Algebraic Curves and Their Moduli

We introduce a new construction of error-correcting codes from algebraic curves over finite fields. Modular curves of genus g\ra\infty over a field of size q_0^2 yield nonlinear codes more efficient than the linear Goppa codes obtained from the same curves. These new codes now have the highest asymptotic transmission rates known for certain ranges of alphabet size and error rate. Both the theory and possible practical use of these new record codes require the development of new tools. On the theoretical side, establishing the transmission rate depends on an error estimate for a theorem of Schanuel applied to the function field of an asymptotically optimal curve. On the computational side, actual use of the codes will hinge on the solution of new problems in the computational algebraic geometry of curves.

Rational functions on algebraic curves, which have a single zero and a single pole, are considered. A pair consisting of such a function and a curve is called an Abel pair; a special case of an Abel pair is a Belyi pair. In the present paper, moduli spaces of Abel pairs for curves of genus one are studied. In particular, a number of Belyi pairs over the fields ℂ and $$ \overline{{\mathbb{F}}_p} $$ is computed. This approach could be fruitfully used in studying Hurwitz spaces and modular curves for fields of finite characteristics.

Codes obtained from algebraic curves have attracted much attention from mathematicians and engineers alike since the remarkable work of Tsfasman et al. [17] who showed that the longstAnding Gilbert-Varshamov lower bound can be exceeded for alphabet sizes larger than 49. The Gilbert-Varshamov bound, established in 1952, is a lower bound on the information rate of good codes. This lower bound was not improved until 1982 with the discovery of good algebraic geometric codes. These codes are obtained from modular curves [17], but consideration of these curves is beyond the scope of this book. van Lint and Springer [21] later derived the same results as Tsfasman et al., but by using less complicated concepts from algebraic geometry. Recall that a linear code with parameters [n, k, d] q is a linear subspace of F q n of dimension k and minimum distance d.

Bruin and Najman [LMS J. Comput. Math. 18 (2015), pp. 578–602], Ozman and Siksek [Math. Comp. 88 (2019), pp. 2461–2484], and Box [Math. Comp. 90 (2021), pp. 321–343] described all the quadratic points on the modular curves of genus 2 ≤ g ( X 0 ( n ) ) ≤ 5 2\leq g(X_0(n)) \leq 5 . Since all the hyperelliptic curves X 0 ( n ) X_0(n) are of genus ≤ 5 \leq 5 and as a curve can have infinitely many quadratic points only if it is either of genus ≤ 1 \leq 1 , hyperelliptic or bielliptic, the question of describing the quadratic points on the bielliptic modular curves X 0 ( n ) X_0(n) naturally arises; this question has recently also been posed by Mazur. We answer Mazur’s question completely and describe the quadratic points on all the bielliptic modular curves X 0 ( n ) X_0(n) for which this has not been done already. The values of n n that we deal with are n = 60 n=60 , 62 62 , 69 69 , 79 79 , 83 83 , 89 89 , 92 92 , 94 94 , 95 95 , 101 101 , 119 119 and 131 131 ; the curves X 0 ( n ) X_0(n) are of genus up to 11 11 . We find all the exceptional points on these curves and show that they all correspond to CM elliptic curves. The two main methods we use are Box’s relative symmetric Chabauty method and an application of a moduli description of Q \mathbb {Q} -curves of degree d d with an independent isogeny of degree m m , which reduces the problem to finding the rational points on several quotients of modular curves.

We define and study the stack $\\mathcal{U}^{ns,a}_{g,g}$ of (possibly singular) projective curves of arithmetic genus $g$ with $g$ smooth marked points forming an ample nonspecial divisor. We define an explicit closed embedding of a natural $\\mathbb{G}_{m}^{g}$ -torsor $\\widetilde{\\mathcal{U}}^{ns,a}_{g,g}$ over $\\mathcal{U}^{ns,a}_{g,g}$ into an affine space, and we give explicit equations of the universal curve (away from characteristics $2$ and $3$ ). This construction can be viewed as a generalization of the Weierstrass cubic and the $j$ -invariant of an elliptic curve to the case $g\\gt 1$ . Our main result is that in characteristics different from $2$ and $3$ the moduli space $\\widetilde{\\mathcal{U}}^{ns,a}_{g,g}$ is isomorphic to the moduli space of minimal $A_{\\infty}$ -structures on a certain finite-dimensional graded associative algebra $E_{g}$ (introduced by Fisette and Polishchuk).

