FUNCTIONAL EQUATION OF CHARACTERISTIC ELEMENTS OF ABELIAN VARIETIES OVER FUNCTION FIELDS (ℓ ≠ p)

2014 ◽  
Vol 10 (03) ◽  
pp. 705-735
Author(s):  
APRAMEYO PAL
Keyword(s):  
Functional Equation ◽  
Number Field ◽  
Function Field ◽  
Functional Equations ◽  
Function Fields ◽  
Growth Behavior ◽  
Iwasawa Theory ◽  
Selmer Groups ◽  
Selmer Group ◽  
Field Case

In this paper we apply methods from the number field case of Perrin-Riou [20] and Zábrádi [32] in the function field setup. In ℤℓ- and GL2-cases (ℓ ≠ p), we prove algebraic functional equations of the Pontryagin dual of Selmer group which give further evidence of the main conjectures of Iwasawa theory. We also prove some parity conjectures in commutative and non-commutative cases. As a consequence, we also get results on the growth behavior of Selmer groups in commutative and non-commutative extension of function fields.

scholarly journals On the Selmer groups of abelian varieties over function fields of characteristic p > 0

2009 ◽  
Vol 146 (1) ◽  
pp. 23-43 ◽  
Author(s):  
TADASHI OCHIAI ◽  
FABIEN TRIHAN
Keyword(s):  
Number Field ◽  
Function Field ◽  
Function Fields ◽  
Abelian Varieties ◽  
Iwasawa Theory ◽  
Selmer Groups ◽  
Characteristic P ◽  
Field Case ◽  
New Phenomena ◽  
Geometric Analogue

AbstractWe study a (p-adic) geometric analogue for abelian varieties over a function field of characteristic p of the cyclotomic Iwasawa theory and the non-commutative Iwasawa theory for abelian varieties over a number field initiated by Mazur and Coates respectively. We will prove some analogue of the principal results obtained in the case over a number field and we study new phenomena which did not happen in the case of number field case. We also propose a conjecture (Conjecture 1.6) which might be considered as a counterpart of the principal conjecture in the case over a number field.

scholarly journals Algebraic functional equations and completely faithful Selmer groups

2015 ◽  
Vol 11 (04) ◽  
pp. 1233-1257
Author(s):  
Tibor Backhausz ◽  
Gergely Zábrádi
Keyword(s):  
Functional Equation ◽  
Elliptic Curve ◽  
Galois Group ◽  
Number Field ◽  
Functional Equations ◽  
Complex Multiplication ◽  
Selmer Groups ◽  
Selmer Group ◽  
Torsion Module ◽  
Ordinary Reduction

Let E be an elliptic curve — defined over a number field K — without complex multiplication and with good ordinary reduction at all the primes above a rational prime p ≥ 5. We construct a pairing on the dual p∞-Selmer group of E over any strongly admissible p-adic Lie extension K∞/K under the assumption that it is a torsion module over the Iwasawa algebra of the Galois group G = Gal(K∞/K). Under some mild additional hypotheses, this gives an algebraic functional equation of the conjectured p-adic L-function. As an application, we construct completely faithful Selmer groups in case the p-adic Lie extension is obtained by adjoining the p-power division points of another non-CM elliptic curve A.

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.

Algebraic functional equation for Hida family

2014 ◽  
Vol 10 (07) ◽  
pp. 1649-1674
Author(s):  
Somnath Jha ◽  
Aprameyo Pal
Keyword(s):  
Functional Equation ◽  
Number Field ◽  
Modular Forms ◽  
Euler System ◽  
Selmer Groups ◽  
Selmer Group ◽  
Main Conjecture ◽  
Hida Family ◽  
General Number ◽  
The Individual

We prove a functional equation for the characteristic ideal of the "big" Selmer group 𝒳(𝒯ℱ/F cyc ) associated to an ordinary Hida family of elliptic modular forms over the cyclotomic ℤp extension of a general number field F, under the assumption that there is at least one arithmetic specialization whose Selmer group is torsion over its Iwasawa algebra. For a general number field, the two-variable cyclotomic Iwasawa main conjecture for ordinary Hida family is not proved and this can be thought of as an evidence to the validity of the Iwasawa main conjecture. The central idea of the proof is to prove a variant of the result of Perrin-Riou [Groupes de Selmer et accouplements; cas particulier des courbes elliptiques, Doc. Math.2003 (2003) 725–760, Extra Volume: Kazuya Kato's fiftieth birthday] by constructing a generalized pairing on the individual Selmer groups corresponding to the arithmetic points and make use of the appropriate specialization techniques of Ochiai [Euler system for Galois deformations, Ann. Inst. Fourier (Grenoble)55(1) (2005) 113–146].

