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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  1. [1] N. Billerey – « Semi-stabilité des courbes elliptiques », Dissertationes Math. (Rozprawy Mat.) 468 (2009). Zbl1238.11067MR2605449
  2. [2] P. Deligne – « Les constantes des équations fonctionnelles des fonctions L », in Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture Notes in Math., vol. 349, Springer, 1973, p. 501–597. Zbl0271.14011MR349635
  3. [3] T. Dokchitser & V. Dokchitser – « Regulator constants and the parity conjecture », Invent. Math.178 (2009), p. 23–71. Zbl1219.11083MR2534092
  4. [4] —, « Self-duality of Selmer groups », Math. Proc. Cambridge Philos. Soc.146 (2009), p. 257–267. Zbl1205.11065MR2475965
  5. [5] —, « Roots numbers and parity of ranks of elliptics curves », J. reine angew. Math. 658 (2011), p. 39–64. Zbl1314.11041MR2831512
  6. [6] A. Kraus – « Sur le défaut de semi-stabilité des courbes elliptiques à réduction additive », Manuscripta Math.69 (1990), p. 353–385. Zbl0792.14014MR1080288
  7. [7] J. Nekovář – « On the parity of ranks of Selmer groups. III », Doc. Math.12 (2007), p. 243–274. Zbl1201.11067MR2350290
  8. [8] —, « On the parity of ranks of Selmer groups. IV », Compos. Math.145 (2009), p. 1351–1359. Zbl1221.11150MR2575086
  9. [9] D. E. Rohrlich – « Elliptic curves and the Weil-Deligne group », in Elliptic curves and related topics, CRM Proc. Lecture Notes, vol. 4, Amer. Math. Soc., 1994, p. 125–157. Zbl0852.14008MR1260960
  10. [10] —, « Galois theory, elliptic curves, and root numbers », Compositio Math.100 (1996), p. 311–349. Zbl0860.11033MR1387669
  11. [11] —, « Galois invariance of local root numbers », Mathematische Annalen351 (2011), p. 979–1003. Zbl1260.11071MR2854120
  12. [12] J-P. Serre – Représentations linéaires des groupes finis, Hermann, 1998. Zbl0926.20003
  13. [13] J. H. Silverman – Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Math., vol. 151, Springer, 1994. Zbl0911.14015MR1312368
  14. [14] —, The arithmetic of elliptic curves, Graduate Texts in Math., vol. 106, Springer, 1994. Zbl0585.14026
  15. [15] J. Tate – « Number theoretic background », in Automorphic forms, representations and L -functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., 1979, p. 3–26. Zbl0422.12007MR546607
  16. [16] J. P. Wintenberger – « Potential modularity of elliptic curves over totally real fields », appendix to [8]. 

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