Let \(k\) be perfect with \(\operatorname{ch} k = p > 0\) and let \(E/k\) be elliptic, with Frobenius \(F \colon E \to E\). Then \(F\) acts on cohomology by \[ F^* \colon H^1(E; \OO_E) \selfmap , \] which is not \(k\)-linear but \(p\)-linear: \(F^*(\lambda a) = \lambda^p F^*(a)\) for \(\lambda \in k\). Since \(E\) is elliptic, \(h^1(\OO_E) = 1\). Say \(E\) has Hasse invariant \(0\), and call \(E\) supersingular, when \(F^* = 0\); otherwise \(F^*\) is bijective, \(E\) has Hasse invariant \(1\), and \(E\) is ordinary.
(Independence of the basis)
Fix a basis vector \(e\) of the line \(H^1(E;\OO_E)\) and write \(F^*(e) = c\, e\). Then \(F^*(\lambda e) = \lambda^p c \, e = (\lambda^{p-1} c)(\lambda e)\), so changing the basis replaces \(c\) by \(\lambda^{p-1}c\), and whether \(c = 0\) is independent of the basis. If \(c \neq 0\) then \(F^*\) is injective, and it is surjective because every element of the perfect field \(k\) has a \(p\)-th root: \(\mu e = F^*\big( (\mu/c)^{1/p} e \big)\). Over an imperfect field with \(c\ne0\), the image of \(F^*\) is \(k^pc\,e\), a proper subset of \(k\,e\).
(The $p$-torsion group scheme)
The Hasse invariant separates the two possible \(p\)-torsion group schemes. In both cases \(E[p]\) has order \(p^2\), while the group of geometric points is \(\ZZ/p\) in the ordinary case and trivial in the supersingular case.
Ordinary. \(E[p] \cong \mu_p \times \ZZ/p\) over \(\bar k\). The connected-étale sequence has étale quotient \(\ZZ/p\) and connected part \(\mu_p\), which is forced by Cartier self-duality of \(E[p]\). So \(E[p](\bar k) \cong \ZZ/p\): \(p\) points, not \(p^2\).
Supersingular. \(E[p]\) is connected with connected Cartier dual — local-local — of order \(p^2\), the unique self-dual non-split extension of \(\alpha_p\) by \(\alpha_p\). It has no nontrivial \(\bar k\)-points at all, so \(E[p](\bar k) = 0\).
For a prime \(\ell \neq p\), \(E[\ell] \cong (\ZZ/\ell)^2\) over \(\bar k\). Multiplication by \(p\) has degree \(p^2\) in both cases; its separable degree is \(p\) when \(E\) is ordinary and \(1\) when \(E\) is supersingular.
(Endomorphisms)
Let \(E_{\bar k}\) be the base change of \(E\) to \(\bar k\). For \(E\) ordinary, \(\operatorname{End}(E_{\bar k},p_0)\) is \(\ZZ\) or an order in an imaginary quadratic field, and it is an order in an imaginary quadratic field when \(k\) is finite. For \(E\) supersingular, \(\operatorname{End}(E_{\bar k},p_0)\) is a maximal order in the quaternion algebra over \(\QQ\) ramified exactly at \(p\) and \(\infty\); it has rank four and is noncommutative. Over \(\CC\), the endomorphism ring of an elliptic curve has rank \(1\) or \(2\), and the \(j\)-invariants of the curves with rank \(2\) are the singular moduli; the name supersingular refers to the larger endomorphism ring, and a supersingular curve is smooth.
Dependencies
PR-CRVGRPThe group law on an elliptic curve, and multiplication by $n$uses
Backlinks
T-CRVHASSEThe Hasse invariant criterion and the Hasse polynomial $h_p(\lambda)$uses