Igor Shparlinski, Macquarie University, Sydney (Australie)

Isomorphism classes of elliptic and hyperelliptic curves in some thin families

(joint work with Mei-Chu Chang, Javier Cilleruelo, Moubariz Garaev. Jose Hernandez and
Ana Zumalacarregui)

For a prime p , we obtain non-trivial upper bounds for the number of hyperelliptic curves
                                 Y² = X^2g+1 + a_2g-1 X^2g-1 + ... + a₁ X + a₀
over F_p, with coefficients in a 2g-dimensional cube
                                 (a₀, ..., a_2g-1) ∈ [R₀+1, R₀+M] × ... × [R_2g-1+1, R_2g-1+M]
that are isomorphic to a given curve and give an almost sharp lower bound on the number
of non-isomorphic hyperelliptic curves with coefficients in that cube.