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 Selmer Groups of Elliptic Curves with Complex Multiplication

2004 ◽  
Vol 56 (1) ◽  
pp. 194-208
Author(s):  
A. Saikia
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Simple Formula ◽  
Quadratic Field ◽  
Complex Multiplication ◽  
Selmer Groups ◽  
Selmer Group ◽  
Finitely Generated ◽  
Ring Of Integers

AbstractSuppose K is an imaginary quadratic field and E is an elliptic curve over a number field F with complex multiplication by the ring of integers in K. Let p be a rational prime that splits as in K. Let Epn denote the pn-division points on E. Assume that F(Epn) is abelian over K for all n ≥ 0. This paper proves that the Pontrjagin dual of the -Selmer group of E over F(Ep∞) is a finitely generated free Λ-module, where Λ is the Iwasawa algebra of . It also gives a simple formula for the rank of the Pontrjagin dual as a Λ-module.

scholarly journals Chebyshev’s bias in function fields

2008 ◽  
Vol 144 (6) ◽  
pp. 1351-1374 ◽  
Author(s):  
Byungchul Cha
Keyword(s):  
Finite Field ◽  
Number Field ◽  
Central Limit ◽  
Polynomial Ring ◽  
Rational Number ◽  
Function Field ◽  
Sufficient Conditions ◽  
Monic Polynomial ◽  
Field Case ◽  
Necessary And Sufficient

AbstractWe study a function field analog of Chebyshev’s bias. Our results, as well as their proofs, are similar to those of Rubinstein and Sarnak in the case of the rational number field. Following Rubinstein and Sarnak, we introduce the grand simplicity hypothesis (GSH), a certain hypothesis on the inverse zeros of Dirichlet L-series of a polynomial ring over a finite field. Under this hypothesis, we investigate how primes, that is, irreducible monic polynomials in a polynomial ring over a finite field, are distributed in a given set of residue classes modulo a fixed monic polynomial. In particular, we prove under the GSH that, like the number field case, primes are biased toward quadratic nonresidues. Unlike the number field case, the GSH can be proved to hold in some cases and can be violated in some other cases. Also, under the GSH, we give the necessary and sufficient conditions for which primes are unbiased and describe certain central limit behaviors as the degree of modulus under consideration tends to infinity, all of which have been established in the number field case by Rubinstein and Sarnak.

SELMER GROUPS AND GENERALIZED CLASS FIELD TOWERS

2012 ◽  
Vol 08 (04) ◽  
pp. 881-909 ◽  
Author(s):  
AHMED MATAR
Keyword(s):  
Galois Group ◽  
Abelian Variety ◽  
Number Field ◽  
Selmer Groups ◽  
Selmer Group ◽  
Class Field ◽  
Finite Set ◽  
Class Field Towers

This paper proves a control theorem for the p-primary Selmer group of an abelian variety with respect to extensions of the form: Maximal pro-p extension of a number field unramified outside a finite set of primes R which does not include any primes dividing p in which another finite set of primes S splits completely. When the Galois group of the extension is not p-adic analytic, the control theorem gives information about p-ranks of Selmer and Tate–Shafarevich groups of the abelian variety. The paper also discusses what can be said in regards to a control theorem when the set R contains all the primes of the number field dividing p.

scholarly journals Squarefree doubly primitive divisors in dynamical sequences

2017 ◽  
Vol 164 (3) ◽  
pp. 551-572 ◽  
Author(s):  
DRAGOS GHIOCA ◽  
KHOA D. NGUYEN ◽  
THOMAS J. TUCKER
Keyword(s):  
Number Field ◽  
Function Field ◽  
Field Case ◽  
Primitive Divisors ◽  
Finite Set ◽  
Function Field Case

AbstractLetKbe a number field or a function field of characteristic 0, letφ∈K(z) with deg(φ) ⩾ 2, and letα∈ ℙ1(K). LetSbe a finite set of places ofKcontaining all the archimedean ones and the primes whereφhas bad reduction. After excluding all the natural counterexamples, we define a subsetA(φ,α) of ℤ⩾0× ℤ>0and show that for all but finitely many (m,n) ∈A(φ,α) there is a prime 𝔭 ∉Ssuch that ord𝔭(φm+n(α)−φm(α)) = 1 andαhas portrait (m,n) under the action ofφmodulo 𝔭. This latter condition implies ord𝔭(φu+v(α)−φu(α)) ⩽ 0 for (u,v) ∈ ℤ⩾0× ℤ>0satisfyingu<morv<n. Our proof assumes a conjecture of Vojta for ℙ1× ℙ1in the number field case and is unconditional in the function field case thanks to a deep theorem of Yamanoi. This paper extends earlier work of Ingram–Silverman, Faber–Granville and the authors.

scholarly journals Self-duality of Selmer groups

2009 ◽  
Vol 146 (2) ◽  
pp. 257-267 ◽  
Author(s):  
TIM DOKCHITSER ◽  
VLADIMIR DOKCHITSER
Keyword(s):  
Galois Group ◽  
Abelian Variety ◽  
Number Field ◽  
Galois Extension ◽  
Selmer Groups ◽  
Selmer Group ◽  
Self Duality ◽  
Tamagawa Numbers

AbstractThe first part of the paper gives a new proof of self-duality for Selmer groups: if A is an abelian variety over a number field K, and F/K is a Galois extension with Galois group G, then the $\Q_p$G-representation naturally associated to the p∞-Selmer group of A/F is self-dual. The second part describes a method for obtaining information about parities of Selmer ranks from the local Tamagawa numbers of A in intermediate extensions of F/K.

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