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# Jacobian of elliptic curve

### Jacobian variety - Wikipedi

1. The Jacobian of a curve over an arbitrary field was constructed by Weil (1948) as part of his proof of the Riemann hypothesis for curves over a finite field. The Abel-Jacobi theorem states that the torus thus built is a variety, the classical Jacobian of a curve, that indeed parametrizes the degree 0 line bundles, that is, it can be identified with its Picard variety of degree 0 divisors modulo linear equivalence
2. An equation for the elliptic curve in an aﬃne neighborhood of ∞ is Z= X 3 +aXZ 2 +bZ 3 (where ∞ = (0,0) with respect to these coordinates) and x/y= X in these new coordinates
3. Jacobian using a basis for the holomorphic differentials on the curve. The construction generalizes that of the formal group of an elliptic curve. In §1, we review some of the basic facts about higher dimensional formal groups. Hazewinkel  is a good reference for the theory of formal groups in general. Section II details the construction of the formal group of the Jacobian. Since this construction depends onl
4. Now, the theorem more precisely states that: Let L be the invertible sheaf O ( Δ) ⊗ pr 1 ∗ O ( − P 0) on X × X. Then, for any k -scheme T and an invertible sheaf M ∈ Pic 0 ( X × T), there is a unique map f: T → X such that M = f ∗ L modulo p ∗ Pic ( T). The proof is as follows. Write M ′ = M ⊗ pr 1 ∗ O ( P 0)
5. to have a split Jacobian for some curves Cin positive characteristic (see e.g. discussion in ). In particular, one can have a Jacobian of the form J(C) = E 1 E 2;the sum of two elliptic curves. One has an isogeny: E 1 E 2!E 1 E 2 de ned as an identity on E 1;and as a power of Frobenius on E 2: We must note that in this example we do not know whether the curve
6. 1 Answer1. The division can't be right. You need to compute the multiplicative inverse modulo FIELD. This operation is quite expensive, and should only be performed once at the end of a scalar multiplication, not after each doubling/addition. Use z^ {-1} = ModPow (z, FIELD-2, FIELD)

Let ζ n be a primitive n th root of unity and use ζ to denote both (i) the automorphism of C defined by ( x, y) ↦ ( x, ζ n y) and (ii) the corresponding automorphism this induces on J. Note that each of the aforementioned divisor classes [ ( x i, 0) − ∞] are fixed by ζ and hence. [ ( x i, 0) − ∞] ∈ J [ 1 − ζ] Elliptic curves are often defined as the points of the affine curve plus an additional symbol , which is then called the point at infinity. The mapping of that point to (1:1:0) is then by definition only. As you noticed, the point at infinity has the property that it results in adding P and -P Conversion to Jacobian coordinates is easy by setting X = x, Y = y and Z = 1: (x,y,1) is a perfectly valid Jacobian representation of the (x,y) point. Conversion from Jacobian coordinates is computationally harder: you have to do a modular inversion, and a few multiplications (you compute U = 1/Z, then x = X·U 2 and y = Y·U 3) The genus 9 curve actually has Jacobian isogenous to the four copies of one elliptic curve plus five copies of another! The 1993 paper of Ekedahl and Serre remains the most thorough general investigation of curves (not just hyperelliptic curves) whose Jacobians are isogenous to products of elliptic curves

For elliptic curves, each element of the jacobian is equivalent to $$P - O$$ for some point $$P$$, and addition on points induces addition in the jacobian. Contents Elliptic Curves Computing in the Jacobian of a hyper-elliptic curve, (1987) Guide to Elliptic Curve Cryptography. Elliptic curves have been intensively studied in number theory and algebraic... Efficient Pairing Computation on Supersingular Abelian Varieties. We present a general technique for the efficient....

