Frey's Formula and utilisation of the Hasse Invariant in "Links between Stable elliptic curves and Diophantine equations."

Question

In the paper "Links between Stable elliptic curves and Diophantine equations" for an elliptic curve $E$ with normal Weierstrass form $$y^2 = x^3 -g_2x -g_3$$ with $g_i \in \mathbb{Z}$ w.l.o.g. Then he states the Hasse invariant $\delta$ for $E$ is defined as $$\delta_E := -\frac{1}{2} g_2 g_3^{-1} \: (mod \: \mathbb{Q}^{*2}) .$$

In all other references to the Hasse Invariant of elliptic curves, Arithmetic of Elliptic Curves by Silverman etc., it is stated that for an elliptic curve over a finite field the invariant is either $0$ or $1$.

Furthermore later in the paper Frey begins to talk about the extension of $\mathbb{Q}$ by the square root of the Hasse invariant i.e. $ \mathbb{Q}(\sqrt{\delta_E})$ where $E$ is the 'Frey curve.' In one case this is trivial but in other not.

Are these two seperate quantities that I am getting confused, or is there additional structure behind the Hasse invariant which I have not encountered?

Answer 1

This is one of those subjects where I used to know a lot more than I do now; and I wasn’t an expert even then. About the Hasse invariant that Frey is talking about, if I ever knew anything, I forgot it entirely. But I do (did) know something serious about the Hasse invariant $H(E)$ for an elliptic curve $E$ in characteristic $p$.

Joe Silverman simplified the story considerably for these Hasse invariants. In fact, $H(E)$ is an element of the base well-defined up to $(p-1)$-th powers. If you set the weight of $g_2$ to be $4$ and of $g_3$ to be $6$, then the H-invariant of $y^2=x^3-g_2x-g_3$ will be a polynomial in which every monomial has weight $p-1$. For instance, in characteristic $5$, it’s $3g_2$, in characteristic $7$, it’s $4g_3$, and in characteristic $13$, it’s $2g_3^2 + 6g_2^3$. I think I recall that N. Katz has a way of looking at this according to which $H(E)=E_{p-1}$, the Eisenstein series. I’ll have to beg one of the better-educated correspondents here to fill in the details.