Previous research has revealed the uniqueness-facilitation effect in the multiple object tracking (MOT) task: simple distinct identities and surface features of moving targets could facilitate attentional tracking. By adapting compound stimuli, the present study investigated whether the global or local properties played the main role in the uniqueness-facilitation effect in the MOT task. The uniqueness of local properties, of global properties or of both local and global properties were considered. Observers’ tracking performance in alternative conditions were compared with that in the homogeneous condition wherein all stimuli have identical local and global properties. Results from two experiments suggest that the global properties played the key role in facilitating tracking. The distinctiveness of local properties can also facilitate tracking with global properties being homogeneous. However, when the stimuli’s global properties are distinct from each other—whether the local properties being unique or not—observers’ tracking performance can achieve the same level as that in the unitary-uniqueness condition wherein the moving objects were distinct unitary letters. These results revealed a global superiority effect in the MOT task. Finally, the facilitation effects of the global and local properties might depend on the stimulus sparsity. When the compound stimuli had fewer local elements, the uniqueness facilitation effect on tracking decreased.

We introduce a number of new results in the context of multi-view geometry from general algebraic curves. We start with the derivation of the extended Kruppa's equations which are responsible for describing the epipolar constraint of two projections of a general (non-planar) algebraic curve. As part of the derivation of those constraints we address the issue of dimension analysis and as a result establish the minimal number of algebraic curves required for a solution of the epipolar geometry as a function of their degree and genus. We then establish new results on the reconstruction of general algebraic curves from multiple views. We address three different representations of curves: (i) the regular point representation for which we show that the reconstruction from two views of a curve of degree d admits two solutions, one of degree d and the other of degree d(d-1), (ii) the dual space representation (tangents) for which we derive a lower bound for the number of views necessary for reconstruction as a function of the curve degree and genus, and (iii) a new representation (to computer vision) based on the set of lines meeting the curve which does not require any curve fitting in image space, for which we also derive lower bounds for the number of views necessary for reconstruction as a function of the curve degree alone.

The generalized Verlinde formulae expressing traces of mapping classes corresponding to automorphisms of certain Riemann surfaces, and the congruence relations on allowed modular representations following from them are presented. The surfaces considered are families of algebraic curves given by suitably chosen equations, the modular curve X(11), and a factor curve of X(8). The examples of modular curves illustrate how the study of arithmetic properties of suitable modular representations can be used to gain information on automorphic properties of Riemann surfaces.

The Uniformization Theorem due to Koebe and Poincare implies that every compact Riemann surface of genus greater or equal to 2 can be endowed with a metric of constant curvature – 1. On the other hand, a compact Riemann surface is a complex algebraic curve and is therefore described by a polynomial equation with complex coefficients. The uniformization problem is then to link explicitly these two descriptions. In [BS05b], Peter Buser and Robert Silhol develop a new uniformization method for compact Riemann surfaces of genus two. Given such a surface S, the method describes a polynomial equation of an algebraic curve conformally equivalent to S. However, in this method appear a complex number τ BS and a function f BS which is holomorphic on the unit disk, both being characterized by some functional equations. This means that τ BS, f BS are given implicitly. P. Buser and R. Silhol then approximate them numerically by a complex number τ and a polynomial p using the approximation method developped in [BS05a]. In cases where the equation of the algebraic curve is known, they notice that these approximations are very good. In this thesis we prove a convergence theorem for the approximation method of P. Buser and R. Silhol, and we propose an adaptation of their method that allows to solve some of the numerical problems to which it is prone. Moreover, we generalize this uniformization method to hyperelliptic Riemann surfaces of genus greater than 2, and we give some examples of numerical uniformization in genus 3.

In this paper it is shown that cycles of planar dynamical systems with polynomial vector fields are in many ways similar to ovals of algebraic curves, and more generally to "separating solutions" of such systems, on general algebraic curves. For example, for separating solutions the analog of Bezout's theorem is valid. The proofs are straightforward. They are based on the version of Rolle's theorem and Bezout's theorem for planar algebraic curves offered below. We note that according to Hilbert's conjecture the total number of limit cycles of a dynamical system with a polynomial vector field is bounded above by the degree of the field. From the validity of the conjecture and the results of the present note it follows that the curve consisting of all the limit cycles of a polynomial dynamical system resembles an algebraic curve. At the present time, even the finiteness of the number of limit cycles has not been proved: Ii'yashenko proved that Dyudaktsproof [i] contains an unfillable gap (cf. [2]) . I. Rolle's Theorem for Dynamical Systems We consider a smooth dynamical system on the plane

In this chapter we present the basic facts about algebraic curves (i.e. projective varieties of dimension 1) which will be needed for our study of elliptic curves. (Actually, since elliptic curves are curves of genus 1, one of our tasks will be to define the genus of a curve.) As in Chapter I, we give references for those proofs which are not included. There are many books where the reader can find more material on the subject of algebraic curves, for example [Har, Ch. IV], [Sha 2], [G–H, Ch. 2], [Wa].KeywordsSmooth CurveElliptic CurfFunction FieldAlgebraic CurfSmooth CurfThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper discusses a solution to the problem of how to approximate an algebraic curve of any degree with piecewise parametric curves. The approximation can be performed using any differentiable parametric curve form. An implementation using rational cubic Bezier curves results in a G 2 piecewise approximation which experiences better than O(h 7) convergence. If the algebraic curve happens to be a cubic curve of genus zero, a parametric curve can be found which exactly fits the given algebraic curve to within floating point precision. An error bound is developed which is much tighter than previously published bounds.

