Schur function expansions of KP τ-functions associated to algebraic curves

Одной из актуальных современных проблем алгебры и теории чисел является проблема существования и поиска фундаментальных S-единиц в гиперэллиптических полях. Проблема существования и поиска S-единиц в гиперэллиптических полях эквивалентна разрешимости норменного уравнения - функционального уравнения Пелля - с некоторыми дополнительными условиями на вид этого уравнения и его решения. Существуетглубокая связь между точками конечного порядка в якобиевом многообразии (якобиане) гиперэллиптической кривой и нетривиальными S-единицами соответствующего гиперэллиптического поля. Эта связь легла в основу предложенного В. П. Платоновым алгебраического подхода к известной фундаментальной проблеме об ограниченности кручения в якобиевых многообразиях гиперэллиптических кривых. Для эллиптических кривых над полем рациональных чисел проблема кручения была решена Мазуром в 1970-ых годах. Для кривых рода 2 и выше над полем рациональных чисел проблема кручения оказалась значительно сложнее, и пока далека от своего полного решения. Основные результаты, полученные к настоящему времени в этом направлении, относятся к описанию подгрупп кручения якобиевых многообразий конкретных гиперэллиптических кривых, а также к описанию некоторых семейств гиперэллиптических кривых рода g >= 2.В данной статье нами найден новый метод исследования разрешимости функциональных норменных уравнений, дающий полное описание гиперэллиптических кривых над полем рациональных чисел, якобиевы многообразия которых обладают точками кручения данных порядков. Наш метод основан на аналитическом изучении представителей дивизоров конечного порядка в группе классов дивизоров степени ноль и их представлений Мамфорда. В качестве иллюстрации работы нашего метода в данной статье непосредственно найдены все параметрические семейства гиперэллиптических кривых рода два над полем рациональных чисел, якобиевы многообразия которых обладают рациональными точками кручения порядков не превосходящих пяти. Более того, наш метод позволяет определить, какому найденному параметрическому семейству принадлежит данная кривая, якобиан которой обладает точкой кручения порядка, не превосходящего пяти.

Let E be a semistable, rank two vector bundle of degree d on a Riemann surface C of genus g > 1, i.e. such that the minimal degree s of a tensor product of E with a line bundle having a nonzero section is nonnegative.We give an analogue of Clifford's lemma by showing that E has at most (d-s)/2 + 6 independent sections, where 6 is 2 or 1 according to whether the Krawtchouk polynomial Kr(n, N) is zero or not at r = (d-s)/2 + 1, n = g, N = 2g-s (the analogous bound for nonsemistable rank two bundles being stronger but easier to prove).This gives an answer to the problem posed by Severi asking for the minimal degree of a directrix of a ruled surface.In some cases, namely if s has maximal value s = g, or if s > gonality(C) -2, or if E is general among those of the same Segre invariant s, or also if the genus is a power of two, we prove the bound holds with 3 = 1.The theory of Krawtchouk polynomials investigates which triples (g, s, d) provide zeros of Kr(n,N).Then, they generate invariants which one may expect to be associated to a Severi bundle, i.e., to a rank two semistable bundle reaching the bound with 3 = 2.According to this theory, there are only a finite number of such triples (g, s, d) for each value of ds, with the exception that there are infinitely many triples with d-s = 2 or 4. We then find all the Severi bundles corresponding to those two exceptional values of ds.We thank David Butler for a personal communication which we have used in proving Proposition 3.1.We also thank Enrique Arrondo, Rafael Hernandez, Jorge Jimenez and Bernardo L6pez for conversations during the preparation of this article.JAVIER CILLERUELO AND IGNACIO SOLS having a nonzero section or, equivalently, the integer s such that the maximal degree of a line subbundle L c-E is d2S.In [16] Nagata proved that such an integer s is at most g and that we can assume s > 2-2g, because otherwise H1(L* ® (E/L)) = O, which implies that E is decomposable and the rank one Clifford bound is enough to bound its sections.Let us recall, furthermore, that the bundle E is stable if s > 0, and semistable if s > 0. If E is special, i.e., if h°(E) is neither zero nor given by the Riemann-Roch formula h°(E) = d + 2 -2g, then we have s < d < 4g -4s.Therefore we center our study in this "Severi range" of values of d.This study is trivial in case E is not semistable, providing then a sharp bound (stronger than the "Severi bound" for semistable bundles in Theorem 0.2 below): PROPOSITION 0.1.Let E be a rank two bundle of Segre invariant s, 2 -2g < s < 0, and degree d on a Riemann surface of genus g.If s < d < -s then h°(E) < [ds] + 1. d + 3 if d is odd If-s < d < 4g-4+s then h(E) < + 2 if d is even, and d s (4) + 1 if d is even, and d g s (4).If4g-4 + s < d < 4g s then hO(E) < [3d] g + 2. Furthermore, for each g, s, d in this range, there are a Riemann surface of genus g, and a rank two bundle of invariant s and degree d reaching the bound.Indeed, by Serre duality it is enough to show the bound in the case d < 2g -2.Let L be a subbundle of maximal degree 0 - L - E - L' -0, i.e. degL = d-2 < 2g-2 and degL' = d-< g -2.Then the desired bound follows from h°(L) < [d] + 1 and h°(L') < max{0, [d+] + 1} (for the case -s< d < 2g-2, we note that d+s _ d-s or d+sd-s (mod 4) depending on whether d is even or odd).For the second assertion, since our assumption on the invariant s implies that g > 2, we can take C to be a hyperelliptic curve, and E = L L', where degL = d, degL' = d+s and h°(L) [ds] + 1, h°(L') = max{0, [d+s] + 1 In Section 1 we present the proof of our main theorem, in which the binary Krawtchouk polynomials Kr(n,N) appear.Let us recall that these polynomials are defined, for each r > 0, by their generating function Kr(n, N)zr = (1z)n(l + z)N-n r

