The order and index of a principal homogeneous space for an elliptic curve over a general local field

1975 ◽  
Vol 27 (1) ◽  
pp. 45-46
Author(s):  
V. I. Andriichuk
Keyword(s):  
Elliptic Curve ◽  
Homogeneous Space ◽  
Local Field ◽  
Principal Homogeneous Space

The Witt ring of an elliptic curve over a local field

1996 ◽  
Vol 306 (1) ◽  
pp. 391-418 ◽  
Author(s):  
J�n Kr. Arason ◽  
Richard Elman ◽  
Bill Jacob
Keyword(s):  
Elliptic Curve ◽  
Local Field ◽  
Witt Ring

scholarly journals The picard invariant of a principal homogeneous space

1974 ◽  
Vol 4 (3) ◽  
pp. 273-286 ◽  
Author(s):  
Lindsay N. Childs ◽  
Andy R. Magid
Keyword(s):  
Homogeneous Space ◽  
Principal Homogeneous Space

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 $E$.

scholarly journals Optimal Quotients of Jacobians With Toric Reduction and Component Groups

2016 ◽  
Vol 68 (6) ◽  
pp. 1362-1381
Author(s):  
Mihran Papikian ◽  
Joseph Rabinoff
Keyword(s):  
Elliptic Curve ◽  
Local Field ◽  
Modular Curves ◽  
Jacobian Variety ◽  
Neron Models ◽  
Néron Models

AbstractLet J be a Jacobian variety with toric reduction over a local field K. Let J → E be an optimal quotient defined over K, where E is an elliptic curve. We give examples in which the functorially induced map on component groups of the Néron models is not surjective. This answers a question of Ribet and Takahashi. We also give various criteria under which is surjective and discuss when these criteria hold for the Jacobians of modular curves.

scholarly journals Principes locaux-globaux pour certaines fibrations en torseurs sous un tore

2014 ◽  
Vol 158 (1) ◽  
pp. 131-145 ◽  
Author(s):  
ARNE SMEETS
Keyword(s):  
Homogeneous Space ◽  
Number Field ◽  
Hasse Principle ◽  
Weak Approximation ◽  
Generic Fibre ◽  
Principal Homogeneous Space ◽  
Smooth Compactification

AbstractLetkbe a number field andTak-torus. Consider a family of torsors underT, i.e. a morphismf:X→ ℙ1kfrom a projective, smoothk-varietyXto ℙ1k, the generic fibreXη→ η of which is a smooth compactification of a principal homogeneous space underT⊗kη. We study the Brauer–Manin obstruction to the Hasse principle and to weak approximation forX, assuming Schinzel's hypothesis. We generalise Wei's recent results [21]. Our results are unconditional ifk=Qand all non-split fibres offare defined overQ. We also establish an unconditional analogue of our main result for zero-cycles of degree 1.

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