scholarly journals Distribution of root numbers of Hecke characters attached to some elliptic curves

2021 ◽  
Vol 33 (3) ◽  
pp. 653-668
Author(s):  
Keunyoung Jeong ◽  
Jigu Kim ◽  
Taekyung Kim
Keyword(s):  
Elliptic Curves ◽  
Root Number ◽  
Root Numbers ◽  
Local Root

Abstract In this paper, we show that an action on the set of elliptic curves with j = 1728 j=1728 preserves a certain kind of symmetry on the local root number of Hecke characters attached to such elliptic curves. As a consequence, we give results on the distribution of the root numbers and their average of the aforementioned Hecke characters.

RATIONAL TORSION ON OPTIMAL CURVES

2005 ◽  
Vol 01 (04) ◽  
pp. 513-531 ◽  
Author(s):  
NEIL DUMMIGAN
Keyword(s):  
Elliptic Curves ◽  
Prime Order ◽  
Rational Points ◽  
Root Number ◽  
Hecke Eigenforms ◽  
Modular Parametrization ◽  
Local Root

Vatsal has proved recently a result which has consequences for the existence of rational points of odd prime order ℓ on optimal elliptic curves over ℚ. When the conductor N is squarefree, ℓ ∤ N and the local root number wp= -1 for at least one prime p | N, we offer a somewhat different proof, starting from an explicit cuspidal divisor on X0(N). We also prove some results linking the vanishing of L(E,1) with the divisibility by ℓ of the modular parametrization degree, fitting well with the Bloch–Kato conjecture for L( Sym2E,2), and with an earlier construction of elements in Shafarevich–Tate groups. Finally (following Faltings and Jordan) we prove an analogue of the result on ℓ-torsion for cuspidal Hecke eigenforms of level one (and higher weight), thereby strengthening some existing evidence for another case of the Bloch–Kato conjecture.

Local Root Numbers for Heisenberg-Representations — Some Explicit Results

2020 ◽  
pp. 2050127
Author(s):  
Sazzad Ali Biswas ◽  
Ernst-Wilhelm Zink
Keyword(s):  
Finite Groups ◽  
Abelian Extension ◽  
Factor Group ◽  
Irreducible Representations ◽  
Heisenberg Groups ◽  
Root Number ◽  
Absolute Galois Group ◽  
Root Numbers ◽  
Number Formula ◽  
Local Root

Heisenberg representations [Formula: see text] of (pro-)finite groups [Formula: see text] are by definition irreducible representations of the two-step nilpotent factor group [Formula: see text] Better known are Heisenberg groups which can be understood as allowing faithful Heisenberg representations. A special feature is that [Formula: see text] will be induced by characters [Formula: see text] of subgroups in multiple ways, where the pairs [Formula: see text] can be interpreted as maximal isotropic pairs. If [Formula: see text] is a [Formula: see text]-adic number field and [Formula: see text] the absolute Galois group then maximal isotropic pairs rewrite as [Formula: see text] where [Formula: see text] is an abelian extension and [Formula: see text] a character. We will consider the extended local Artin-root-number [Formula: see text] for those [Formula: see text] which are essentially tame and express it by a formula not depending on the various maximal isotropic pairs [Formula: see text] for [Formula: see text]

On elliptic curves with complex multiplication and root numbers

2020 ◽  
pp. 1-16
Author(s):  
Wan Lee ◽  
Myungjun Yu
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Quadratic Field ◽  
Complex Multiplication ◽  
Imaginary Quadratic Field ◽  
Root Number ◽  
Root Numbers ◽  
Quadratic Twists

Let [Formula: see text] be an elliptic curve defined over a number field [Formula: see text]. Suppose that [Formula: see text] has complex multiplication over [Formula: see text], i.e. [Formula: see text] is an imaginary quadratic field. With the aid of CM theory, we find elliptic curves whose quadratic twists have a constant root number.

scholarly journals On the Nielsen root theory of n-valued maps

2021 ◽  
Vol 23 (4) ◽  
Author(s):  
Robert F. Brown
Keyword(s):  
Fixed Point ◽  
Lower Bound ◽  
Lie Group ◽  
Hausdorff Distance ◽  
Root Number ◽  
Root Numbers ◽  
Fixed Point Result

AbstractLet $$\phi :X \multimap Y$$ ϕ : X ⊸ Y be an n-valued map of connected finite polyhedra and let $$a \in Y$$ a ∈ Y . Then, $$x \in X$$ x ∈ X is a root of $$\phi $$ ϕ at a if $$a \in \phi (x)$$ a ∈ ϕ ( x ) . The Nielsen root number $$N(\phi : a)$$ N ( ϕ : a ) is a lower bound for the number of roots at a of any n-valued map homotopic to $$\phi $$ ϕ . We prove that if X and Y are compact, connected triangulated manifolds without boundary, of the same dimension, then given $$\epsilon > 0$$ ϵ > 0 , there is an n-valued map $$\psi $$ ψ homotopic to $$\phi $$ ϕ within Hausdorff distance $$\epsilon $$ ϵ of $$\phi $$ ϕ such that $$\psi $$ ψ has finitely many roots at a. We conjecture that if X and Y are q-manifolds without boundary, $$q \ne 2$$ q ≠ 2 , then there is an n-valued map homotopic to $$\phi $$ ϕ that has $$N(\phi : a)$$ N ( ϕ : a ) roots at a. We verify the conjecture when $$X = Y$$ X = Y is a Lie group by employing a fixed point result of Schirmer. As an application, we calculate the Nielsen root numbers of linear n-valued maps of tori.

scholarly journals Gamma Factors, Root Numbers, and Distinction

2018 ◽  
Vol 70 (3) ◽  
pp. 683-701 ◽  
Author(s):  
Nadir Matringe ◽  
Omer Offen
Keyword(s):  
Linear Group ◽  
General Linear Group ◽  
Quadratic Extension ◽  
Root Number ◽  
Converse Problem ◽  
Root Numbers ◽  
Special Values ◽  
Distinguished Representations ◽  
Local Invariants ◽  
Gamma Factor

AbstractWe study a relation between distinction and special values of local invariants for representations of the general linear group over a quadratic extension of p-adic fields. We show that the local Rankin–Selberg root number of any pair of distinguished representation is trivial, and as a corollary we obtain an analogue for the global root number of any pair of distinguished cuspidal representations. We further study the extent to which the gamma factor at 1/2 is trivial for distinguished representations as well as the converse problem.

Sign in / Sign up

Export Citation Format

Share Document