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The remaining cases of the Kramer–Tunnell conjecture

  • Kęstutis Česnavičius (a1) and Naoki Imai (a2)


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 $E$ .



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[BH06] Bushnell, C. J. and Henniart, G., The local Langlands conjecture for GL(2), Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 335 (Springer, Berlin, 2006), xii+347; MR 2234120 (2007m:22013).
[Čes16] Česnavičius, K., The p-parity conjecture for elliptic curves with a p-isogeny , J. reine angew. Math. (2016), to appear. Preprint, arXiv:1207.0431.
[Del75] Deligne, P., Courbes elliptiques: formulaire d’après J. Tate , in Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture Notes in Mathematics, vol. 476 (Springer, Berlin, 1975), 5373 (French); MR 0387292 (52 #8135).
[Del84] Deligne, P., Les corps locaux de caractéristique p, limites de corps locaux de caractéristique 0 , in Représentations des groupes réductifs sur un corps local, Travaux en Cours (Hermann, Paris, 1984), 119157 (French); MR 771673 (86g:11068).
[DD08] Dokchitser, T. and Dokchitser, V., Root numbers of elliptic curves in residue characteristic 2 , Bull. Lond. Math. Soc. 40 (2008), 516524; MR 2418807 (2009k:11093).
[DD11] Dokchitser, T. and Dokchitser, V., Root numbers and parity of ranks of elliptic curves , J. reine angew. Math. 658 (2011), 3964; MR 2831512.
[DD16] Dokchitser, T. and Dokchitser, V., Euler factors determine local Weil representations , J. reine angew. Math. (2016). Preprint, arXiv:1112.4889.
[Hel09] Helfgott, H. A., On the behaviour of root numbers in families of elliptic curves, Preprint (2009), arXiv:math/0408141.
[Ima15] Imai, N., Local root numbers of elliptic curves over dyadic fields , J. Math. Sci. Univ. Tokyo 22 (2015), 247260; MR 3329196.
[Kis99] Kisin, M., Local constancy in p-adic families of Galois representations , Math. Z. 230 (1999), 569593; MR 1680032 (2000f:14034).
[KT82] Kramer, K. and Tunnell, J., Elliptic curves and local 𝜀-factors , Compositio Math. 46 (1982), 307352; MR 664648 (83m:14031).
[Roh94] Rohrlich, D. E., Elliptic curves and the Weil-Deligne group , in Elliptic curves and related topics, CRM Proceedings & Lecture Notes, vol. 4 (American Mathematical Society, Providence, RI, 1994), 125157; MR 1260960 (95a:11054).
[Ser79] Serre, J.-P., Local fields, Graduate Texts in Mathematics, vol. 67 (Springer, New York, 1979), viii+241; translated from the French by Marvin Jay Greenberg; MR 554237 (82e:12016).
[Sil94] Silverman, J. H., Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 151 (Springer, New York, 1994), xiv+525; MR 1312368 (96b:11074).
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Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
  • URL: /core/journals/compositio-mathematica
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