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 $\Zp$-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.
Let $\mathcal{O}$ be the ring of integers of a finite extension of $\mathbb{Q}_p$. We prove two control theorems for fine Selmer groups of general cofinitely generated modules over $\mathcal{O}$. We apply these control theorems to compare the fine Selmer group attached to a modular form $f$ over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ to its counterpart attached to the conjugate modular form $\overline{f}$.
For any abelian variety $A$ over a number field, we construct an extension of the Tate-Shafarevich group by the Bloch-Tamagawa space using the recent work of Lichtenbaum and Flach. This gives a new example of a Zagier sequence for the Selmer group of $A$.
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.
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.
We investigate fine Selmer groups for elliptic curves and for Galois representations over a number field. More specifically, we discuss Conjecture A, which states that the fine Selmer group of an elliptic curve over the cyclotomic extension is a finitely generated $\mathbb{Z}_p$-module. The relationship between this conjecture and Iwasawa's classical $μ=0$ conjecture is clarified. We also present some partial results towards the question whether Conjecture A is invariant under isogenies.
Let A be an abelian variety defined over anumber field F. In this paper, we will investigate the growth of p-rank of the fine Selmer group in three situations. In particular, in each of these situations, we show that there is a strong analogy between the growth of the p-rank of the fine Selmer group and the growth of the p-rank of the class groups.
Let $p$ be an odd prime and $F_\infty$ be a $\mathbb{Z}_p$-extension of a number field $F$. Given an elliptic curve $E$ over $F$, we study the structure of the fine Selmer group over $F_\infty$. It is shown that under certain conditions, the fine Selmer group is a cofinitely generated module over $\mathbb{Z}_p$ and furthermore, we obtain an upper bound for its corank (i.e., the $λ$-invariant), in terms of various local and global invariants.
We compare the Pontryagin duals of fine Selmer groups of two congruent $p$-adic Galois representations over admissible pro-$p$, $p$-adic Lie extensions $K_\infty$ of number fields $K$. We prove that in several natural settings the $π$-primary submodules of the Pontryagin duals are pseudo-isomorphic over the Iwasawa algebra; if the coranks of the fine Selmer groups are not equal, then we can still prove inequalities between the $μ$-invariants. In the special case of a $\mathbb{Z}_p$-extension $K_\infty/K$, we also compare the Iwasawa $λ$-invariants of the fine Selmer groups, even in situations …
In this paper, we will study the pseudo-nullity of the fine Selmer group and its related question. Namely, we investigate certain situations, where one can deduce the pseudo-nullity of the dual fine Selmer group of a general Galois module over a admissible $p$-adic Lie extension $F_{\infty}$ from the knowledge that pseudo-nullity of the Galois group of the maximal abelian unramified pro-$p$ extension of $F_{\infty}$ at which every prime of $F_{\infty}$ 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 …
Let $E/\mathbb{Q}$ be an elliptic curve, $p$ an odd prime and $K_{\infty}/K$ the anticyclotomic $\mathbb{Z}_p$-extension of a quadratic imaginary field $K$. In a previous article the author conjectured that the fine $p^{\infty}$-Selmer group $R_{p^{\infty}}(E/K_{\infty})$ is confinitely generated over $\mathbb{Z}_p$. In this note we prove this conjecture assuming some hypotheses on $E$ and $p$.