It has been conjectured that for N sufficiently large, there are no quadratic polynomials in Q[z] with rational periodic points of period N. Morton proved there were none with N=4, by showing that the genus 2 algebraic curve that classifies periodic points of period 4 is birational to X$_1$(16), whose rational points had been previously computed. We prove there are none with N=5. Here the relevant curve has genus 14, but it has a genus 2 quotient, whose rational points we compute by performing a 2-descent on its Jacobian and applying a refinement of the method of Chabauty and Coleman. We hope that our computation will serve as a model for others who need to compute rational points on hyperelliptic curves. We also describe the three possible Galois-stable 5-cycles, and show that there exist Galois-stable N-cycles for infinitely many N. Furthermore, we answer a question of Morton by showing that the genus 14 curve and its quotient are not modular. Finally, we mention some partial results for N=6.

A subexponential algorithm for discrete logarithms over hyperelliptic curves of large genus over GF( q)

Arithmetic in the divisor class group of a hyperelliptic curve is a fundamental component of algebraic geometry packages implemented in computer algebra systems such as Magma and Sage. In this paper, we present an adaptation of Shanks’ NUCOMP algorithm for split model hyperelliptic curves of arbitrary genus that uses balanced divisors and includes a number of enhancements to optimize its efficiency in that setting. Our version of NUCOMP offers better performance than Cantor’s algorithm in the balanced divisor setting. Compared with Magma’s built-in arithmetic, our Magma implementation shows significant speed-ups for curves of all but the smallest genera, with the improvement increasing as the genus grows.

We compute rational points on real hyperelliptic curves of genus 3 defined on whose Jacobian have Mordell-Weil rank r=0. We present an implementation in sagemath of an algorithm which describes the birational transformation of real hyperelliptic curves into imaginary hyperelliptic curves and the Chabauty-Coleman method to find C (). We run the algorithms in Sage on 47 real hyperelliptic curves of genus 3.

Hyperelliptic curves (HEC) look promising for cryptographic applications, because of their short operand size compared to other public-key schemes. The operand sizes seem well suited for small processor architectures, where memory and speed are constrained. However, the group operation has been believed to be too complex and, thus, HEC have not been used in this context so far. In recent years, a lot of effort has been made to speed up group operation of genus-2 HEC. In this paper, we increase the efficiency of the genus-2 and genus-3 hyperelliptic curve cryptosystems (HECC). For certain genus-3 curves, we can gain almost 80 percent performance for a group doubling. This work not only improves Gaudry and Harley's algorithm, but also improves the original algorithm introduced by Cantor [1987]. Contrary to common belief, we show that it is also practical for certain curves to use Cantor's algorithm to obtain the highest efficiency for the group operation. In addition, we introduce a general reduction method for polynomials according to Karatsuba. We implemented our most efficient group operations on Pentium and ARM microprocessors.

Schoof's classic algorithm allows point-counting for elliptic curves over finite fields in polynomial time. This algorithm was subsequently improved by Atkin, using factorizations of modular polynomials, and by Elkies, using a theory of explicit isogenies. Moving to Jacobians of genus-2 curves, the current state of the art for point counting is a generalization of Schoof's algorithm. While we are currently missing the tools we need to generalize Elkies' methods to genus 2, recently Martindale and Milio have computed analogues of modular polynomials for genus-2 curves whose Jacobians have real multiplication by maximal orders of small discriminant. In this article, we prove Atkin-style results for genus-2 Jacobians with real multiplication by maximal orders, with a view to using these new modular polynomials to improve the practicality of point-counting algorithms for these curves.

With recent developments in hyperelliptic curve cryptographic system (HECC) processing and the proposed work, the myth that HECC is not suited for practical applications in constrained networks is dispelled. This article proposes dynamic authenticated contributory group key agreement (DACGKA) protocol using HEC‐DH. The DACGKA is less costly and ideally appropriate for resource‐restricted networks such as VANETs, MANETs, and WSNs because of its lesser key sizes and advancements in operational processing. The suggested protocol is implemented across a finite field using an HEC of genus 1 (ECC), genus 2, and genus 3, with promising results, including a significant reduction in key size and exchange overhead, an increase in key safety, and a broader range of applicability. According to a comparison of the proposed approach on different genus curves, HECC over genus 2 with a suitable key length can be applied for memory and power restricted devices to increase data secrecy and system efficiency. Further, HECC can outperform ECC for low‐cost, secure group communication applications. As a part of security analysis, first, we proved that each coordinate of HEC‐DH secret value is as hard as the entire DH value, and then concrete security proofs based on the HEC‐DLP were established.