Point addition in projective and Jacobian coordinates, however, doesn't require to compute any modular inverse. Hence, the cost of adding two elliptic curve points in projective or Jacobian coordinates is much smaller than in affine representation Curves with Jacobian varieties that have many elliptic curve factors in their de-compositions up to isogeny have been studied in many different contexts. Ekedahl and Serre found examples of curves whose Jacobians split completely into elliptic curves (not necessarily isogenous to one another)(see also,[14, §5])

elliptic curve cryptosystems, a na¨ıve implementation of the point multiplication is particularly susceptible to such attacks as the classical formulæ for doubling a point and for adding two (distinct) points are diﬀerent. Hence, according to the implemented crypto-algorithm, a simple power analysis (SPA)y can yield the value of multiplier d used in the computation of Q = dP on an. Return the dimension of this Jacobian. OUTPUT: Integer. EXAMPLES: sage: k.<a> = GF(9); R.<x> = k[] sage: HyperellipticCurve(x^3 + x - 1, x+a).jacobian().dimension() 1 sage: g = HyperellipticCurve(x^6 + x - 1, x+a).jacobian().dimension(); g 2 sage: type(g) <... 'sage.rings.integer.Integer'>. point(mumford, check=True) ¶ freedom in trying to deform the elliptic curve into the Jacobian of a higher genus curve without destroying the underlying discrete logarithm problem. In particular, if an isogeny f : A!Bhas a kernel in some part of A that is small, say a few elements of order 2, then one could hope that by studying fone could transfer discrete logarithms with more freedom than if we restricted to isomorphisms. Elliptic curve cryptosystem is proposed by Koblitz  and Miller  which is public key cryptosystem that it can be constructed on the group of points of an elliptic curve over a finite field instead of finite field. Elliptic curves based on the difficulty of the Elliptic Curve Discrete Logarithm Problem (ECDLP)

1. Particularly, if an elliptic curve $$E/\mathbb {F}_{q^n}$$ is given, one might try to use the idea of cover attacks to reduce the problem to the corresponding problem in the Jacobian of a curve of.
2. Particularly, if an elliptic curve E/\mathbb {F}_ {q^n} is given, one might try to use the idea of cover attacks to reduce the problem to the corresponding problem in the Jacobian of a curve of genus g \ge n over \mathbb {F}_q
3. For Jacobians of (hyper)elliptic curves there exists an index calculus attack on DLP. If the genus of the curve becomes too high, the attack will be more efficient than Pollard's rho. Today it is known that even a genus of cannot assure security. Hence we are left with elliptic curves and hyperelliptic curves of genus 2
4. In a recent IACR ePrint note 1, Samuel Dobson and Steven Galbraith proposed the use of the Jacobian group of a hyperelliptic curve of genus 3 as a candidate group of unknown order. The authors conjecture that at the same level of security, Jacobian groups of genus 3 hyperelliptic curves outperform the ideal or form class group in terms of both operational speed and representation size. Given.

quite complicated, in general, but for elliptic curves, the curve and its Jacobian both have dimension one, and in fact the Jacobian is isomorphic to the curve itself. 2.1 The group law for Weierstrass curves Recall from Lecture 1 that the group law for an elliptic curve de ned by a Weierstrass equation y2 = x3 + Ax+ B is determined by the following rule: Three points on a line sum to zero. Abstract. In recent years there has been an interest in constructing examples of closed Riemann surfaces whose jacobian varieties are isogenous to a product of many elliptic factors and some other jacobian varieties

At this time, Jacobians of hyperelliptic curves are handled differently than elliptic curves: sage: J = H . jacobian (); J Jacobian of Hyperelliptic Curve over Finite Field of size 37 defined by y^2 = x^5 + 12*x^4 + 13*x^3 + 15*x^2 + 33*x sage: J = J ( J . base_ring ()); J Set of rational points of Jacobian of Hyperelliptic Curve over Finite Field of size 37 defined by y^2 = x^5 + 12*x^4 + 13*x^3 + 15*x^2 + 33* An elliptic curve is given by a Weierstrass model y^2 + a 1 xy + a 3 y = x^3 + a 2 x^2 + a 4 x + a 6, whose discriminant is non-zero. Affine points on E are represented as two-component vectors [x,y]; the point at infinity, i.e. the identity element of the group law, is represented by the one-component vector Cover attacks for elliptic curves with prime order. We give a new approach to the elliptic curve discrete logarithm problem over cubic extension fields $\mathbb {F}_ {q^3}$. It is based on a transfer: First an $\mathbb {F}_q$-rational $(\ell,\ell,\ell)$-isogeny from the Weil restriction of the elliptic curve under consideration with respect to.