Pseudonullity for fine Selmer groups over multiple $\mathbb{Z}_p^{d}$-extensions
Morningside Center of Mathematics, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
Let $A$ be an abelian variety defined over a number field $F$. We prove a control theorem for the fine Selmer group of the abelian variety $A$ which essentially says that the kernel and cokernel of the natural restriction maps in a given $\mathbb{Z}_p$-extension $F_\infty/F$ are finite and bounded. We emphasise that our result does not have any constraints on the reduction of $A$ and the ramification of $F_\infty/F$. As a first consequence of the control theorem, we show that the fine Tate-Shafarevich group over an arbitrary $\mathbb{Z}_p$-extension has trivial $Λ$-corank. We then derive an as…
In this article, we study the pseudo-isomorphism class of the dual fine Selmer group $X$ attached to a $p$-adic Galois deformation whose deformation ring $Λ$ is isomorphic to the ring of formal power series. By using the "Kolyvagin system" arising from a given Euler system $c$, we shall construct ideals $C_i(c)$ of $Λ$, and prove that the ideals $C_i(c)$ approximate the higher Fitting ideals of $X$ under suitable hypothesis. In particular, we shall prove that the ideals $C_i(c)$ arising from the Euler system of Beilinson-Kato elements determine the pseudo-isomorphism classes of the dual fine S…
Let $E/\mathbb{Q}$ be an elliptic curve, $p$ a prime and $K_{\infty}/K$ the anticyclotomic $\mathbb{Z}_p$-extension of a quadratic imaginary field $K$ satisfying the Heegner hypothesis. In this paper we make a conjecture about the fine Selmer group over $K_{\infty}$. We also make a conjecture about the structure of the module of Heegner points in $E(K_{\mathfrak{p}_{\infty}})/p$ where $K_{\mathfrak{p}_{\infty}}$ is the union of the completions of the fields $K_n$ at a prime of $K_{\infty}$ above $p$. We prove that these conjectures are equivalent. When $E$ has supersingular reduction at $p$ we…
Fix two distinct odd primes $p$ and $q$. We study "$p\ne q$" Iwasawa theory in two different settings. Let $K$ be an imaginary quadratic field of class number 1 such that both $p$ and $q$ split in $K$. We show that under appropriate hypotheses, the $p$-part of the ideal class groups is bounded over finite subextensions of an anticyclotomic $\mathbb{Z}_q$-extension of $K$. Let $F$ be a number field and let $A_{/F}$ be an abelian variety with $A[p]\subseteq A(F)$. We give sufficient conditions for the $p$-part of the fine Selmer groups of $A$ over finite subextensions of a $\mathbb{Z}_q$-extensi…
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…
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.
In this paper, we study completely faithful torsion $\mathbb{Z}_p[[G]]$-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 $\mathbb{Z}_p[[G]]$-modules, i.e., non-trivial torsion modules whose global annihilator ideal is zero. We then show that such $\mathbb{Z}_p[[G]]$-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 gr…
This note shows how to use the framework of Euler characteristic formulae to study Selmer groups of abelian varieties in certain dihedral or anticyclotomic extensions of CM fields via Iwasawa main conjectures, and in particular how to verify the p-part of the refined Birch and Swinnerton-Dyer conjecture in this setting. When the Selmer group is cotorsion with respect to the associated Iwasawa algebra, we obtain the p-part of formula predicted by the refined Birch and Swinnerton-Dyer conjecture. When the Selmer group is not cotorsion with respect to the associated Iwasawa algebra, we give a con…
Let $E$ be an elliptic curve defined over a number field $K$ with supersingular reduction at all primes of $K$ above $p$. If $K_{\infty}/K$ is a $\mathbb{Z}_p$-extension such that $E(K_{\infty})[p^{\infty}]$ is finite and $H^2(G_S(K_{\infty}), E[p^{\infty}])=0$, then we prove that the $Λ$-torsion subgroup of the Pontryagin dual of $\text{Sel}_{p^{\infty}}(E/K_{\infty})$ is pseudo-isomorphic to the Pontryagin dual of the fine Selmer group of $E$ over $K_{\infty}$. This is the Galois-cohomological analog of a flat-cohomological result of Wingberg.
In this note, we compare the dual Selmer groups of an abelian variety with that of its dual over certain large Galois field. We give formula which relates the generalized Iwasawa $μ$-invariants associated with their dual Selmer groups under the natural isogeny.
In this paper, we study the Fitting ideals of Selmer groups over finite subextensions in the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ of an elliptic curve over $\mathbb{Q}$. Especially, we present a proof of the "weak main conjecture" à la Mazur and Tate for elliptic curves with good (supersingular) reduction at an odd prime $p$. We also prove the "strong main conjecture" suggested by the second named author under the validity of the $\pm$-main conjecture and the vanishing of a certain error term. The key idea is the explicit comparison among "finite layer objects", "$\pm$-objects",…
On Conjectures of $μ$-Invariant for Selmer Groups at $p=2$
The University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）；Morningside Center of Mathematics（中国科学院数学与系统科学研究院；中国科学院数学与系统科学研究所）；Academy of Mathematics and Systems Sciences, Chinese Academy of Science
Let us consider a $p$-adic Lie extension of a number field $K$ which fits into the setting of non-commutative Iwasawa theory formulated by Coates-Fukaya-Kato-Sujatha-Venjakob. For the first main result, we will prove the control theorem of Selmer group associated to a motive, which generalizes previous results by the second author and Greenberg. For the second main result, we prove the functional equation of the dual Selmer groups, which generalizes previous results by Greenberg, Perrin-Riou and Zabradi. Note that our proof of the functional equation is different from the proof of Zabradi even…
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…
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.
