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Statistics for traces of cyclic trigonal curves over finite fields. (English) Zbl 1201.11063

The authors study the variation of the trace of the Frobenius endomorphism associated to a cyclic trigonal curve of genus \(g\) over \(\mathbb{F}_q\) \((q \equiv 1\pmod 3)\) as the curve varies in an irreducible component of the moduli space. They show that for \(q\) fixed and \(g\) increasing, the limiting distribution of the trace of Frobenius equals the sum of \(q+1\) independent random variables taking the value \(0\) with probability \(2/(q+2)\) and \(1,\,e^{2\pi i/3},\,e^{4\pi i/3}\) each with probability \(q/(3(q+2))\). This extends the work of P. Kurlberg and Z. Rudnick [J. Number Theory 129, No. 3, 580–587 (2009; Zbl 1221.11141)] who considered the same limit for hyperelliptic curves (although not on the moduli space, which makes a slight difference explained in Theorem 1.1). They also show that when both \(g\) and \(q\) go to infinity, the normalized trace has a standard complex Gaussian distribution and how to generalize these results to \(p\)-fold covers of the projective line.

MSC:

11G20 Curves over finite and local fields
14G15 Finite ground fields in algebraic geometry

Citations:

Zbl 1221.11141
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