Crosspost from math.stackexchange, since it's quite possible I might not get a response there.

Remark 4.23.4 in Chapter IV of Hartshorne's Algebraic Geometry references a paper by Elkies that explains that$$\mathfrak{B} = \{p \text{ prime}: X_{(p)} \text{ is nonsingular over }k_{(p)}, \text{ and }X_{(p)}\text{ has Hasse invariant }0\}$$is infinite.

The existence of infinitely many superisngular primes for every elliptic curve over $\mathbb{Q}$, Invent. Math. 89 (1987) 561-567.

However, in the paper, the main theorem is stated as follows.

Let $E$ be any elliptic curve defined over $\mathbb{Q}$, and let $S$ be a finite set of primes. Then $E$ has a supersingular prime outside $S$.

Can anyone comment on the precise relationship between these two statements? Are they identical?


1 Answer 1


The term "$X_{(p)}$ has Hasse invariant 0" means that "$X_{(p)}$ is supersingular". See the definition in Hartshorne's book 2 pages before the quoted remark. So the quoted remark says that there are infinitely many supersingular primes for an elliptic curve over $\mathbb{Q}$, i.e., for any finite set $S$ of primes there is a supersingular prime outside $S$.


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