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On the Hasse invariants of elliptic curves. (English) Zbl 0841.11031

Let \(p\) be a prime, and \(\mathbb{F}_p\) be the field of order \(p\). Binomial coefficients have an \(\mathbb{F}_p\)-analogue defined in terms of Jacobi sums. This paper establishes congruences modulo powers of \(p\) between these finite field binomial coefficients and ordinary ones. This, in turn, can be used to obtain a congruence modulo \(p^2\) for the Hasse invariant of an elliptic curve, and the paper does so for the curve \(y^2= x^3+1\) when \(p\equiv 1\pmod 3\). Finally, a congruence \(\text{mod} p\) for the Hasse invariant of an arbitrary elliptic curve is derived in terms of hypergeometric series over \(\mathbb{F}_p\), working through \(\mathbb{F}_p\)-binomial coefficients.
Reviewer: J.Jones (Tempe)

MSC:

11G20 Curves over finite and local fields
14H52 Elliptic curves
11T24 Other character sums and Gauss sums
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