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In this article we give an explicit construction of the moduli space of trigonal superelliptic curves with level 3 structure. The construction is given in terms of point sets on the projective line and leads to a closed formula for the number of connected (and irreducible) components of the moduli space. The results of the article generalise the description of the moduli space of hyperelliptic curves with level 2 structure, due to Dolgachev and Ortland, Runge and Tsuyumine. [ABSTRACT FROM AUTHOR]

Conditionally on the a b c conjecture, we generalize the previous work of Clark and the author to show that a superelliptic curve C : y n = f (x) of sufficiently high genus has infinitely many twists violating the Hasse Principle if and only if f (x) has no ℚ -rational roots. We also show unconditionally that a curve defined by C : y p N = f (x) (for p prime and N > 1) has infinitely many twists violating the Hasse Principle over any number field k such that k contains the p th roots of unity and f (x) has no k -rational roots. [ABSTRACT FROM AUTHOR]

For the superelliptic curves of the form (x + 1) ⋯ (x + i − 1) (x + i + 1) ⋯ (x + k) = y ℓ with x , y ∈ ℚ , y ≠ 0 , k ≥ 3 , 1 ≤ i ≤ k , ℓ ≥ 2 , a prime, Das, Laishram, Saradha and Edis showed that the superelliptic curve has no rational points for ℓ ≥ e 3 k . In fact, the double exponential bound, obtained in these papers is far from the reality. In this paper, we study the superelliptic curves for small values of k. In particular, we explicitly solve the above equation for 4 ≤ k ≤ 8. [ABSTRACT FROM AUTHOR]

We present an algorithm for computing the Berkovich skeleton of a superelliptic curve yn = f(x) over a valued field. After defining superelliptic weighted metric graphs, we show that each one is realizable by an algebraic superelliptic curve when n is prime. Lastly, we study the locus of superelliptic weighted metric graphs inside the moduli space of tropical curves of genus g. [ABSTRACT FROM AUTHOR]

The theory of elliptic and hyperelliptic curves has been of crucial importance in the development of algebraic geometry. Almost all fundamental ideas were first obtained and generalized from computations and constructions carried out for elliptic or hyperelliptic curves. In this long survey, we show that this theory can be extended naturally to all superel-liptic curves. We focus on automorphism groups, stratification of the moduli space Mg, binary forms, invariants of curves, weighted projective spaces, minimal models for superel-liptic curves, field of moduli versus field of definition, theta functions, Jacobian varieties, addition law in the Jacobian, isogenies among Jacobians, etc. Many recent developments on the theory of superelliptic curves are provided as well as many open problems. [ABSTRACT FROM AUTHOR]

In this paper, we study the number of F q n -rational points on the affine curve X d , a , b given by the equation y d = a x Tr (x) + b , where Tr denote the trace function from F q n to F q and d is a positive integer. In particular, we present bounds for the number of F q -rational points on X d , a , b and, for the cases where d satisfies a natural condition, explicit formulas for the number of rational points are obtained. Particularly, a complete characterization is given for the case d = 2. As a consequence of our results, we compute the number of elements α in F q n such that α and Tr (α) are quadratic residues in F q n . [ABSTRACT FROM AUTHOR]

The commutative ring R(P(t))=C[t±1,u|u2=P(t)], where P(t)=∑i=0naiti=∏k=1n(t-αi) with αi∈C pairwise distinct, is the coordinate ring of a hyperelliptic curve when n>4. The Lie algebra R(P(t))=Der(R(P(t))) of derivations is called the hyperelliptic Lie algebra associated to P(t) and is a particular type of multipoint Krichever-Novikov algebra. In this paper, we describe the universal central extension of Der(R(P(t))) in terms of certain families of polynomials which in a particular case are associated Legendre polynomials. Moreover we describe certain families of polynomials that arise in the study of the group of units for the ring R(P(t)),  where P(t)=t4-2bt2+1. In this study, pairs of Chebyshev polynomials (Un,Tn) arise as particular cases of a pairs (rn,sn) with rn+snP(t) a unit in R(P(t)). We explicitly describe these polynomial pairs as coefficients of certain generating functions and show that certain of these polynomials satisfy particular second-order linear differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the set NR ( G ) of natural numbers which are not in the numerical semigroup generated by a compound sequence G . We generalize a result of Tuenter which completely characterizes NR ( G ) . We use this result to compute Sylvester sums, and we give a direct application to the computation of weights of higher-order Weierstrass points on some families of complex algebraic curves. [ABSTRACT FROM AUTHOR]

We investigate the proportion of superelliptic curves that have a Q p point for every place p of Q. We show that this proportion is positive and given by the product of local densities, we provide lower bounds for this proportion in general, and for superelliptic curves of the form y 3 = f (x , z) for an integral binary form f of degree 6, we determine this proportion to be 96.94%. More precisely, we give explicit rational functions in p for the proportion of such curves over Z p having a Q p -point. [ABSTRACT FROM AUTHOR]

Given $k\geqslant 2$, we show that there are at most finitely many rational numbers $x$ and $y\neq 0$ and integers $\ell \geqslant 2$ (with $(k,\ell )\neq (2,2)$) for which $$\begin{eqnarray}x(x+1)\cdots (x+k-1)=y^{\ell }.\end{eqnarray}$$ In particular, if we assume that $\ell$ is prime, then all such triples $(x,y,\ell )$ satisfy either $y=0$ or $\ell <\exp (3^{k})$. [ABSTRACT FROM AUTHOR]

Let $$\eta : C_{f,N}\rightarrow \mathbb {P}^1$$ be a cyclic cover of $$\mathbb {P}^1$$ of degree $$N$$ which is totally and tamely ramified for all the ramification points. We determine the group of fixed points of the cyclic covering group $${{\mathrm{Aut}}}(\eta )\simeq \mathbb {Z}/ N \mathbb {Z}$$ acting on the Jacobian $$J_N:={{\mathrm{Jac}}}(C_{f,N})$$ . For each prime $$\ell $$ distinct from the characteristic of the base field, the Tate module $$T_\ell J_N$$ is shown to be a free module over the ring $$\mathbb {Z}_\ell [T]/(\sum _{i=0}^{N-1}T^i)$$ . We also study the subvarieties of $$J_N$$ and calculate the degree of the induced polarization on the new part $$J_N^\mathrm {new}$$ of the Jacobian. [ABSTRACT FROM AUTHOR]

In this paper, we obtain all rational points (x,y) on the superelliptic curves \[ y^{k}=x (x+2), \]\[ y^{k}=x (x+2)(x+3), \]\[ y^{k}=x (x+1)(x+3). \] [ABSTRACT FROM PUBLISHER]

We study the Schlesinger system of partial differential equations in the case when the unknown matrices of arbitrary size (p × p) are triangular and the eigenvalues of each matrix form an arithmetic progression with a rational difference q , the same for all matrices. We show that such a system possesses a family of solutions expressed via periods of meromorphic differentials on the Riemann surfaces of superelliptic curves. We determine the values of the difference q , for which our solutions lead to explicit polynomial or rational solutions of the Schlesinger system. As an application of the (2 × 2) -case, we obtain explicit sequences of rational solutions and of one-parameter families of rational solutions of Painlevé VI equations. Using similar methods, we provide algebraic solutions of particular Garnier systems. • Schlesinger systems for triangular matrices with arithmetic constraints are studied. • Solutions are constructed via periods of differentials on superelliptic curves. • Rational solutions of Painlevé VI equations are derived. • Algebraic solutions of Garnier systems are provided. [ABSTRACT FROM AUTHOR]