In this paper, we present a construction of locally recoverable codes (LRCs) with multiple recovery sets using algebraic curves with many rational points. By leveraging separable morphisms between smooth projective curves and expanding the class of curves previously considered, we significantly generalize and enhance the framework. Our approach corrects certain inaccuracies in the existing literature while extending results to a broader range of curves, thereby achieving better parameters and wider applicability. In addition, the constructions presented here result in LRCs with large availability.

This paper studies hyperelliptic curves _d corresponding to y^2=φ_d(x) over finite fields, with φ_d(x) a Chebyshev polynomial. Starting from the case where d=ℓ is an odd prime number, new cases (d,q) are presented where _d is maximal over the finite field _q^2 of cardinality q^2. In addition, new conditions ruling out the possibility that _d/_q^2 is maximal for given (d,q), are presented. The arguments involve a mix of results on slopes of Frobenius, explicit descriptions of abelian subvarieties of the jacobian of _d with complex multiplication, and a technique from the theory of 2-descent on jacobians of hyperelliptic curves. In particular, the method used here to prove maximality in characteristics p≡ 1 4 for d≡ 1 4 a prime number, deserves attention, as it differs from earlier maximality arguments for other curves. Using the new results as well as extensive calculations with Magma, we pose some questions. A positive answer would completely classify the pairs (q,d) resulting in maximality.

Let K be a complete, non-archimedean valued field with a residue field of characteristic different from 2. A Whittaker group G is a discontinuous subgroup of PGL(2,K), freely generated by elements s_0,...,s_g of order two, each defined by a pair of fixed points {a_0,b_0},...,{a_g,b_g}. These fixed points are called ``in good position''. A subgroup W in G of index 2 is a Schottky group and produces a hyperelliptic Mumford curve Omega/W --> Omega/G = P^1, called `Whittaker curve', of genus g and with branch locus B in P^1(K). An explicit parametrization of Whittaker curves in terms of theta functions for W and G and the data of the fixed points, is developed. In particular, this allows one to express the branched points (and other data such as p-adic periods and p-adic heights) in terms of values of theta functions. A central theme of this paper is the relation between the fixed points and the branch locus. For a given configuration (P,m) of $g+1$ pairs of points in P^1, one defines a rigid space Fix_{P,m} of fixed points in good position with that configuration and a rigid space of branched points $ Branch_{P,m} in that configuration. A main result is that the natural morphism FB: Fix_{P,m} --> Branch_{P,m} is a rigid etale covering with Galois group {\pm 1}^{d-1} for some d>0. For all cases of genus g=2,3 (and for some more), an approximation of FB is computed which confirms the main result. Classification of Whittaker groups and analytic reductions of Whittaker curves is another important issue of this paper. The background material in this paper complements the work of L.~Gerritzen, G.~Van Steen, F.~Herrlich and others. It involves re-examination of some proofs, the derivation of properties of semi-stable analytic reductions and studying good position of fixed points.

There is an abundance of equations of Painleve type besides the classical Painleve equations. Classifications have been computed by the Japanese school. Here we consider Painleve type equations induced by isomonodromic families of linear ODE's having at most z = 0 and z = infinity as singularities. Requiring that the formal data at the singularities produce isomonodromic families parametrized by a single variable t leads to a small list of hierarchies of cases. The study of these cases involves Stokes matrices and moduli for linear ODE's on the projective line. Case studies reveal interesting families of linear ODE's and Painleve type equations. However, rather often the complexity (especially of the Lax pair) is too high for either the computations or for the output. Apart from classical Painleve equations one rediscovers work of Harnad, Noumi and Yamada. A hierarchy, probably new, related to the classical P3(D8), is discovered. Finally, an amusing "companion" of P1 is presented.

We investigate the automorphism groups of the algebraic curves 𝒞_d : y^d = φ_d(x), where φ_d(x) denotes the Chebyshev polynomial of degree d, defined over a field k with p:=char(k) ∤ 2d. We determine the full automorphism group of 𝒞_d in all the cases considered in this paper, namely for d=4, and more generally when 2d = p^r+1 or 4d = p^r+1. For all other d>4, Expectation predicts what the automorphism group should be. As an application, we show that certain maximal curves of the same genus are not isomorphic.

We study hyperelliptic curves arising from Chebyshev polynomials. The aim of this paper is to characterize the pairs (q,d) such that the hyperelliptic curve C over a finite field Fq2 given by y2=φd(x) is maximal over the finite field Fq2 of cardinality q2. Here φd(x) denotes the Chebyshev polynomial of degree d. The same question is studied for the curves given by y2=(x±2)φd(x), and also for y2=(x2−4)φd(x). Our results generalize some of the statements in [12].

The aim of this paper is to present families of elliptic surfaces defined over function fields of even characteristic having arbitrarily large Mordell-Weil rank. More precisely, we study the elliptic surface arising from the quartic twist of a supersingular elliptic curve defined over the field with two elements, using the function field of a maximal curve C admitting an order 4 automorphism. For the resulting elliptic surface, we provide a rank formula for its Mordell-Weil group in terms of the genera of C and another curve covered by C.

