We study the variation of Iwasawa invariants of the anticyclotomic Selmer groups of congruent modular forms under the Heegner hypothesis. In particular, we show that even if the Selmer groups we study may have positive coranks, the mu-invariant vanishes for one modular form if and only if it vanishes for the other, and that their lambda-invariants are related by an explicit formula. This generalizes results of Greenberg-Vatsal for the cyclotomic extension, as well as results of Pollack-Weston and Castella-Kim-Longo for the anticyclotomic extension when the Selmer groups in question are cotorsion.
We study the Selmer group associated to a $p$-ordinary newform $f \in S_{2r}(Γ_0(N))$ over the anticyclotomic $\mathbb{Z}_p$-extension of an imaginary quadratic field $K/\mathbb{Q}$. Under certain assumptions, we prove that this Selmer group has no proper $Λ$-submodules of finite index. This generalizes work of Bertolini in the elliptic curve case. We also offer both a correction and an improvement to an earlier result on Iwasawa invariants of congruent modular forms by the present authors.
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.
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…
Let $E/\mathbb{Q}$ be an elliptic curve, $p$ a prime where $E$ has ordinary reduction and $K_{\infty}/K$ the anticyclotomic $\mathbb{Z}_p$-extension of a quadratic imaginary field $K$ satisfying the Heegner hypothesis. We give sufficient conditions on $E$ and $p$ in order to ensure that $\text{Sel}_{p^{\infty}}(E/K_{\infty})$ is a cofree $Λ$-module of rank one. We also show that these conditions imply that $\text{rank}(E(K_n))=p^n$ for all $n \geq 0$ and that the $p$-primary subgroup of the Tate-Shafarevich group of $E/K_n$ is trivial for all $n \geq 0$.
Let $A$ be an abelian variety defined over a number field $F$ with supersingular reduction at all primes of $F$ above $p$. We establish an equivalence between the weak Leopoldt conjecture and the expected value of the corank of the classical Selmer group of $A$ over a $p$-adic Lie extension (not neccesasily containing the cyclotomic $\Zp$-extension). As an application, we obtain the exactness of the defining sequence of the Selmer group. In the event that the $p$-adic Lie extension is one-dimensional, we show that the dual Selmer group has no nontrivial finite submodules. Finally, we show that…
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 give a new proof to a theorem of Bertolini which determines the value of the $Λ$-corank of $\text{Sel}(E/K_{\infty})$ in the case where $E$ has ordinary reduction at $p$. In the case where $E$ has supersingular reduction at $p$ we make a conjecture about the structure of the module of Heegner points mod $p$. Assuming this conjecture we give a new proof to a theorem of Ciperiani which determines…
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.
Let $p$ be an odd prime and $K$ an imaginary quadratic field where $p$ splits. Under appropriate hypotheses, Bertolini showed that the Selmer group of a $p$-ordinary elliptic curve over the anticyclotomic $\mathbb Z_p$-extension of $K$ does not admit any proper $Λ$-submodule of finite index, where $Λ$ is a suitable Iwasawa algebra. We generalize this result to the plus and minus Selmer groups (in the sense of Kobayashi) of $p$-supersingular elliptic curves. In particular, in our setting the plus/minus Selmer groups have $Λ$-corank one, so they are not $Λ$-cotorsion. As an application of our ma…
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 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.
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 $p$.
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…
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 …
We prove analogues of the major algebraic results of Greenberg-Vatsal for Selmer groups of $p$-ordinary newforms over $\mathbf{Z}_p$-extensions which may be neither cyclotomic nor anticyclotomic, under a number of technical hypotheses, including a cotorsion assumption on the Selmer groups. The main complication which arises in our work is the possible presence of finite primes which can split completely in the $\mathbf{Z}_p$-extension being considered, resulting in the local cohomology groups that appear in the definition of the Selmer groups being significantly larger than they are in the cas…
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 $q$ be a prime with $q \equiv 7 \mod 8$, and let $K=\mathbb{Q}(\sqrt{-q})$. Then $2$ splits in $K$, and we write $\mathfrak{p}$ for either of the primes $K$ above $2$. Let $K_\infty$ be the unique $\mathbb{Z}_2$-extension of $K$ unramified outside $\mathfrak{p}$ with $n$-th layer $K_n$. For certain quadratic extensions $F/K$, we prove a simple exact formula for the $λ$-invariant of the Galois group of the maximal abelian 2-extension unramified outside $\mathfrak{p}$ of the field $F_\infty = FK_\infty$. Equivalently, our result determines the exact $\mathbb{Z}_2$-corank of certain Selmer gr…
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…
For a prime $\ell$ and an abelian variety $A$ over a global field $K$, the $\ell$-parity conjecture predicts that, in accordance with the ideas of Birch and Swinnerton-Dyer, the $\mathbb{Z}_{\ell}$-corank of the $\ell^{\infty}$-Selmer group and the analytic rank agree modulo $2$. Assuming that $\mathrm{char} K > 0$, we prove that the $\ell$-parity conjecture holds for the base change of $A$ to the constant quadratic extension if $\ell$ is odd, coprime to $\mathrm{char} K$, and does not divide the degree of every polarization of $A$. The techniques involved in the proof include the étale cohomo…
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…
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…
