the Pontryagin dual of) the Selmer group attache…
In this paper we prove that the p-adic L-function that interpolates the Rankin-Selberg product of a general modular form and a CM form of higher weight divides the characteristic ideal of the corresponding Selmer group. This is one divisibility of the Iwasawa main conjecture for the p-adic L-function. We prove this conjecture using the congruences between Klingen Eisensteinseries and cusp forms on the group GU(3; 1), following the strategy of a recent work of C. Skinner and E. Urban. This theorem can be used to deduce a converse of Gross-Zagier-Kolyvagin theorem and the precise BSD formula in …
