Iwasawa theory
View article
p‐adic heights of Heegner points and Beilinson–Flach classes Open
We give a new proof of Howard's ‐adic Gross–Zagier formula Compos. Math . 141 (2005) 811–846. MR 2148200 (2006f:11074)], which we extend to the context of indefinite Shimura curves over attached to nonsplit quaternion algebras. This formul…
View article
Arithmetic properties of signed Selmer groups at non-ordinary primes Open
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 -cyclotomic extensions. First, …
View article
EULER SYSTEMS FOR HILBERT MODULAR SURFACES Open
We construct an Euler system—a compatible family of global cohomology classes—for the Galois representations appearing in the geometry of Hilbert modular surfaces. If a conjecture of Bloch and Kato on injectivity of regulator maps holds, t…
View article
On Iwasawa theory, zeta elements for 𝔾m, and the equivariant Tamagawa number conjecture Open
We develop an explicit “higher-rank” Iwasawa theory for zeta elements associated to the multiplicative group over abelian extensions of number fields. We show this theory leads to a concrete new strategy for proving special cases of the eq…
View article
Iwasawa theory for the symmetric square of a modular form Open
We construct a compatible family of global cohomology classes (an Euler system) for the symmetric square of a modular form, and apply this to bounding Selmer groups of the symmetric square Galois representation and its twists.
View article
Global group laws and equivariant bordism rings Open
For every abelian compact Lie group AA, we prove that the homotopical AA-equivariant complex bordism ring, introduced by tom Dieck (1970), is isomorphic to the AA-equivariant Lazard ring, introduced by Cole–Greenlees–Kriz (2000). This sett…
View article
CODIMENSION TWO CYCLES IN IWASAWA THEORY AND ELLIPTIC CURVES WITH SUPERSINGULAR REDUCTION Open
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 analyt…
View article
Iwasawa theory for Rankin-Selberg products ofp-nonordinary eigenforms Open
Let f and g be two modular forms which are non-ordinary at p. The theory of Beilinson-Flach elements gives rise to four rank-one non-integral Euler systems for the Rankin-Selberg convolution f⊗g, one for each choice of p-stabilisations of …
View article
The Iwasawa Main Conjecture for elliptic curves at odd supersingular primes Open
In this paper, we prove the Iwasawa main conjecture for elliptic curves at an odd supersingular prime p. Some consequences are the p-parts of the leading term formulas in the Birch and Swinnerton-Dyer conjectures for analytic rank 0 or 1.
View article
On the Iwasawa main conjectures for modular forms at non-ordinary primes Open
In this paper, we prove under mild hypotheses the Iwasawa main conjectures of Lei--Loeffler--Zerbes for modular forms of weight $2$ at non-ordinary primes. Our proof is based on the study of the two-variable analogues of these conjectures …
View article
Anticyclotomic Iwasawa theory of elliptic modular forms at non-ordinary primes Open
This article is a continuation of our previous work on the Iwasawa theory of an elliptic modular form over an imaginary quadratic field $K$, where the modular form in question was assumed to be ordinary at a fixed odd prime $p$. We formula…
View article
Comparing the $\pi$-primary submodules of the dual Selmer groups Open
In this paper, we compare the structure of Selmer groups of certain classes\nof Galois representations over an admissible $p$-adic Lie extension. Namely, we\nshow that the $\\pi$-primary submodules of the Pontryagin dual of the Selmer\ngro…
View article
A proof of Perrin-Riou’s Heegner point mainconjecture Open
Let $E/\mathbf{Q}$ be an elliptic curve of conductor $N$, let $p>3$ be a prime where $E$ has good ordinary reduction, and let $K$ be an imaginary quadratic field satisfying the Heegner hypothesis. In 1987, Perrin-Riou formulated an Iwasawa…
View article
\(\Lambda\)-adic Gross-Zagier formula for elliptic curves at supersingular primes Open
In 2013, Kobayashi proved an analogue of Perrin-Riou's \(p\)-adic Gross-Zagier formula for elliptic curves at supersingular primes. In this talk, we will explain an extension of Kobayashi's result to the \(\Lambda\)-adic setting. The main …
View article
Iwasawa Theory and $F$-Analytic Lubin-Tate $(\varphi,\Gamma)$-Modules Open
Let K be a finite extension of \bold{Q}p . We use the theory of (\varphi,\Gamma) -modules in the Lubin-Tate setting to construct some corestriction-compatible families of classes in the cohomology of V , for certain representations V of \m…
View article
On the Non Commutative Iwasawa Main Conjecture for Abelian Varieties over Function Fields Open
