Isogenies of elliptic curves and the Morava stabilizer group

Mark Behrens; Tyler Lawson

doi:10.1016/j.jpaa.2005.09.007

Isogenies of elliptic curves and the Morava stabilizer group

Mark Behrens, Tyler Lawson

Research output: Contribution to journal › Article › peer-review

9 Scopus citations

Abstract

Let S₂ be the p-primary second Morava stabilizer group, C a supersingular elliptic curve over over(F, -)_p, O the ring of endomorphisms of C, and ℓ a topological generator of Z_p^× (or Z₂^× / {± 1} if p = 2). We show that for p > 2 the group Γ ⊆ O [1 / ℓ]^× of quasi-endomorphisms of degree a power of ℓ is dense in S₂. For p = 2, we show that Γ is dense in an index 2 subgroup of S₂.

Original language	English (US)
Pages (from-to)	37-49
Number of pages	13
Journal	Journal of Pure and Applied Algebra
Volume	207
Issue number	1
DOIs	https://doi.org/10.1016/j.jpaa.2005.09.007
State	Published - Sep 2006
Externally published	Yes

Bibliographical note

Funding Information:
Behrens and Lawson were supported by the NSF.

Keywords

Morava stabilizer group
Quaternion algebras
Supersingular elliptic curves

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abstract = "Let S2 be the p-primary second Morava stabilizer group, C a supersingular elliptic curve over over(F, -)p, O the ring of endomorphisms of C, and ℓ a topological generator of Zp× (or Z2× / {± 1} if p = 2). We show that for p > 2 the group Γ ⊆ O [1 / ℓ]× of quasi-endomorphisms of degree a power of ℓ is dense in S2. For p = 2, we show that Γ is dense in an index 2 subgroup of S2.",

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AU - Lawson, Tyler

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N2 - Let S2 be the p-primary second Morava stabilizer group, C a supersingular elliptic curve over over(F, -)p, O the ring of endomorphisms of C, and ℓ a topological generator of Zp× (or Z2× / {± 1} if p = 2). We show that for p > 2 the group Γ ⊆ O [1 / ℓ]× of quasi-endomorphisms of degree a power of ℓ is dense in S2. For p = 2, we show that Γ is dense in an index 2 subgroup of S2.

AB - Let S2 be the p-primary second Morava stabilizer group, C a supersingular elliptic curve over over(F, -)p, O the ring of endomorphisms of C, and ℓ a topological generator of Zp× (or Z2× / {± 1} if p = 2). We show that for p > 2 the group Γ ⊆ O [1 / ℓ]× of quasi-endomorphisms of degree a power of ℓ is dense in S2. For p = 2, we show that Γ is dense in an index 2 subgroup of S2.

KW - Morava stabilizer group

KW - Quaternion algebras

KW - Supersingular elliptic curves

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Isogenies of elliptic curves and the Morava stabilizer group

Abstract

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Keywords

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