Its Jacobian is an elliptic curve Edeﬁned over the same ﬁeld K. However it is only if Chas a K-rational point that Cand Eare isomorphic over K. Starting with equations for Cwe would like to compute a Weierstrass equation for E. Let Dbe a K-rational divisor on Cof degree n 1. It is natural to split into cases according to the value of n. If n = 1 then Chas a K-rational point, and our task. dimension of the Jacobian is equal to the genus g of the curve, which means that in gen-eral the Jacobian is a much more complicated object than the curve itself (which always has dimension one). Writing explicit equations for the Jacobian as a projective variety is quite complicated, in general, but for elliptic curves, the curve and its Jacobian both have dimension one, and in fact the Jacobian is isomorphic to the curve itself This project is a study in the latest trend in cryptography - HECC(Hyper-Elliptic Curve Cryptography). An investigation in the existing methods of group operation in jacobian of a curve. A few attempts made to improve the present status of the eld also are included. Chapter 1 Introduction Basic research is what I'm doing when I don't know what I'm doing. { Wernher Von Braun 1.1 The. The periods of the generalized Jacobian of a complex elliptic curve Di Bartolo, Alfonso; Falcone, Giovanni 2015-01-01 00:00:00 We show that the toroidal Lie group G = 2 / , where is the lattice generated by (1, 0), (0, and ( , ), with , is isomorphic to the generalized Jacobian of the complex elliptic curve C with modulus , de ned by any divisor class ( ) + ( ) of C ful lling.

### Video: algebraic geometry - Jacobian Variety of an Elliptic Curve

ABSTRACT. We say that an algebraic curve has split jacobian if its jacobian is isogenous to a product of elliptic curves. If X is a curve of genus 2, and /: X — E a map from X to an elliptic curve, then X has split jacobian. It is not true that a complement to E in the jacobian of X is uniquely determined Provides Racket implementations of elliptic curve arithmetic over prime fields in Jacobian coordinates, efficient integer multiplication in the elliptic curve group, affine/Jacobian coordinate conversion, and point serialization. Provides parameters for several popular cryptographic elliptic curves. This library should not be used to process information that must be kept secret. No effort has.

Jacobian of hyper-elliptic curve. Ask Question Asked 3 years, 10 months ago. Active 3 years, 10 months ago. Viewed 171 times 2. 2 $\begingroup$ I'm writing a survey paper about Weierstrass subgroup of plane quartic curves. I found that. The goal of this paper is to describe a practical and efficient algorithm for computing in the Jacobian of a large class of algebraic curves over a finite field. For elliptic and hyperelliptic.. • the generic ﬁber of f is a smooth curve of genus 1 (elliptic ﬁbration), • the ﬁbers of E do not contain smooth rational curves of self-intersection −1 (relative minimality), • we have a global zero section e : B → E (Jacobian elliptic ﬁbration), • the j-function which to each smooth ﬁber E b ⊂ E assigns its j-invarian ### encryption - Elliptic curve addition in Jacobian

† Elliptic curves can have points with coordinates in any ﬂeld, such as Fp, Q, R, or C. † Elliptic curves with points in Fp are ﬂnite groups. † Elliptic Curve Discrete Logarithm Prob-lem (ECDLP) is the discrete logarithm problem for the group of points on an elliptic curve over a ﬂnite ﬂeld. † The best known algorithm to solve the ECDLP is exponential, which is why elliptic. RSA . Elliptic curve cryptography is lower power consumption, faster computation and small bandwidth . Elliptic curve cryptography is based on the coordinate system such as affine, projective, Jacobean, Chudnovsky-Jacobian and modified Jacobean coordinate systems . EC Elliptic curve point addition in mixed Jacobian-affine coordinates So once the disclosure was out and after taking few week of rest I was ready to dig deeper this issue. First thing I did was to download the OpenJDK source code and started inspecting. After quite a bit of investigation I ended up here: At the same time I found out that for some reason NSS (hence Firefox) shares the same code.

