Poisson type phenomena for points on hyperelliptic curves modulo \(p\). (English) Zbl 1320.11056

Mak, Kit-Ho; Zaharescu, Alexandru

Funct. Approximatio, Comment. Math. 47, No. 1, 65-78 (2012).

Summary: Let \(p\) be a large prime, and let \(C\) be a hyperelliptic curve over \(\mathbb{F}_p\). We study the distribution of the \(x\)-coordinates in short intervals when the \(y\)-coordinates lie in a prescribed interval, and the distribution of the distance between consecutive \(x\)-coordinates with the same property. Next, let \(g(P,P_0)\) be a rational function of two points on \(C\). We study the distribution of the above distances with an extra condition that \(g(P_i,P_{i+1})\) lies in a prescribed interval, for any consecutive points \(P_i,P_{i+1}\).

MSC:

11G20	Curves over finite and local fields
11T99	Finite fields and commutative rings (number-theoretic aspects)
14G15	Finite ground fields in algebraic geometry

Keywords:

Poisson distribution; hyperelliptic curves

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