Given a genus 2 curve $C$ defined over a finite field $𝔽_{q}$ of odd characteristic such that $2 | # Jac (C) (𝔽_{q})$ , we study the growth of the 2-adic valuation of the cardinality of the Jacobian over a tower of quadratic extensions of $𝔽_{q}$ . In the cases of simpler regularity, we determine the exponents of the 2-Sylow subgroup of $Jac (C) (𝔽_{q^{2^{k}}})$ .

Communicated by E. Gorla

Keywords:

AMSC: 11G20, 14H40, 14H45

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