We establish the Iwasawa main conjecture for semistable abelian varieties over a function field of characteristic p under certain restrictive assumptions. Namely we consider p -torsion free p -adic Lie extensions of the base field which co…
View article
Aspects of Iwasawa theory over function fields Open
We consider $\mathbb{Z}_p^{\mathbb{N}}$-extensions $\mathcal{F}$ of a global function field $F$ and study various aspects of Iwasawa theory with emphasis on the two main themes already (and still) developed in the number fields case as wel…
View article
THE <i>p</i>-ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES, II: NONSPLIT PRIMES Open
The formula of the title relates p -adic heights of Heegner points and derivatives of p -adic L -functions. It was originally proved by Perrin-Riou for p -ordinary elliptic curves over the rationals, under the assumption that p splits in t…
View article
Values of the Riemann Zeta Function at the Odd Positive Integers and Iwasawa Theory Open
There are still many arithmetic mysteries surrounding the values of the Riemann zeta function at the odd positive integers greater than one. For example, the matter of their irrationality, let alone transcendence, remains largely unknown. …
View article
Integral Iwasawa theory of Galois representations for non-ordinary primes Open
In this paper, we study the Iwasawa theory of a motive whose Hodge-Tate weights are $0$ or $1$ (thence in practice, of a motive associated to an abelian variety) at a non-ordinary prime, over the cyclotomic tower of a number field that is …
View article
Iwasawa main conjecture for the Carlitz cyclotomic extension and applications Open
We prove an Iwasawa Main Conjecture for the class group of the p-cyclotomic extension F of the function field Fq(θ) (p is a prime of Fq[θ]), showing that its Fitting ideal is generated by a Stickelberger element. We use this and a link bet…
View article
Fitting ideals of p-ramified Iwasawa modules over totally real fields Open
We completely calculate the Fitting ideal of the classical p -ramified Iwasawa module for any abelian extension K / k of totally real fields, using the shifted Fitting ideals recently developed by the second author. This generalizes former…
View article
ON THE DISTRIBUTION OF IWASAWA INVARIANTS ASSOCIATED TO MULTIGRAPHS Open
Let $\ell $ be a prime number. The Iwasawa theory of multigraphs is the systematic study of growth patterns in the number of spanning trees in abelian $\ell $ -towers of multigraphs. In this context, growth patterns are realized by certain…
View article
Euler characteristics and their congruences for multisigned Selmer groups Open
The notion of the truncated Euler characteristic for Iwasawa modules is a generalization of the the usual Euler characteristic to the case when the Selmer groups are not finite. Let p be an odd prime, $E_{1}$ and $E_{2}$ be elliptic curves…
View article
Norm-Compatible Systems of Galois Cohomology Classes for $\mathbf{GSp}_6$ Open
We construct global cohomology classes in the middle degree plus one cohomology group of the Shimura variety of the symplectic group $\mathbf{GSp}_6$ compatible when one varies the level at $p$. These classes are expected constituents of a…
View article
Fitting invariants in equivariant Iwasawa theory Open
The main conjectures in Iwasawa theory predict the relationship between the Iwasawa modules and the $p$-adic $L$-functions. Using a certain proved formulation of the main conjecture, Greither and Kurihara described explicitly the (initial)…
View article
Characteristic ideals and Selmer groups Open
Let A be an abelian variety defined over a global field F of positive characteristic p and let \mathcalF/F be a Z_p^\infty-extension, unramified outside a finite set of places of F. Assuming that all ramified places are totally ramified, w…
View article
Variation of anticyclotomic Iwasawa invariants in Hida families Open
Building on the construction of big Heegner points in the quaternionic setting, and their relation to special values of Rankin-Selberg $L$-functions, we obtain anticyclotomic analogues of the results of Emerton-Pollack-Weston on the variat…
View article
Statistics for Anticyclotomic Iwasawa Invariants of Elliptic Curves Open
We study the average behaviour of the Iwasawa invariants for Selmer groups of elliptic curves, considered over anticyclotomic $\mathbb{Z}_p$-extensions in both the definite and indefinite settings. The results in this paper lie at the inte…
View article
Anticyclotomic p-ordinary Iwasawa Theory of Elliptic Modular Forms Open
This is the first in a series of articles where we will study the Iwasawa theory of an elliptic modular form f along the anticyclotomic Zp-tower of an imaginary quadratic field K where the prime p splits completely. Our goal in this portio…