### Torsion in the jacobian of a super elliptic curv

Jacobian curve. From formulasearchengine. Jump to navigation Jump to search. Template:Multiple issues In mathematics, the Jacobi curve is a representation of an elliptic curve different from the usual one (Weierstrass equation). Sometimes it is used in cryptography instead of the Weierstrass form because it can provide a defence against. 18.783 Elliptic Curves Lecture #2 Spring 2017 02/15/2017 2 Elliptic curves as abelian groups In Lecture 1 we deﬁned an elliptic curve as a smooth projective curve of genus 1 with THE JACOBIAN AND FORMAL GROUP OF A CURVE OF GENUS 2 OVER AN ARBITRARY GROUND FIELD E. V. Flynn, Mathematical Institute, University of Oxford §0. Introduction The ability to perform practical computations on particular cases has greatly inﬂu- enced the theory of elliptic curves. First, it has allowed a rich sub-branch of the Math-ematics of Computation to develop, devoted to elliptic curves. ELLIPTIC-CURVE SINGLE-SCALAR MULTIPLICATION 5 To emphasize the importance of speedups in elliptic-curve addition, we an-alyzed not only Jacobian and Jacobian-3 using the best speeds known, but also Std-Jacobian using the speeds most commonly quoted in the literature. Th Jacobian of a high genus curve than on a comparably sized group of points of an elliptic curve. In this paper we have explored in details the main operations like scalar multiplication, group operations on Jacobian, finite field operations which are prime steps for efficient implementation of ECC and HECC. II. Elliptic Curve

elliptic curves. Given two elliptic curves E1 and E2 with rational 4-torsion points, we construct a hyperelliptic curve C of genus 3 with two maps ϕi: C ! Ei (i = 1,2) of degree 2. Then the Jacobian J(C) of C is isogenous to E1 £ E2 £ E3 with a third elliptic curve E3. We vary E1 and E2 in such a way that they are isomorphic, and ﬁnd the case where E3 is also isomorphic. Note that there are many examples of nonhyperelliptic curves Introduces S-M tradeoffs. In particular: for Jacobian doubling with a4=-3, reports 3M+5S (improving previous 4 squares, 4 mults, 8 reduces to 5 squares, 3 mults, 7 reduces); for Jacobian addition, reports 11M+5S (could again trade mult for square). Nigel P. Smart. The Hessian form of an elliptic curve. Pages 118-125 in: Cetin Kaya Koc.

### elliptic curves - Point at infinity for Jacobian

• 1.1.3 Discrete logarithm problem in the Jacobian of curves Jacobian have a certain dimension gand their number of points over F qis approximately qg. In order to have fast operations, it is good to have qas small as possible and so g has to be big. However, attacks become more powerful for large gas the following table giving the complexity of the best index calculus attack shows g= 1 g= 2 g.
• ed by Mazur in . Theorem 2.1. Let E be an elliptic curve deﬁned over Q. Then.
• Comparatively Study of ECC and Jacobian Elliptic Curve Cryptography. Anagha P. Zele, Avinash P. Wadhe - Elliptic curves were first proposed as a basis for public key cryptography in the mid-1980s independently by Koblitz and Miller. Elliptic curve cryptography (ECC) algorithm is practical than existing security algorithms. Because of this fact, it showed real attraction to portable devices.
• The original GHS attack against elliptic curves has been extended to various classes of curves and generalized to the cover attack. These attacks can be described as to compute the DLP in the Jacobian of a curve C0 deﬁned over the d degree extension ﬁeld kd:= Fqd of k:= Fq by mapping it to the DLP in the Jacobian of a covering curve.
• In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O.An elliptic curve is defined over a field K and describes points in K 2, the Cartesian product of K with itself. If the field has characteristic different from 2 and 3 then the curve can be described as a plane algebraic curve which, after a linear change of variables.