Let E be an elliptic curve defined over a number field K. Michael Larsen conjectured that for any finitely generated subgroup G of Gal(\bar K/K), the Mordell-Weil rank of E is unbounded in number fields fixed by G. We prove that the conjecture holds over K=Q for both the analytic rank and the p-infinity Selmer rank of E for every odd prime p. For arbitrary E/K, we show that Larsen's conjecture follows from the standard conjectures for ranks of elliptic curves, provided K has a real place or E has non-integral j-invariant.
Variation of Iwasawa Invariants for Ordinary Representations
Harish Chandra Research Institute, A CI of Homi Bhabha National Institute（哈拉·钱德拉研究所，霍米·巴巴国立学院下属机构）；Indian Institute of Science Education and Research Mohali（印度科学教育与研究学院莫哈利分校）；Indian Institute of Science, Bengaluru（印度科学学院班加罗尔校区）
Let $E$ be an elliptic curve over an imaginary quadratic field $K$, and $p$ be an odd prime such that the residual representation $E[p]$ is reducible. The $μ$-invariant of the fine Selmer group of $E$ over the anticyclotomic $\mathbb{Z}_p$-extension of $K$ is studied. We do not impose the Heegner hypothesis on $E$, thus allowing certain primes of bad reduction to decompose infinitely in the anticyclotomic $\mathbb{Z}_p$-extension. It is shown that the fine $μ$-invariant vanishes if certain explicit conditions are satisfied. Further, a partial converse is proven.
A_k = {1, a_2, ..., a_k} is an h-basis for n if every positive integer not exceeding n can be expressed as the sum of no more than h values a_i; we write n = n_h(A_k). An extremal h-basis A_k is one for which n is as large as possible, and then we write n = n_h(k). The "local" Postage Stamp Problem is concerned with properties of particular sets A_k, and it is clear that sets where n_h(A_k) does not exceed a_k are of little interest. We define h_0(k) to be the smallest value of h for which n_h(A_k) exceeds a_k; such sets are called "admissible". We say that a value n can be "generated" by A_k …
In this note we define the notion of Tate-Shafarevich group and Selmer group of the Chow group of an abelian variety defined over a number field. In this context we give positive answer to the question of Colliot-Thélène that the Chow group of zero cycles modulo $n$ is finite over any number field.
In this note we revisit classic work of Soundararajan on class groups of imaginary quadratic fields. Let $A,B,g \ge 3$ be positive integers such that $\gcd(A,B)$ is square-free. We refine Soundararajan's result to show that if $4 \nmid g$ or if $A$ and $B$ satisfy certain conditions, then the number of negative square-free $D \equiv A \pmod{B}$ down to $-X$ such that the ideal class group of $\mathbb{Q} (\sqrt{D})$ contains an element of order $g$ is bounded below by $X^{\frac{1}{2} + ε(g) - ε}$, where the exponent is the same as in Soundararajan's theorem. Combining this with a theorem of Fre…
In this article, we study a relation between certain quotients of ideal class groups and the cyclotomic Iwasawa module $X_\infty$ of the Pontrjagin dual of the fine Selmer group of an elliptic curve $E$ defined over $\mathbb{Q}$. We consider the Galois extension field $K^E_n$ of $\mathbb{Q}$ generated by coordinates of all $p^n$-torsion points of $E$, and introduce a quotient $A^E_n$ of the $p$-sylow subgroup of the ideal class group of $K^E_n$ cut out by the modulo $p^n$ Galois representation $E[p^n]$. We describe the asymptotic behavior of $A^E_n$ by using the Iwasawa module $X_\infty$. In p…
