Abstract
In this paper, we show that there are solutions of degree r of the equation of Pell–Abel on some real hyperelliptic curve of genus g if and only if r>g. This result, which is known to the experts, has consequences, which seem to be unknown to the experts. First, we deduce the existence of a primitive k-differential on an hyperelliptic curve of genus g with a unique zero of order k(2g-2) for every (k,g)≠(2,2). Moreover, we show that there exists a non Weierstrass point of order n modulo a Weierstrass point on a hyperelliptic curve of genus g if and only if n>2g.
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