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Let $K$ be a local field and $f(x)in K[x]$ be a non-constant polynomial. The local zeta function $Z_f(s, chi)$ was first introduced by Weil, then studied in detail by Igusa. When ${rm char}(K)=0$, Igusa proved that $Z_f(s, chi)$ is a rational function of $q^{-s}$ by using the resolution of singularities. Later on, Denef gave another proof of this remarkable result. However, if ${rm char}(K)>0$, the question of rationality of $Z_f(s, chi)$ is still kept open. Actually, there are only a few known results so far. In this paper, we investigate the local zeta functions of two-variable polynomial $g(x, y)$, where $g(x, y)=0$ is the superelliptic curve with coefficients in a non-archimedean local field of positive characteristic. By using the notable Igusas stationary phase formula and with the help of some results due to Denef and Z${rm acute{u}}$${rmtilde{n}}$iga-Galindo, and developing a detailed analysis, we prove the rationality of these local zeta functions and also describe explicitly all their candidate poles.
We give an explicit description of the stable reduction of superelliptic curves of the form $y^n=f(x)$ at primes $p$ whose residue characteristic is prime to the exponent $n$. We then use this description to compute the local $L$-factor of the curve and the exponent of conductor at $p$.
Let $p$ be a prime, let $r$ and $q$ be powers of $p$, and let $a$ and $b$ be relatively prime integers not divisible by $p$. Let $C/mathbb F_{r}(t)$ be the superelliptic curve with affine equation $y^b+x^a=t^q-t$. Let $J$ be the Jacobian of $C$. By w
In this paper we study the Coleman-Oort conjecture for superelliptic curves, i.e., curves defined by affine equations $y^n=F(x)$ with $F$ a separable polynomial. We prove that up to isomorphism there are at most finitely many superelliptic curves of
We define generalised zeta functions associated to indefinite quadratic forms of signature (g-1,1) -- and more generally, to complex symmetric matrices whose imaginary part has signature (g-1,1) -- and we investigate their properties. These indefinit
We describe the practical implementation of an average polynomial-time algorithm for counting points on superelliptic curves defined over $mathbb Q$ that is substantially faster than previous approaches. Our algorithm takes as input a superelliptic c