scholarly journals Malle's conjecture for for

2021 ◽  
Vol 157 (1) ◽  
pp. 83-121
Author(s):  
Jiuya Wang
Keyword(s):  
Abelian Group ◽  
Number Field ◽  
Arbitrary Number ◽  
Number Fields ◽  
Counter Example ◽  
Fundamental Domains ◽  
Good Uniformity

We propose a framework to prove Malle's conjecture for the compositum of two number fields based on proven results of Malle's conjecture and good uniformity estimates. Using this method, we prove Malle's conjecture for $S_n\times A$ over any number field $k$ for $n=3$ with $A$ an abelian group of order relatively prime to 2, for $n= 4$ with $A$ an abelian group of order relatively prime to 6, and for $n=5$ with $A$ an abelian group of order relatively prime to 30. As a consequence, we prove that Malle's conjecture is true for $C_3\wr C_2$ in its $S_9$ representation, whereas its $S_6$ representation is the first counter-example of Malle's conjecture given by Klüners. We also prove new local uniformity results for ramified $S_5$ quintic extensions over arbitrary number fields by adapting Bhargava's geometric sieve and averaging over fundamental domains of the parametrization space.

scholarly journals On Unit Groups of Absolute Abelian Number Fields of Degree pq

1960 ◽  
Vol 16 ◽  
pp. 73-81
Author(s):  
Hideo Yokoi
Keyword(s):  
Abelian Group ◽  
Number Field ◽  
Unit Group ◽  
Number Fields ◽  
Free Subgroup ◽  
Finite Degree ◽  
Maximal Torsion ◽  
Torsion Free Subgroup ◽  
Torsion Free ◽  
Arbitrary Abelian Group

In this note, we denote by Q the rational number field, by EΩ the whole unit group of an arbitrary number field Ω of finite degree, and by rΩ the rank of where generally G* for an arbitrary abelian group G means a maximal torsion-free subgroup of G. (NK/ΩEK)* is shortly denoted by and (G1 : G2) is, as usual, the index of a subgroup G2 in G1.

scholarly journals Integral moments of automorphic L-functions

2008 ◽  
Vol 8 (2) ◽  
pp. 335-382 ◽  
Author(s):  
Adrian Diaconu ◽  
Paul Garrett
Keyword(s):  
Representation Theory ◽  
Number Field ◽  
Spectral Decomposition ◽  
Arbitrary Number ◽  
Number Fields ◽  
Rational Numbers ◽  
Poincaré Series ◽  
Poincare Series ◽  
Global And Local ◽  
Adele Group

AbstractWe obtain second integral moments of automorphic L-functions on adele groups GL2 over arbitrary number fields, by a spectral decomposition using the structure and representation theory of adele groups GL1 and GL2. This requires reformulation of the notion of Poincaré series, replacing the collection of classical Poincaré series over GL2(ℚ) or GL2(ℚ(i)) with a single, coherent, global object that makes sense over a number field. This is the first expression of integral moments in adele-group terms, distinguishing global and local issues, and allowing uniform application to number fields. When specialized to the field of rational numbers ℚ, we recover the classical results on moments.

scholarly journals Hyperbolic tessellations and generators of for imaginary quadratic fields

2021 ◽  
Vol 9 ◽  
Author(s):  
David Burns ◽  
Rob de Jeu ◽  
Herbert Gangl ◽  
Alexander D. Rahm ◽  
Dan Yasaki
Keyword(s):  
Number Field ◽  
Number Fields ◽  
Quadratic Fields ◽  
Quadratic Number ◽  
Imaginary Quadratic Fields ◽  
Quadratic Number Fields ◽  
Formula Group ◽  
Imaginary Quadratic Number ◽  
Direct Calculations

Abstract We develop methods for constructing explicit generators, modulo torsion, of the $K_3$ -groups of imaginary quadratic number fields. These methods are based on either tessellations of hyperbolic $3$ -space or on direct calculations in suitable pre-Bloch groups and lead to the very first proven examples of explicit generators, modulo torsion, of any infinite $K_3$ -group of a number field. As part of this approach, we make several improvements to the theory of Bloch groups for $ K_3 $ of any field, predict the precise power of $2$ that should occur in the Lichtenbaum conjecture at $ -1 $ and prove that this prediction is valid for all abelian number fields.

scholarly journals On the complex dynamics of birational surface maps defined over number fields

2016 ◽  
Vol 0 (0) ◽  
Author(s):  
Mattias Jonsson ◽  
Paul Reschke
Keyword(s):  
Number Field ◽  
Complex Dynamics ◽  
Energy Condition ◽  
Height Function ◽  
Number Fields ◽  
Projective Surface ◽  
Complex Projective Surface ◽  
Canonical Height ◽  
Dynamical Degree ◽  
Complex Projective

AbstractWe show that any birational selfmap of a complex projective surface that has dynamical degree greater than one and is defined over a number field automatically satisfies the Bedford–Diller energy condition after a suitable birational conjugacy. As a consequence, the complex dynamics of the map is well behaved. We also show that there is a well-defined canonical height function.

