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

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.

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

International Journal of Number Theory ◽

10.1142/s1793042113501194 ◽

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.

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ON THE GROWTH OF LINEAR RECURRENCES IN FUNCTION FIELDS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972720001094 ◽

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.

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CONTROL THEOREMS FOR ELLIPTIC CURVES OVER FUNCTION FIELDS

International Journal of Number Theory ◽

10.1142/s1793042109002067 ◽

2009 ◽

Vol 05 (02) ◽

pp. 229-256 ◽

Cited By ~ 6

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.

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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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Selmer groups of abelian varieties in extensions of function fields

Mathematische Zeitschrift ◽

10.1007/s00209-008-0351-4 ◽

2008 ◽

Vol 261 (4) ◽

pp. 787-804 ◽

Cited By ~ 1

Author(s):

Amílcar Pacheco

Keyword(s):

Function Fields ◽

Abelian Varieties ◽

Selmer Groups

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Infinitely $p$ -Divisible Points on Abelian Varieties Defined over Function Fields of Characteristic $p\gt 0$

Notre Dame Journal of Formal Logic ◽

10.1215/00294527-2143943 ◽

2013 ◽

Vol 54 (3-4) ◽

pp. 579-589 ◽

Cited By ~ 3

Author(s):

Damian Rössler

Keyword(s):

Function Fields ◽

Abelian Varieties ◽

Characteristic P

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Chebyshev’s bias in function fields

Compositio Mathematica ◽

10.1112/s0010437x08003631 ◽

2008 ◽

Vol 144 (6) ◽

pp. 1351-1374 ◽

Cited By ~ 4

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.

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Squarefree doubly primitive divisors in dynamical sequences

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004117000500 ◽

2017 ◽

Vol 164 (3) ◽

pp. 551-572 ◽

Cited By ~ 1

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.

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