Supersingular prime (algebraic number theory)

Template:Short description 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 $E$ is defined over the rational numbers, then a prime $p$ is supersingular for E if the reduction of $E$ modulo $p$ is a supersingular elliptic curve over the residue field ${𝔽}_p$.Template:Sfn

Equivalently, for a prime $p>3$ of good reduction for $E$, the prime is supersingular for $E$ if and only if the trace of the Frobenius endomorphism $a_p=p+1-\mathrm{\#}E({𝔽}_p)$ is zero, that is, $\mathrm{\#}E({𝔽}_p)=p+1$.Template:Sfn This condition means that the reduction of $E$ modulo $p$ has the maximum possible endomorphism ring—an order in a quaternion algebra—rather than an order in an imaginary quadratic field.Template:Sfn

Distribution

Complex multiplication case

When $E$ has complex multiplication (CM) by an order in an imaginary quadratic field $K$, the distribution of supersingular primes is well understood. A classical result of Deuring (1941) implies that a prime $p$ of good reduction is supersingular for $E$ if and only if $p$ is inert or ramified in $K$.Template:Sfn 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 $1/2$.Template:Sfn

Non-CM case

When $E$ 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 $E$.Template:SfnTemplate:Sfn Serre later obtained the unconditional upper bound

${\pi }_{E,0}(x)\ll \frac{x}{{(\log x)}^{3/2-\epsilon }}$

for any $\epsilon >0$, where ${\pi }_{E,0}(x)$ denotes the number of supersingular primes up to $x$, and showed that under the Generalized Riemann Hypothesis (GRH) one could achieve the bound ${\pi }_{E,0}(x)\ll x^{3/4}$.Template:SfnTemplate:Sfn The unconditional exponent was subsequently improved by Wan (1990), who showed

${\pi }_{E,0}(x)\ll \frac{x}{{(\log x)}^{2-\epsilon }}$

by incorporating sieve-theoretic techniques.Template:Sfn

Despite this rarity, Noam Elkies proved in 1987 that every elliptic curve over $\mathbb{Q}$ has infinitely many supersingular primes.Template:Sfn His proof uses the theory of complex multiplication and Deuring's lifting lemma: given any finite set $S$ of primes, one can find a negative fundamental discriminant $-D$ such that the Hilbert class polynomial $P_D(X)$ evaluated at the j-invariant $j(E)$ has a prime factor outside $S$, and this prime is necessarily supersingular for $E$.Template:Sfn Elkies later extended this result to elliptic curves defined over any number field with at least one real embedding.Template:Sfn

Lang–Trotter conjecture

Template:Harvtxt conjectured that the number of supersingular primes less than a bound $X$ satisfies

${\pi }_{E,0}(X)\sim C_E\frac{\sqrt{X}}{\log X}$

as $X\to \mathrm{\infty }$, where $C_E>0$ is an explicit constant depending on $E$.Template:Sfn This prediction arises from a probabilistic heuristic: by the Hasse bound, $|a_p|\le 2\sqrt{p}$, so the "probability" that $a_p=0$ for a random prime $p$ is roughly $1/\sqrt{p}$, and summing this over primes up to $X$ gives an expected count on the order of $\sqrt{X}/\log X$.Template:Sfn Template:As of, this conjecture remains open.Template:Sfn

Example

Consider the elliptic curve $E:y^2+y=x^3-x^2-10x-20$, which is the modular curve $X_0(11)$ with j-invariant $j=-2^{12}/11$. The supersingular primes for this curve begin: 2, 19, 29, 199, 569, 809, 1289, 1439, 2539, 3319, ... Template:OEIS.Template:Sfn These are exactly the primes $p$ for which the coefficient of $q^p$ in $\eta (\tau {)}^2\eta (11\tau {)}^2$ vanishes modulo $p$, where $\eta$ is the Dedekind eta function.

Generalization

More generally, if $K$ is any global field—that is, a finite extension of $\mathbb{Q}$ or of ${𝔽}_p(t)$—and $A$ is an abelian variety defined over $K$, then a supersingular prime $𝔭$ for A is a finite place of $K$ such that the reduction of $A$ modulo $𝔭$ is a supersingular abelian variety.Template:Sfn It is conjectured that every abelian variety over a number field has infinitely many supersingular primes, but this is known only in special cases.Template:Sfn

See also

• Supersingular prime (moonshine theory)
• Supersingular elliptic curve
• Lang–Trotter conjecture
• Sato–Tate conjecture

References

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