The Local Root Number of Elliptic Curves

2002 ◽  
pp. 73-83
Author(s):  
Shin-Ichi Kobayashi
Keyword(s):  
Elliptic Curves ◽  
Root Number ◽  
Local Root

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.

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.

Twisted Hasse-Weil L-Functions and the Rank of Mordell-Weil Groups

1997 ◽  
Vol 49 (4) ◽  
pp. 749-771 ◽  
Author(s):  
Lawrence Howe
Keyword(s):  
Functional Equation ◽  
Lower Bound ◽  
Irreducible Representation ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Root Number

AbstractFollowing a method outlined by Greenberg, root number computations give a conjectural lower bound for the ranks of certain Mordell–Weil groups of elliptic curves. More specifically, for PQn a PGL2(Z/pnZ)–extension of Q and E an elliptic curve over Q, define the motive E ⊗ ρ, where ρ is any irreducible representation of Gal(PQn /Q). Under some restrictions, the root number in the conjectural functional equation for the L-function of E ⊗ ρ is easily computable, and a ‘–1’ implies, by the Birch and Swinnerton–Dyer conjecture, that ρ is found in E(PQn) ⊗ C. Summing the dimensions of such ρ gives a conjectural lower bound ofp2n–p2n–1–p–1for the rank of E(PQn).

scholarly journals The remaining cases of the Kramer–Tunnell conjecture

2016 ◽  
Vol 152 (11) ◽  
pp. 2255-2268
Author(s):  
Kęstutis Česnavičius ◽  
Naoki Imai
Keyword(s):  
Elliptic Curve ◽  
Local Field ◽  
Positive Characteristic ◽  
Quadratic Extension ◽  
Base Change ◽  
Local Fields ◽  
Root Number ◽  
Characteristic Case ◽  
Local Root

For an elliptic curve $E$ over a local field $K$ and a separable quadratic extension of $K$, motivated by connections to the Birch and Swinnerton-Dyer conjecture, Kramer and Tunnell have conjectured a formula for computing the local root number of the base change of $E$ to the quadratic extension in terms of a certain norm index. The formula is known in all cases except some where $K$ is of characteristic $2$, and we complete its proof by reducing the positive characteristic case to characteristic $0$. For this reduction, we exploit the principle that local fields of characteristic $p$ can be approximated by finite extensions of $\mathbb{Q}_{p}$: we find an elliptic curve $E^{\prime }$ defined over a $p$-adic field such that all the terms in the Kramer–Tunnell formula for $E^{\prime }$ are equal to those for $E$.

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]

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