scholarly journals Root number in integer parameter families of elliptic curves

2021 ◽  
Author(s):  
Julie Desjardins
Keyword(s):  
Elliptic Curves ◽  
Root Number

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).

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.

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 Quadratic twists of elliptic curves with 3-Selmer rank 1

2014 ◽  
Vol 10 (05) ◽  
pp. 1191-1217 ◽  
Author(s):  
Zane Kun Li
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Galois Cohomology ◽  
Number Fields ◽  
Root Number ◽  
Positive Proportion ◽  
Quadratic Twists

A weaker form of a 1979 conjecture of Goldfeld states that for every elliptic curve E/ℚ, a positive proportion of its quadratic twists E(d) have rank 1. Using tools from Galois cohomology, we give criteria on E and d which force a positive proportion of the quadratic twists of E to have 3-Selmer rank 1 and global root number -1. We then give four nonisomorphic infinite families of elliptic curves Em,n which satisfy these criteria. Conditional on the rank part of the Birch and Swinnerton-Dyer conjecture, this verifies the aforementioned conjecture for infinitely many elliptic curves. Our elliptic curves are easy to give explicitly and we state precisely which quadratic twists d to use. Furthermore, our methods have the potential of being generalized to elliptic curves over other number fields.

scholarly journals Elliptic curves with all cubic twists of the same root number

2014 ◽  
Vol 136 ◽  
pp. 22-27 ◽  
Author(s):  
Dongho Byeon ◽  
Nayoung Kim
Keyword(s):  
Elliptic Curves ◽  
Root Number

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.

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