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We give an explicit version of a classical theorem of Stickelberger on the representation of certain integers by binary quadratic forms. This is achieved by generalizing Stickelberger's original congruences via an extension of a recent result of Young.

The Coleman-Chabauty bound is an upper bound for the number of rational points on a curve of genus g ≥ 2 whose Jacobian has Mordell-Weil rank r less than g . The bound is given in terms of the genus of the curve and the number of F p -points on the reduced curve, for all primes p of good reduction such that p > 2 g . In this Note we show that the hypothesis on the Mordell-Weil rank is essential. We do so by exhibiting, for each prime p ≥ 5, an explicit family of curves of genus ( p − 1)/2 (and rank at least ( p − 1)/2) for which the bound in question does not hold. Our examples show that the difference between the number of rational points and the bound in question can in fact be linear in the genus. Under mild assumptions, our curves have rank at least twice their genus.

In his paper ("Oevres Scientifiques, Collected Papers, Vol. III, pp. 329-342, Springer-Verlag, Berlin/New York, 1979), Well asserted (without proof) that the automorphism group of the Fermat hypersurface of exponent N and dimension r − 1 over an algebraically closed field of characteristic prime to N is the semidirect product of the symmetric group on r + 1 letters and the direct sum of r copies of the cyclic group of order N. It turns out that the assertion is false in positive characteristic. In this paper, we present a proof of Weil′s assertion for the case of the Fermat curves (r = 2) in characteristic 0.