scholarly journals Stickelberger series and Main Conjecture for function fields

2021 ◽  
Vol 65 ◽  
pp. 459-498
Author(s):  
Andrea Bandini ◽  
Edoardo Coscelli
Keyword(s):  
Function Fields ◽  
Main Conjecture

Non-communtative Iwasawa main conjecture

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.

scholarly journals The Iwasawa Main Conjecture for constant ordinary abelian varieties over function fields

2016 ◽  
Vol 112 (6) ◽  
pp. 1040-1058 ◽  
Author(s):  
King Fai Lai ◽  
Ignazio Longhi ◽  
Ki-Seng Tan ◽  
Fabien Trihan
Keyword(s):  
Function Fields ◽  
Abelian Varieties ◽  
Main Conjecture

scholarly journals CONTROL THEOREMS FOR ELLIPTIC CURVES OVER FUNCTION FIELDS

2009 ◽  
Vol 05 (02) ◽  
pp. 229-256 ◽  
Author(s):  
A. BANDINI ◽  
I. LONGHI
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Galois Extension ◽  
Function Field ◽  
Function Fields ◽  
Global Field ◽  
Selmer Groups ◽  
Characteristic P ◽  
Main Conjecture ◽  
Fitting Ideal

Let F be a global field of characteristic p > 0, 𝔽/F a Galois extension with [Formula: see text] and E/F a non-isotrivial elliptic curve. We study the behavior of Selmer groups SelE(L)l (l any prime) as L varies through the subextensions of 𝔽 via appropriate versions of Mazur's Control Theorem. In the case l = p, we let 𝔽 = ∪ 𝔽d where 𝔽d/F is a [Formula: see text]-extension. We prove that Sel E(𝔽d)p is a cofinitely generated ℤp[[ Gal (ℤd/F)]]-module and we associate to its Pontrjagin dual a Fitting ideal. This allows to define an algebraic L-function associated to E in ℤp[[Gal(ℤ/F)]], providing an ingredient for a function field analogue of Iwasawa's Main Conjecture for elliptic curves.

scholarly journals The Iwasawa Main Conjecture for semistable abelian varieties over function fields

2015 ◽  
Vol 282 (1-2) ◽  
pp. 485-510 ◽  
Author(s):  
King Fai Lai ◽  
Ignazio Longhi ◽  
Ki-Seng Tan ◽  
Fabien Trihan
Keyword(s):  
Function Fields ◽  
Abelian Varieties ◽  
Main Conjecture

scholarly journals Anticyclotomic p-ordinary Iwasawa theory of elliptic modular forms

2018 ◽  
Vol 30 (4) ◽  
pp. 887-913 ◽  
Author(s):  
Kâzım Büyükboduk ◽  
Antonio Lei
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Quadratic Field ◽  
Iwasawa Theory ◽  
Imaginary Quadratic Field ◽  
Main Conjecture ◽  
Elliptic Modular Form

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.

scholarly journals ON THE GROWTH OF LINEAR RECURRENCES IN FUNCTION FIELDS

2020 ◽  
pp. 1-10
Author(s):  
CLEMENS FUCHS ◽  
SEBASTIAN HEINTZE
Keyword(s):  
Number Field ◽  
Function Field ◽  
Function Fields ◽  
Linear Recurrence ◽  
Recurrence Sequence ◽  
Linear Recurrences ◽  
Field Case ◽  
Power Sum ◽  
Linear Recurrence Sequence

Abstract Let $ (G_n)_{n=0}^{\infty } $ be a nondegenerate linear recurrence sequence whose power sum representation is given by $ G_n = a_1(n) \alpha _1^n + \cdots + a_t(n) \alpha _t^n $ . We prove a function field analogue of the well-known result in the number field case that, under some nonrestrictive conditions, $ |{G_n}| \geq ( \max _{j=1,\ldots ,t} |{\alpha _j}| )^{n(1-\varepsilon )} $ for $ n $ large enough.

scholarly journals Arithmetic of higher-dimensional orbifolds and a mixed Waring problem

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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