Concentration of Points and Isomorphism Classes of Hyperelliptic Curves over a Finite Field in Some Thin Families
نویسندگان
چکیده
For a prime p and a polynomial f ∈ Fp[X], we obtain upper bounds on the number of solutions of the congruences f(x) ≡ y (mod p) and f(x) ≡ y (mod p), where (x, y) belongs to an arbitrary square with side length M . Further, we obtain non-trivial upper bounds for the number of hyperelliptic curves Y 2 = X + a2g−1X 2g−1 + . . .+ a1X + a0 over Fp, with coefficients in a 2g-dimensional cube (a0, . . . , a2g−1) ∈ [R0 + 1, R0 +M ]× . . .× [R2g−1 + 1, R2g−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.
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