Welding is one of the most widely used joining techniques for manufacturing complex engineering components. However, due to transient temperature cycles in the adjacent materials being joined, the local microstructure is altered and plastic strain in the surrounding materials is introduced resulting in a variation of mechanical properties across the weldment. When an extracted cross-weld test specimen is subjected to an external uniaxial load, there is interaction between the various constituents of the weld (such as the weld fused zone, heat affected zone and parent metal) resulting in development of constraint in the weaker material due to the adjacent stronger material. This affects the distribution of stress along and within the test specimen. The level of constraint developed is dependent on the strength mismatch between adjacent materials which may be graded across the heat affected zone and the specimen cross-sectional geometry. With development of digital image correlation (DIC) and its application to measuring deformation of solids, it has become feasible to measure local stress-strain properties of cross-weld samples and also specimens of variable geometry at both room and high temperature. Local deformation measured by DIC is correlated with nominal stress based on an iso-stress assumption, where it is assumed that the various regions of the weld are arranged in series and that the stress is uniform over the cross-section area at any given location in the specimen. Thus, local mechanical properties are inferred by mapping the measured local strain against the global stress. In this context the effect of constraint is important because it introduces a local stress distribution that will differ from the global net section stress and can therefore introduce errors in determination of local stress-strain properties. The influence of these factors raises questions concerning the interpretation of apparent local mechanical properties of cross-welds measured using DIC. Specifically, there is uncertainty as to whether the measured variation in local strain versus average cross-sectional stress (average global stress) represents the local mechanical properties of the material with negligible error. The severity of potential constraint errors will be influenced by the specimen geometry (particularly the tensile specimen thickness to width ratio for 2D DIC strain measurements) and the gradient of material property inhomogeneity along the specimen gauge length. In the current study, measurement of deformation in varying stress fields using DIC was carried out. Two distinct methods of introducing varying stress fields were employed: in the first case, the stress field was varied by designing an hour-glass shaped specimen with the specimen width varying as a function of the gauge length. In the second case, stress fields were varied due to a change in the specimen thickness in mismatched dissimilar metal joints. To reduce the complexity in the developed stress fields in the region near the interface, two materials were joined using solid state diffusion bonding giving a step change in the material properties across a planar interface. Test specimens spanning the interface were studied by finite element modelling and experimental tests with DIC deformation monitoring. Using DIC and a single hour-glass shaped specimen, multiple creep curves at different stress levels along the specimen gauge length have been successfully extracted for a test temperature of 525oC. The creep data obtained at 525oC from the hour-glass specimen test have been fitted to RCC-MR, Garofalo and Graham-Walles creep deformation models. A better correlation is achieved between the experimental data and the RCC-MR model predicted values when the local deformations from the single specimen are plotted as a function of time and true stress at the end of the specimen’s initial loading. From both elastic-plastic FE models and experimental local stress-strain curves extracted from DIC measurements, it can be concluded that a thin (1mm thick or less) 6mm wide test specimen should be used to extract the local mechanical properties from tensile specimens that have varying material properties along the gauge length, such as found in cross-weld test specimens. This is because the thinner specimen reduces the size of the strain concentration region adjacent to the interface where errors in measured local mechanical properties can occur. As a ‘rule of thumb’, the intrinsic stress-strain properties of the material should not be inferred from regions within 1mm of the interface for these 1mm thick samples because of concentration errors. The results obtained demonstrate the capability of DIC as a measurement technique for characterising the local deformation of specimens with varying stress fields. The findings also highlight some aspects of the technique which need to be considered when interpreting data from DIC measurements of local deformation from specimens with varying stress fields.

We present a new systematic method to construct Abelian functions on Jacobian varieties of plane, algebraic curves. The main tool used is a symmetric generalisation of the bilinear operator defined in the work of Baker and Hirota. We give explicit formulae for the multiple applications of the operators, use them to define infinite sequences of Abelian functions of a prescribed pole structure and deduce the key properties of these functions. We apply the theory on the two canonical curves of genus three, presenting new explicit examples of vector space bases of Abelian functions. These reveal previously unseen similarities between the theories of functions associated to curves of the same genus.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011.