Let l be a prime and G a pro-l group with torsion-free abelianization. We
produce group-theoretic analogues of the Johnson/Morita cocycle for G -- in the
case of surface groups, these cocycles appear to refine existing constructions
when l=2. We apply this to the pro-l etale fundamental groups of smooth curves
to obtain Galois-cohomological analogues, and discuss their relationship to
work of Hain and Matsumoto in the case the curve is proper. We analyze many of
the fundamental properties of these classes and use them to give an example of
a non-hyperelliptic curve whose Ceresa class has torsion image under the l-adic
Abel-Jacobi map.
We study when Hurwitz curves are supersingular. Specifically, we show that the curve H n,ℓ : X n Y ℓ + Y n Z ℓ + Z n X ℓ = 0, with n and ℓ relatively prime, is supersingular over the finite field Fp if and only if there exists an integer i such that p i ≡ −1 mod (n 2 −nℓ+ℓ 2 ). If this holds, we prove that it is also true that the curve is maximal over F p 2i . Further, we provide a complete table of supersingular Hurwitz curves of genus less than 5 for characteristic less than 37.
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