Invariance of the parity conjecture for p -Selmer groups of elliptic curves in a D 2 p n -extension

Thomas de La Rochefoucauld

Bulletin de la Société Mathématique de France (2011)

  • Volume: 139, Issue: 4, page 571-592
  • ISSN: 0037-9484

Abstract

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We show a p -parity result in a D 2 p n -extension of number fields L / K ( p ≥ 5 ) for the twist 1 ⊕ η ⊕ τ : W ( E / K , 1 ⊕ η ⊕ τ ) = ( - 1 ) 1 ⊕ η ⊕ τ , X p ( E / L ) , where E is an elliptic curve over K , η and τ are respectively the quadratic character and an irreductible representation of degree 2 of Gal ( L / K ) = D 2 p n , and X p ( E / L ) is the p -Selmer group. The main novelty is that we use a congruence result between ε 0 -factors (due to Deligne) for the determination of local root numbers in bad cases (places of additive reduction above 2 and 3). We also give applications to the p -parity conjecture (using the machinery of the Dokchitser brothers).

How to cite

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de La Rochefoucauld, Thomas. "Invariance of the parity conjecture for $p$-Selmer groups of elliptic curves in a $D_{2p^{n}}$-extension." Bulletin de la Société Mathématique de France 139.4 (2011): 571-592. <http://eudml.org/doc/272747>.

@article{deLaRochefoucauld2011,
abstract = {We show a $p$-parity result in a $D_\{2p^\{n\}\}$-extension of number fields $L/K$ ($p\ge 5$) for the twist $1\oplus \eta \oplus \tau $: $W(E/K,1\oplus \eta \oplus \tau )=(-1)^\{\left\langle 1\oplus \eta \oplus \tau ,X_\{p\}(E/L)\right\rangle \}$, where $E$ is an elliptic curve over $K$, $\eta $ and $\tau $ are respectively the quadratic character and an irreductible representation of degree $2$ of $\mathrm \{Gal\}(L/K)=D_\{2p^\{n\}\}$, and $X_\{p\}(E/L)$ is the $p$-Selmer group. The main novelty is that we use a congruence result between $ \varepsilon _\{0\}$-factors (due to Deligne) for the determination of local root numbers in bad cases (places of additive reduction above 2 and 3). We also give applications to the $p$-parity conjecture (using the machinery of the Dokchitser brothers).},
author = {de La Rochefoucauld, Thomas},
journal = {Bulletin de la Société Mathématique de France},
keywords = {elliptic curves; Birch and Swinnerton-Dyer conjecture; parity conjecture; regulator constants; epsilon factors; root numbers},
language = {eng},
number = {4},
pages = {571-592},
publisher = {Société mathématique de France},
title = {Invariance of the parity conjecture for $p$-Selmer groups of elliptic curves in a $D_\{2p^\{n\}\}$-extension},
url = {http://eudml.org/doc/272747},
volume = {139},
year = {2011},
}

TY - JOUR
AU - de La Rochefoucauld, Thomas
TI - Invariance of the parity conjecture for $p$-Selmer groups of elliptic curves in a $D_{2p^{n}}$-extension
JO - Bulletin de la Société Mathématique de France
PY - 2011
PB - Société mathématique de France
VL - 139
IS - 4
SP - 571
EP - 592
AB - We show a $p$-parity result in a $D_{2p^{n}}$-extension of number fields $L/K$ ($p\ge 5$) for the twist $1\oplus \eta \oplus \tau $: $W(E/K,1\oplus \eta \oplus \tau )=(-1)^{\left\langle 1\oplus \eta \oplus \tau ,X_{p}(E/L)\right\rangle }$, where $E$ is an elliptic curve over $K$, $\eta $ and $\tau $ are respectively the quadratic character and an irreductible representation of degree $2$ of $\mathrm {Gal}(L/K)=D_{2p^{n}}$, and $X_{p}(E/L)$ is the $p$-Selmer group. The main novelty is that we use a congruence result between $ \varepsilon _{0}$-factors (due to Deligne) for the determination of local root numbers in bad cases (places of additive reduction above 2 and 3). We also give applications to the $p$-parity conjecture (using the machinery of the Dokchitser brothers).
LA - eng
KW - elliptic curves; Birch and Swinnerton-Dyer conjecture; parity conjecture; regulator constants; epsilon factors; root numbers
UR - http://eudml.org/doc/272747
ER -

References

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  12. [12] J-P. Serre – Représentations linéaires des groupes finis, Hermann, 1998. Zbl0926.20003
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