scholarly journals Stickelberger series and Main Conjecture for function fields

2021 ◽  
Vol 65 ◽  
pp. 459-498
Author(s):  
Andrea Bandini ◽  
Edoardo Coscelli
2020 ◽  
Vol 16 (09) ◽  
pp. 2041-2094
Author(s):  
Malte Witte

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.


2016 ◽  
Vol 112 (6) ◽  
pp. 1040-1058 ◽  
Author(s):  
King Fai Lai ◽  
Ignazio Longhi ◽  
Ki-Seng Tan ◽  
Fabien Trihan

2009 ◽  
Vol 05 (02) ◽  
pp. 229-256 ◽  
Author(s):  
A. BANDINI ◽  
I. LONGHI

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.


2015 ◽  
Vol 282 (1-2) ◽  
pp. 485-510 ◽  
Author(s):  
King Fai Lai ◽  
Ignazio Longhi ◽  
Ki-Seng Tan ◽  
Fabien Trihan

2018 ◽  
Vol 30 (4) ◽  
pp. 887-913 ◽  
Author(s):  
Kâzım Büyükboduk ◽  
Antonio Lei

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.


Author(s):  
CLEMENS FUCHS ◽  
SEBASTIAN HEINTZE

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.


Author(s):  
Tim Browning ◽  
Shuntaro Yamagishi

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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