On the non-abelian Brumer–Stark conjecture and the equivariant Iwasawa main conjecture

Abstract 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 {\mathbb{Z}_{p}} -tower of an imaginary quadratic field K where the prime p splits completely. Our goal in this portion is to prove the Iwasawa main conjecture for suitable twists of f assuming that f is p-ordinary, both in the definite and indefinite setups simultaneously, via an analysis of Beilinson–Flach elements.

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Arithmetic of higher-dimensional orbifolds and a mixed Waring problem

Mathematische Zeitschrift ◽

10.1007/s00209-021-02695-w ◽

2021 ◽

Author(s):

Tim Browning ◽

Shuntaro Yamagishi

Keyword(s):

Circle Method ◽

Rational Points ◽

Mean Value ◽

Mean Value Theorem ◽

Waring’S Problem ◽

Waring Problem ◽

Waring's Problem ◽

Main Conjecture ◽

Higher Dimensional ◽

Thin Set

AbstractWe study the density of rational points on a higher-dimensional orbifold $$(\mathbb {P}^{n-1},\Delta )$$ ( P n - 1 , Δ ) when $$\Delta $$ Δ is a $$\mathbb {Q}$$ Q -divisor involving hyperplanes. This allows us to address a question of Tanimoto about whether the set of rational points on such an orbifold constitutes a thin set. Our approach relies on the Hardy–Littlewood circle method to first study an asymptotic version of Waring’s problem for mixed powers. In doing so we make crucial use of the recent resolution of the main conjecture in Vinogradov’s mean value theorem, due to Bourgain–Demeter–Guth and Wooley.

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THE IWASAWA MAIN CONJECTURE FOR HILBERT MODULAR FORMS

Forum of Mathematics Sigma ◽

10.1017/fms.2015.16 ◽

2015 ◽

Vol 3 ◽

Cited By ~ 9

Author(s):

XIN WAN

Keyword(s):

Modular Forms ◽

Base Change ◽

Hilbert Modular Forms ◽

Selmer Group ◽

Hilbert Modular Form ◽

Main Conjecture ◽

Hilbert Modular ◽

Trivial Character ◽

The One ◽

Ordinary Reduction

Following the ideas and methods of a recent work of Skinner and Urban, we prove the one divisibility of the Iwasawa main conjecture for nearly ordinary Hilbert modular forms under certain local hypotheses. As a consequence, we prove that for a Hilbert modular form of parallel weight, trivial character, and good ordinary reduction at all primes dividing$p$, if the central critical$L$-value is zero then the$p$-adic Selmer group of it has rank at least one. We also prove that one of the local assumptions in the main result of Skinner and Urban can be removed by a base-change trick.

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A New Approach to the Main Conjecture on Algebraic-Geometric MDS Codes

Codes, Designs and Geometry ◽

10.1007/978-1-4613-1423-3_11 ◽

1996 ◽

pp. 111-116

Author(s):

Judy L. Walker

Keyword(s):

Mds Codes ◽

New Approach ◽

Main Conjecture

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Connectivity for Random Graphs from a Weighted Bridge-Addable Class

The Electronic Journal of Combinatorics ◽

10.37236/2596 ◽

2012 ◽

Vol 19 (4) ◽

Cited By ~ 4

Author(s):

Colin McDiarmid

Keyword(s):

Random Graph ◽

Random Graphs ◽

Planar Graphs ◽

Additional Assumption ◽

General Distribution ◽

Expected Number ◽

Recent Interest ◽

Main Conjecture

There has been much recent interest in random graphs sampled uniformly from the $n$-vertex graphs in a suitable structured class, such as the class of all planar graphs. Here we consider a general bridge-addable class $\cal A$ of graphs -- if a graph is in $\cal A$ and $u$ and $v$ are vertices in different components then the graph obtained by adding an edge (bridge) between $u$ and $v$ must also be in $\cal A$. Various bounds are known concerning the probability of a random graph from such a class being connected or having many components, sometimes under the additional assumption that bridges can be deleted as well as added. Here we improve or amplify or generalise these bounds (though we do not resolve the main conjecture). For example, we see that the expected number of vertices left when we remove a largest component is less than 2. The generalisation is to consider `weighted' random graphs, sampled from a suitable more general distribution, where the focus is on the bridges.

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Introduction to Skinner-Urban’s Work on the Iwasawa Main Conjecture for GL2

Contributions in Mathematical and Computational Sciences - Iwasawa Theory 2012 ◽

10.1007/978-3-642-55245-8_2 ◽

2014 ◽

pp. 35-61 ◽

Cited By ~ 1

Author(s):

Xin Wan

Keyword(s):

Main Conjecture

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Non-communtative Iwasawa main conjecture

International Journal of Number Theory ◽

10.1142/s1793042120501067 ◽

2020 ◽

Vol 16 (09) ◽

pp. 2041-2094

Author(s):

Malte Witte

Keyword(s):

Functional Equation ◽

Galois Group ◽

Function Field ◽

Function Fields ◽

Abelian Varieties ◽

Main Conjecture

We formulate and prove an analogue of the non-commutative Iwasawa Main Conjecture for [Formula: see text]-adic representations of the Galois group of a function field of characteristic [Formula: see text]. We also prove a functional equation for the resulting non-commutative [Formula: see text]-functions. As corollaries, we obtain non-commutative generalizations of the main conjecture for Picard-[Formula: see text]-motives of Greither and Popescu and a main conjecture for abelian varieties over function fields in precise analogy to the [Formula: see text] main conjecture of Coates, Fukaya, Kato, Sujatha and Venjakob.

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Euler characteristics and their congruences in the positive rank setting

Canadian Mathematical Bulletin ◽

10.4153/s0008439520000429 ◽

2020 ◽

pp. 1-18

Author(s):

Anwesh Ray ◽

R. Sujatha

Keyword(s):

Elliptic Curves ◽

Euler Characteristic ◽

Selmer Groups ◽

Euler Characteristics ◽

Rank Zero ◽

Homology Groups ◽

Main Conjecture ◽

Congruence Relations ◽

Higher Rank ◽

Positive Rank

Abstract 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. The results provide evidence for the p-adic Birch and Swinnerton-Dyer formula without assuming the main conjecture.

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