An elementary proof for the number of supersingular elliptic curves
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An elementary proof for the number of supersingular elliptic curves Luís R. A. Finotti1
© Instituto de Matemática e Estatística da Universidade de São Paulo 2020
Abstract Building on Finotti in (Acta Arith 139(3):265–273, 2009), we give an elementary proof for the well known result that there exactly ⌈(p − 1)∕4⌉ − ⌊(p − 1)∕6⌋ supersingular elliptic curves in characteristic p. We use a related polynomial instead of the supersingular polynomial itself to simplify the proof and this idea might be helpful to prove other results related to the supersingular polynomial. Keywords Arithmetic geometry · Elliptic curves · Supersingular polynomial Mathematics Subject Classification Primary 11G20; Secondary 11G07
1 Introduction An elliptic curve over a field of characteristic p > 0 is ordinary if its p-torsion is isomorphic to ℤ∕pℤ . Otherwise, its p-torsion is trivial and we say that the elliptic curve is supersingular. It’s a well known result that there are only finitely many supsersingular elliptic curves up to isomorphism, and in fact there are exactly ⌈(p − 1)∕4⌉ − ⌊(p − 1)∕6⌋ supersingular elliptic curves in characteristic p ≥ 5 . More precisely, if k is an algebraically closed field of characteristic p > 0 , or, more generally, if k contains 𝔽p2 , then there are exactly ⌈(p − 1)∕4⌉ − ⌊(p − 1)∕6⌋ supersingular elliptic curves over k. (See for instance Chapter V of [6].) Hence, for a fixed characteristic p > 0 , we define the supersingular polynomial (in characteristic p), denoted by ssp (X) , as the monic polynomial that has simple roots exactly at the j-invariants of all supersingular elliptic curves, i.e.,
Communicated by Claudio Gorodski. * Luís R. A. Finotti [email protected] 1
Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA
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ssp (X) =
∏
(X − j).
(1.1)
j supersig.
In [3], it was proved that the supersingular polynomial can be explicitly written as )( r2 ( )( ) ( )r ∑ � r i 27 i i−r1� 2 X (X − 1728)r2 −i , − ssp (X) = − (1.2) i 3i − r 9 i=r 4 1
def
def
def
def
def
where r = (p − 1)∕2 , r1 = ⌈r∕3⌉ , r2 = ⌊r∕2⌋ , r1� = ⌊r∕3⌋ , and r2� = ⌈r∕2⌉ . Note, in particular, that ssp (X) ∈ 𝔽p [X] . More on the supersingular polynomial, including different formulas, can be found in Kaneko and Zagier’s [4], Brillhart and Morton’s [1], and Morton’s [5]. We also note that the published formula in [3] has a typo, but Eq. (1.2) is correct. Note that in principle we are working over an algebraically closed field k of characteristic p > 0 , so the supersingular j-invariants in Eq. (1.1) are taken to be in k. On the other hand, since ssp ∈ 𝔽p [x] , the polynomial itself does not depend on k, but simply on its characteristic. Formula (1.2) above, which was nearly deduced by Deuring in [2], was fully derived in [3] by using the fact the an elliptic curve is supersingular if, and only if, its Hasse invariant is zero. (This result is due to Deuring and Hasse.) This was enough to obtain an expression quite close to the o
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