Number fields with large minimal index containing quadratic subfields

2018 ◽  
Vol 14 (09) ◽  
pp. 2333-2342 ◽  
Author(s):  
Henry H. Kim ◽  
Zack Wolske
Keyword(s):  
Number Field ◽  
Number Fields ◽  
Upper Bounds ◽  
Minimal Index

In this paper, we consider number fields containing quadratic subfields with minimal index that is large relative to the discriminant of the number field. We give new upper bounds on the minimal index, and construct families with the largest possible minimal index.

scholarly journals ON SELMER RANK PARITY OF TWISTS

2016 ◽  
Vol 102 (3) ◽  
pp. 316-330 ◽  
Author(s):  
MAJID HADIAN ◽  
MATTHEW WEIDNER
Keyword(s):  
Elliptic Curves ◽  
Abelian Variety ◽  
Number Field ◽  
Arbitrary Number ◽  
Hyperelliptic Curve ◽  
Abelian Varieties ◽  
Hyperelliptic Curves ◽  
Torsion Subgroup ◽  
Tate Conjecture ◽  
Quadratic Twists

In this paper we study the variation of the $p$-Selmer rank parities of $p$-twists of a principally polarized Abelian variety over an arbitrary number field $K$ and show, under certain assumptions, that this parity is periodic with an explicit period. Our result applies in particular to principally polarized Abelian varieties with full $K$-rational $p$-torsion subgroup, arbitrary elliptic curves, and Jacobians of hyperelliptic curves. Assuming the Shafarevich–Tate conjecture, our result allows one to classify the rank parities of all quadratic twists of an elliptic or hyperelliptic curve after a finite calculation.

scholarly journals A Density of Ramified Primes

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Stephanie Chan ◽  
Christine McMeekin ◽  
Djordjo Milovic
Keyword(s):  
Number Field ◽  
Class Number ◽  
Joint Distribution ◽  
Number Fields ◽  
Character Sums ◽  
Prime Ideals ◽  
Odd Degree ◽  
Narrow Class

AbstractLet K be a cyclic number field of odd degree over $${\mathbb {Q}}$$ Q with odd narrow class number, such that 2 is inert in $$K/{\mathbb {Q}}$$ K / Q . We define a family of number fields $$\{K(p)\}_p$$ { K ( p ) } p , depending on K and indexed by the rational primes p that split completely in $$K/{\mathbb {Q}}$$ K / Q , in which p is always ramified of degree 2. Conditional on a standard conjecture on short character sums, the density of such rational primes p that exhibit one of two possible ramified factorizations in $$K(p)/{\mathbb {Q}}$$ K ( p ) / Q is strictly between 0 and 1 and is given explicitly as a formula in terms of the degree of the extension $$K/{\mathbb {Q}}$$ K / Q . Our results are unconditional in the cubic case. Our proof relies on a detailed study of the joint distribution of spins of prime ideals.

Prime splitting in abelian number fields and linear combinations of Dirichlet characters

2014 ◽  
Vol 10 (04) ◽  
pp. 885-903 ◽  
Author(s):  
Paul Pollack
Keyword(s):  
Number Field ◽  
Nonzero Element ◽  
Number Fields ◽  
Upper Bounds ◽  
Prime Ideals ◽  
Linear Combinations ◽  
Dirichlet Characters ◽  
Abelian Number Field ◽  
Splitting Type ◽  
A Finite Group

Let 𝕏 be a finite group of primitive Dirichlet characters. Let ξ = ∑χ∈𝕏 aχ χ be a nonzero element of the group ring ℤ[𝕏]. We investigate the smallest prime q that is coprime to the conductor of each χ ∈ 𝕏 and that satisfies ∑χ∈𝕏 aχ χ(q) ≠ 0. Our main result is a nontrivial upper bound on q valid for certain special forms ξ. From this, we deduce upper bounds on the smallest unramified prime with a given splitting type in an abelian number field. For example, let K/ℚ be an abelian number field of degree n and conductor f. Let g be a proper divisor of n. If there is any unramified rational prime q that splits into g distinct prime ideals in ØK, then the least such q satisfies [Formula: see text].

scholarly journals Nearly ordinary Galois deformations over arbitrary number fields

2008 ◽  
Vol 8 (1) ◽  
pp. 99-177 ◽  
Author(s):  
Frank Calegari ◽  
Barry Mazur
Keyword(s):  
Arbitrary Number ◽  
Quadratic Field ◽  
Galois Representations ◽  
Galois Representation ◽  
Number Fields ◽  
Base Change ◽  
Deformation Spaces ◽  
Totally Real ◽  
Set Of Points ◽  
Dense Set

AbstractLet K be an arbitrary number field, and let ρ : Gal($\math{\bar{K}}$/K) → GL2(E) be a nearly ordinary irreducible geometric Galois representation. In this paper, we study the nearly ordinary deformations of ρ. When K is totally real and ρ is modular, results of Hida imply that the nearly ordinary deformation space associated to ρ contains a Zariski dense set of points corresponding to ‘automorphic’ Galois representations. We conjecture that if K is not totally real, then this is never the case, except in three exceptional cases, corresponding to: (1) ‘base change’, (2) ‘CM’ forms, and (3) ‘even’ representations. The latter case conjecturally can only occur if the image of ρ is finite. Our results come in two flavours. First, we prove a general result for Artin representations, conditional on a strengthening of the Leopoldt Conjecture. Second, when K is an imaginary quadratic field, we prove an unconditional result that implies the existence of ‘many’ positive-dimensional components (of certain deformation spaces) that do not contain infinitely many classical points. Also included are some speculative remarks about ‘p-adic functoriality’, as well as some remarks on how our methods should apply to n-dimensional representations of Gal($\math{\bar{\QQ}}$/ℚ) when n > 2.

Sign in / Sign up

Export Citation Format

Share Document