Related papers: A Note on the Main Conjecture over Q

Let $f$ be a newform of weight $k=2r$ and level $N$ with trivial nebentypus. Let $\mathfrak{p}\nmid 2N$ be a maximal ideal of the ring of integers of the coefficient field of $f$ such that the self-dual twist of the mod-$\mathfrak{p}$…

We study a natural question in the Iwasawa theory of algebraic curves of genus $>1$. Fix a prime number $p$. Let $X$ be a smooth, projective, geometrically irreducible curve defined over a number field $K$ of genus $g>1$, such that the…

Let $K$ be an imaginary quadratic field and $p$ be an odd prime number. Let $E/\mathbb{Q}$ be an elliptic curve with good ordinary reduction at $p$. We study the Iwasawa theory of $E$ over the anticyclotomic $\mathbb{Z}_p$-extension of $K$…

Let $\mathbb{K}$ be an imaginary quadratic field such that $2$ splits into two primes $\mathfrak{p}$ and $\bar{\mathfrak{p}}$. Let $\mathbb{K}_{\infty}$ be the unique $\mathbb{Z}_2$-extension of $\mathbb{K}$ unramified outside…

We prove one divisibility relation of the anticyclotomic Iwasawa Main Conjecture for a higher weight ordinary modular form $f$ and an imaginary quadratic field satisfying a "relaxed" Heegner hypothesis. Let $\Lambda$ be the anticyclotomic…

In this article, we construct zeta morphisms for the universal deformations of odd absolutely irreducible two dimensional mod p Galois representations satisfying some mild assumptions, and prove that our zeta morphisms interpolate Kato's…

Let $E/\mathbb{Q}$ be an elliptic curve and $p > 2$ be a prime of good ordinary reduction for $E$. Assume that the residue representation associated with $(E, p)$ is irreducible. In this paper, we prove more cases on several Iwasawa main…

In this paper, we define a two-variable analogue of Perrin-Riou's p-adic regulator map for the Iwasawa cohomology of a crystalline representation of the absolute Galois group of $\mathbf{Q}_p$, over a Galois extension of $\mathbf{Q}_p$…

Let $p$ be a prime number. In this article we present a theorem, suggested by Peter Scholze, which states that the absolute Galois group of $\mathbf{Q}_p$ is the \'etale fundamental group of a certain object $Z$ which is defined over an…

Let p be an odd prime. We give an unconditional proof of the equivariant Iwasawa main conjecture for totally real fields for an infinite class of one-dimensional non-abelian p-adic Lie extensions. Crucially, this result does not depend on…

Let p > 2 be prime. We state and prove (under mild hypotheses on the residual representation) a geometric refinement of the Breuil-M\'ezard conjecture for 2-dimensional mod p representations of the absolute Galois group of Qp. We also state…

We prove a `Weierstrass Preparation Theorem' and develop an explicit descent formalism in the context of Whitehead groups of non-commutative Iwasawa algebras. We use these results to describe the precise connection between the main…

Let $E/\mathbb{Q}$ an elliptic curve with good supersingular reduction at a prime $p\geq 5$, and $K$ an imaginary quadratic field such that the root number of $E$ over $K$ equals $-1$. When $p$ splits in $K$, Castella and Wan formulated the…

We study a geometric analogue of the Iwasawa Main Conjecture for abelian varieties in the two following cases: constant ordinary abelian varieties over $Z_p^d$-extensions of function fields ($d\geq 1$) ramified at a finite set of places,…

In this article we generalize results of Clozel and Ray (for $SL_2$ and $SL_n$ respectively) to give explicit ring-theoretic presentation in terms of a complete set of generators and relations of the Iwasawa algebra of the pro-$p$ Iwahori…

Recently, D. Burns and C. Greither (Invent. Math., 2003) deduced an equivariant version of the main conjecture for abelian number fields. This was the key to their proof of the equivariant Tamagawa number conjecture. A. Huber and G. Kings…

For an odd prime p, we determine a minimal set of topological generators of the pro-p Iwahori subgroup of a split reductive group G over Z\_p. In the simple adjoint case and for any sufficiently large regular prime p, we also construct…

In this paper and a forthcoming joint one with Y. Hachimori we study Iwasawa modules over an infinite Galois extension K of a number field k whose Galois group G=G(K/k) is isomorphic to the semidirect product of two copies of the p-adic…

This note is a development of our two previous papers, arXiv:1212.3392v1 and 1306.3660v1. The fundamental question is whether there exists a Galois theory, in which the Galois group is a quantum group. For a linear equations with respect to…

We describe the set of points of the trianguline variety over a given local Galois representation. Global analogues describing companion points in eigenvariety by [Bre14] and [HN17], can be thought of as a rational analogue to the weight…