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Gross lattices of supersingular elliptic curves

5 Mar 2025arXiv:2503.03478links table onlyarchive 2025-07-28

Chenfeng He, Gaurish Korpal, Ha T. N. Tran, Christelle Vincent

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Let p be a prime, E be a supersingular elliptic curve defined over 𝔽̅ₚ, and 𝒪 be its (geometric) endomorphism ring. Earlier results of Chevyrev-Galbraith and Goren-Love have shown that the successive minima of the Gross lattice of 𝒪 characterize the isomorphism class of 𝒪. In this paper, we extend this work and show that the value of the third successive minimum D₃ of the Gross lattice gives necessary and sufficient conditions for the curve to have its j-invariant in the field 𝔽ₚ or in the set 𝔽_(p²) ∖𝔽ₚ, as well as finer information about the endomorphism ring of E when its j-invariant belongs to 𝔽ₚ and p ≡3 4. We end our article with an investigation of the geometry of Gross lattices of supersingular elliptic curves.

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