Public key cryptography is the famous cryptography technique used in many corporate sectors for developing software to provide security services. Hyperelliptic Curve Cryptosystem (HECC) is one of the public key cryptographic technique, an expansion of Elliptic Curve Cryptography which offers the similar level of security compared with other cryptosystems such as RSA, ECC and DSA. HECC supervise the ECC due to shorter operand size. This paper will analysis the performance of HECC over genus curve 2, 3, 4, 5, 6 and compares the result with ECC. It also inspects the use of Hyperelliptic Curve Cryptography technique and affords complete survey on its performance for many applications.

In this paper we give a passage formula between different invariants of genus 3 hyperelliptic curves: in particular between Tsuyumine and Shioda invariants. This is needed to get modular expressions for Shioda invariants, that is, for example, useful for proving the correctness of numerically computed equations of CM genus 3 hyperelliptic curves. On the other hand, we also get Shioda invariants described in terms of differences of roots of the equation defining the hyperelliptic curve, that has applications for studying the reduction type of the curve. Under certain conditions on its Jacobian, we give a criterion for determining the type of bad reduction of a genus 3 hyperelliptic curve.

It has been proved (see [S. J. Kim, On the existence of Weierstrass gap sequences on trigonal curves. J. Pure Appl. Algebra 63 (1990), 171-180. MR1043748 (91b:14036) Zbl 0712.14019] and [M. Brundu, G. Sacchiero, On the varieties parametrizing trigonal curves with assigned Weierstrass points. Comm. Algebra 26 (1998), 3291-3312. MR1641619 (99g:14040) Zbl 0937.14016]. ) that a Weierstrass point on a trigonal curve, which is not a ramification point for the morphism induced by a trigonal series, has a gap sequence GP of the form GP = (1,2, …, r, r + 1 + α, …, g + α) for some integers r and α satisfying ≤ r ≤ g − 1 and 1 ≤ α ≤ 2r + 1 − g. Such points will be called one jump Weierstrass points of type (r,α). It was further proved [S. J. Kim, On the existence of Weierstrass gap sequences on trigonal curves. J. Pure Appl. Algebra 63 (1990), 171-180. MR1043748 (91b:14036) Zbl 0712.14019] that any one jump numerical sequence of type (r, α), with r and α in the range above, is the Weierstrass gap sequence of a trigonal curve. Here we prove that the property of having an extremal, in some sense, one jump Weierstrass point characterizes trigonal curves. More precisely, we show that if α belongs to the range α4 &lt; α ≤ 2r + 1 − g for a suitable α4, any curve with a Weierstrass point of type (r,α) is a triple covering of a smooth curve of genus p with , and that there exist examples of such coverings. Therefore when 2r + 1 − g − α &lt; 3, such a curve is indeed trigonal. As a consequence, any fourgonal curve of genus g ≥ 10 having a one jump Weierstrass point satisfies α ≤ α4 with few exceptions. Finally, we exhibit examples of fourgonal curves with a Weierstrass point of type (r, α) with ≤ r ≤ and 1 ≤ α ≤ α4.

Hyperelliptic curves of small genus have the advantage of providing a group of com- parable size as that of elliptic curves, while working over a field of smaller size. Pairing-friendly hyperelliptic curves are those whose order of the Jacobian is divisible by a large prime, whose em- bedding degree is small enough for computations to be feasible, and whose minimal embedding field is large enough for the discrete logarithm problem in it to be difficult. We give a sequence of Fq-isogeny classes for a family of Jacobians of genus two curves over Fq, for q = 2 m , and their corresponding small embedding degrees. We give examples of the parameters for such curves with embedding degree k < (logq) 2 , such as k = 8,13,16,23,26,37,46,52. For secure and efficient implementation of pairing-based cryptography on genus g curves over Fq, it is desirable that the ratio = g log2 q log2 N be approximately 1, where N is the order of the subgroup with embedding degree k. We show that for our family of curves, is often near 1 and never more than 2. We also give a sequence of Fq-isogeny classes for a family of Jacobians of genus 2 curves over Fq whose minimal embedding field is much smaller than the finite field indicated by the embedding degree k. That is, the extension degrees in this example differ by a factor of m, where q = 2 m , demonstrating that the embedding degree can be a far from accurate measure of security. As a result,

Higher syzygies of hyperelliptic curves

We propose the two heaviest Weierstrass n-semigroups for non- hyperelliptic curves (one on trigonal curves with maximal Maroni invariant and with totally ramification points and one on bielliptic curves and with as points some of the ramification points). We compute them when n = 2.