In algebraic number theory, a supersingular prime for a given elliptic curve is a prime number with a certain relationship to that curve. If the curve E {\displaystyle E} is defined over the rational numbers, then a prime p {\displaystyle p} is supersingular for E if the reduction of E {\displaystyle E} modulo p {\displaystyle p} is a supersingular elliptic curve over the residue field F p {\displaystyle \mathbb {F} _{p}} . Equivalently, for a prime p > 3 {\displaystyle p>3} of good reduction for E {\displaystyle E} , the prime is supersingular for E {\displaystyle E} if and only if the trace of the Frobenius endomorphism a p = p + 1 − # E ( F p ) {\displaystyle a_{p}=p+1-\#E(\mathbb {F} _{p})} is zero, that is, # E ( F p ) = p + 1 {\displaystyle \#E(\mathbb {F} _{p})=p+1} . This condition means that the reduction of E {\displaystyle E} modulo p {\displaystyle p} has the maximum possible endomorphism ring—an order in a quaternion algebra—rather than an order in an imaginary quadratic field.
Distribution
Complex multiplication case When E {\displaystyle E} has complex multiplication (CM) by an order in an imaginary quadratic field K {\displaystyle K} , the distribution of supersingular primes is well understood. A classical result of Deuring (1941) implies that a prime p {\displaystyle p} of good reduction is supersingular for E {\displaystyle E} if and only if p {\displaystyle p} is inert or ramified in K {\displaystyle K} . By the Chebotarev density theorem, these primes constitute exactly half of all primes, so the set of supersingular primes for a CM curve has natural density 1 / 2 {\displaystyle 1/2} .
Non-CM case When E {\displaystyle E} does not have complex multiplication, the situation is more subtle. In 1968, Serre proved that the set of supersingular primes has asymptotic density zero by applying the Chebotarev density theorem to the number fields generated by coordinates of the torsion points of E {\displaystyle E} . Serre later obtained the unconditional upper bound
π E , 0 ( x ) ≪ x ( log x ) 3 / 2 − ε {\displaystyle \pi _{E,0}(x)\ll {\frac {x}{\left(\log x\right)^{3/2-\varepsilon }}}}
for any ε > 0 {\displaystyle \varepsilon >0} , where π E , 0 ( x ) {\displaystyle \pi _{E,0}(x)} denotes the number of supersingular primes up to x {\displaystyle x} , and showed that under the Generalized Riemann Hypothesis (GRH) one could achieve the bound π E , 0 ( x ) ≪ x 3 / 4 {\displaystyle \pi _{E,0}(x)\ll x^{3/4}} . The unconditional exponent was subsequently improved by Wan (1990), who showed
π E , 0 ( x ) ≪ x ( log x ) 2 − ε {\displaystyle \pi _{E,0}(x)\ll {\frac {x}{\left(\log x\right)^{2-\varepsilon }}}}
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