We construct explicit families of hyperelliptic curves over whose Jacobians admit complex multiplication (CM). Each curve in these families is defined by v^2 = (u+2) φ_d(u), d = 2^e or d=p ≥ 3 prime, where φ_d(x) is the Chebyshev polynomial of degree d. We prove that the Jacobians are simple and determine the associated CM-fields explicitly. Our approach exploits the interplay between Chebyshev polynomials and Galois coverings, providing concrete examples of abelian varieties with CM and explicit criteria for their construction.

Isomonodromy for the fifth Painlevé equation ${\rm P}_5$ is studied in detail in the context of certain moduli spaces for connections, monodromy, the Riemann-Hilbert morphism, and Okamoto-Painlevé spaces. This involves explicit formulas for Stokes matrices and parabolic structures. The rank 4 Lax pair for ${\rm P}_5$, introduced by Noumi-Yamada et al., is shown to be induced by a natural fine moduli space of connections of rank 4. As a by-product one obtains a polynomial Hamiltonian for ${\rm P}_5$, equivalent to the one of Okamoto.

We study elliptic surfaces corresponding to an equation of the specific type y2=x3+f(t)x, defined over the finite field Fq for a prime power q≡3mod4. It is shown that if s4=f(t) defines a curve that is maximal over Fq2 then the rank of the group of sections defined over Fq on the elliptic surface is determined in terms of elementary properties of the rational function f(t). Similar results are shown for elliptic surfaces given by y2=x3+g(t) using prime powers q≡5mod6 and curves s6=g(t). Finally, for each of the forms used here, existence of curves with the property that they are maximal over Fq2 is discussed, as well as various examples.

This note recalls an early 13th century result on congruent numbers by Leonardo Pisano (“Fibonacci”), and shows how it relates to a specific much studied K3 surface and to an elliptic fibration on this surface. As an aside, the discussion reveals how, via explicit maps of degree two, the surface is covered by the Fermat quartic surface and also covers one of the two famous ‘most algebraic K3 surfaces’.

This paper discusses prime numbers that are (resp. are not) congruent numbers. Particularly the only case not fully covered by earlier results, namely primes of the form p = 8k +1, receives attention.

The problem of algebraic dependence of solutions to (non-linear) first order autonomous equations over an algebraically closed field of characteristic zero is given a 'complete' answer, obtained independently of model theoretic results on differentially closed fields. Instead, the geometry of curves and generalized Jacobians provides the key ingredient. Classification and formal solutions of autonomous equations are treated. The results are applied to answer a question on D-n-finiteness of solutions of first order differential equations.

This exposition reviews what exactly Gauss asserted and what did he prove in the last chapter of {\sl Disquisitiones Arithmeticae} about dividing the circle into a given number of equal parts. In other words, what did Gauss claim and actually prove concerning the roots of unity and the construction of a regular polygon with a given number of sides. Some history of Gauss's solution is briefly recalled, and in particular many relevant classical references are provided which we believe deserve to be better known.

This note presents explicit equations (up to birational equivalence over $\mathbb{F}_2$) for a complete, smooth, absolutely irreducible curve $X$ over $\mathbb{F}_2$ of genus $50$ satisfying $#X(\mathbb{F}_2)=40$. In his 1985 Harvard lecture notes on curves over finite fields, J-P.~Serre already showed the existence of such a curve: he used class field theory to describe the function field $\mathbb{F}_2(X)$ as a certain abelian extension of the function field $\mathbb{F}_2(E)$ of some elliptic curve $E/\mathbb{F}_2$. Although various more recent texts recall Serre's construction, explicit equations as well as a description of intermediate curves $X\to Y\to E$ over $\mathbb{F}_2$ seem to be new. We also describe explicit equations for a curve over $\mathbb{F}_2$ of genus $8$ with $11$ rational points, and for a curve over $\mathbb{F}_2$ of genus $22$ with $21$ rational points.

The methods of [vdP-Sa, vdP1, vdP2] are applied to the fourth Painlevé equation. One obtains a Riemann—Hilbert correspondence between moduli spaces of rank two connections on ℙ^1 and moduli spaces for the monodromy data. The moduli spaces for these connections are identified with Okamoto—Painlevé varieties and the Painlevé property follows. For an explicit computation of the full group of Bäcklund transformations, rank three connections on ℙ^1 are introduced, inspired by the symmetric form for PIV, studied by M. Noumi and Y. Yamada.

For the hyperelliptic curve C_p with equation y^2=x(x-2p)(x-p)(x+p)(x+2p) with p a prime number, we discuss bounds for the rank of its Jacobian over Q, find many cases having 2-torsion in the associated Shafarevich-Tate group, and we present some results on rational points of C_p.

This note reformulates Mazur’s result on the possible orders of rational torsion points on elliptic curves over $$\mathbb {Q}$$ in a way that makes sense for arbitrary genus one curves, regardless whether or not the curve contains a rational point. The main result is that explicit examples are provided of ‘pointless’ genus one curves over $$\mathbb {Q}$$ corresponding to the torsion orders 7, 8, 9, 10, 12 (and hence, all possibilities) occurring in Mazur’s theorem. In fact three distinct methods are proposed for constructing such examples, each involving different in our opinion quite nice ideas from the arithmetic of elliptic curves or from algebraic geometry.

In this paper, we give explicit equations for homogeneous spaces corresponding to a rational isogeny of degree $3$. An explicit set of elliptic curves with elements of order $3$ in their Tate-Shafarevich group is constructed. Combining this gives explicit examples of plane cubics over $\Q$ that have a point everywhere locally, but not globally.