Osaka Journal of Mathematics

Leading coefficients of isogenies of degree $p$ over $\mathbb{Q}_{p}$

Mayumi Kawachi

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Let $E$ be an elliptic curve over $\mathbb{Q}_{p}$ which has potentially supersingular good reduction. Let $L/\mathbb{Q}_{p}$ be a totally ramified extension such that $E$ has good reduction over $L$ and $\tilde{E}$ be the reduction of $E$ mod $\pi$, where $\pi$ is a prime element of the ring of integers $\mathcal{O}_{L}$ of $L$. Let $\hat{E}$ be the formal group over $\mathcal{O}_{L}$ associated to $E/\mathcal{O}_{L}$. The multiplication by $p$ map $[p]\colon \hat{E} \to \hat{E}$ is written by power series $[p](x) = px +c_{2}x^{2} + \cdots + c_{p}x^{p}+ \cdots + c_{p^{2}} x^{p^{2}} + \cdots{} \in \mathcal{O}_{L}[[x]]$. By using the liftings over $\mathcal{O}_{L}$ of the Dieudonné module of $p$-divisible group $\tilde{E}(p)$ over $\mathbb{F}_{p}$, we determine the values of $v_{L}(c_{p})$.

Article information

Osaka J. Math., Volume 48, Number 3 (2011), 691-708.

First available in Project Euclid: 26 September 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G07: Elliptic curves over local fields [See also 14G20, 14H52]
Secondary: 14L05: Formal groups, $p$-divisible groups [See also 55N22]


Kawachi, Mayumi. Leading coefficients of isogenies of degree $p$ over $\mathbb{Q}_{p}$. Osaka J. Math. 48 (2011), no. 3, 691--708.

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  • J.E. Cremona: Algorithms for Modular Elliptic Curves, second edition, Cambridge Univ. Press, Cambridge, 1997.
  • J.-M. Fontaine: Groupes $p$-Divisibles sur les Corps Locaux, Astérisque 4748, Soc. Math. France, Paris, 1977.
  • M. Hazewinkel: On norm maps for one dimensional formal groups, III, Duke Math. J. 44 (1977), 305–314.
  • T. Honda: Formal groups and zeta-functions, Osaka J. Math. 5 (1968), 199–213.
  • T. Honda: On the theory of commutative formal groups, J. Math. Soc. Japan 22 (1970), 213–246.
  • M. Kawachi: Isogenies of degree $p$ of elliptic curves over local fields and Kummer theory, Tokyo J. Math. 25 (2002), 247–259.
  • J.H. Silverman: The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics 106, Springer, New York, 1985.
  • J. Tate: $p$-divisible groups; in Proc. Conf. Local Fields (Driebergen, 1966), Springer, Berlin, 158–183.
  • J. Tate: Algorithm for determining the type of a singular fiber in an elliptic pencil; in Modular Functions of One Variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture Notes in Math. 476, Springer, Berlin, 33–52, 1975.
  • M. Volkov: Les représentations $l$-adiques associées aux courbes elliptiques sur $\mathbb{Q}_{p}$, J. Reine Angew. Math. 535 (2001), 65–101.