### How to calculate point addition using Jacobian coordinate

THE TROPICAL JACOBIAN OF AN ELLIPTIC CURVE IS THE GROUP S(Q) Darryl G. Wade Department of Mathematics Master of Science We establish consistent definitions for divisors, principal divisors, and Jacobians of a tropical elliptic curve and show that for a tropical elliptic cubic C, the associated Jacobian (or zero divisor class group) is the group S(Q) One can use Jacobian to build elliptic curves: sage: x, y = polygen(QQ, 'x, y') sage: Jacobian(y*y-x**4-3*x+5) Elliptic Curve defined by y^2 = x^3 + 20*x + 9 over Rational Field FrédéricC ( 2014-11-27 11:21:03 -0600 ) edi Points on elliptic curves have a natural addition operation on them which allows us to study their arithmetic as well as their geometry. Higher genus curves unfortunately do not have such a natural structure. We can, however, define a variety associated to each curve which has a natural group structure, called the Jacobian variety of the curve, and we can use it to study properties of the. THE TROPICAL JACOBIAN OF AN ELLIPTIC CURVE IS THE GROUP S. 1 (Q) Darryl G. Wade Department of Mathematics Master of Science We establish consistent de nitions for divisors, principal divisors, and Jacobians of a tropical elliptic curve and show that for a tropical elliptic cubic C, the associated Jacobian (or zero divisor class group) is the group S . 1 (Q). ACKNOWLEDGMENTS All of the. Lawrence C. Washington, Elliptic Curves: Number Theory and Cryptography. Series Editor KENNETH H. ROSEN DISCRETE MATHEMATICS AND ITS APPLICATIONS Boca Raton London New York Singapore Henri Cohen Gerhard Frey Roberto Avanzi, Christophe Doche, Tanja Lange, Kim Nguyen, and Frederik Vercauteren Handbook of Elliptic and Hyperelliptic Curve Cryptography. Published in 2006 by Chapman & Hall/CRC.

### When is a product of elliptic curves isogenous to the

Elliptic curves are the genus 1 case of the Jacobian groups of hyperelliptic curves. The latter are the analogues of the class groups of quadratic number fields (henceforth we shall say simply class group). While many explicit formulas for addition on an elliptic curve have appeared (for practical examples, see Chudnovsky and Chudnovsky , and Montgomery ), and numerous algorithms for. Andrew V. Sutherland We consider the problem of efficient computation in the Jacobian of a hyperelliptic curve of genus 3 defined over a field whose characteristic is not 2. For curves with a rational Weierstrass point, fast explicit formulas are well known and widely available 2.7 Elliptic Functions in General 81 2.8 The p-Function 84 2.9 Elliptic Integrals, Complete and Incomplete 87 2.10 Two Mechanical Applications 89 2.11 The Projective Cubic 92 2.12 The Problem of Inversion 93 2.13 The Function Field 95 2.14 Addition on the Cubic 98 2.15 Abel's Theorem 104 2.16 Jacobian Functions: Reprise 109 2.17 Covering Tori 11 theory of generalized jacobian of the elliptic curve. E. Finally an example, which shows how to apply our method is given in the last section. 2. Usual and Generalized Jacobians D´ech`ene [1, 2, 3] has proposed generalized jacobians as a source of groups for public key cryptosystems based on the hardness of the Discrete Logarithm Problem. We will obtain a relation that presents a way to. hyperelliptic curve over Fp whose Jacobian is isogenous to a power of one ordinary ellip-tic curve. 1. Introduction Let E be an elliptic curve over a ﬁeld L. For various choices of L, it is known that E(L) is a ﬁnitely generated group. This is the case if, for example, - L is a number ﬁeld (by the Mordell-Weil Theorem, see , ), or, more generally, - L is ﬁnitely generated. Here we construct families of Jacobian elliptic surfaces that degenerate in a homologically trivial fashion, so that the period matrix specializes to a matrix lying in the interior of the period domain and there is an explicit formula for the derivative of the period matrix that involves no derivatives but is instead a matrix of rank one, just as for curves TECHNISCHE UNIVERSITÄT MÜNCHEN FAKULTÄT FÜR INFORMATIK Lehrstuhl für Efﬁziente Algorithmen The core operations in elliptic-curve cryptography are single-scalar multiplica-tion (m,P 7→mP), double-scalar multiplication (m,n,P,Q 7→mP + nQ), etc. In the Crypto '85 paper that introduced elliptic-curve cryptography, Miller proposed carrying out these operations in Jacobian coordinates: Each point i

0(N) is replaced by the Jacobian of a Shimura curve. To define these analogues, it is helpful to give a character-ization of din which pdoes not appear explicitly. For this, note that the map j~: A~ (3J 0(N)~ which is dual tojmay be viewed as a homomorphism A 3J 0(N), since Jacobians of curves (and elliptic curves in particular) are canonically. elliptic-curve-solidity . elliptic-curve-solidity is an open source implementation of Elliptic Curve arithmetic operations written in Solidity.. DISCLAIMER: This is experimental software. Use it at your own risk!. The solidity contracts have been generalized in order to support any elliptic curve based on prime numbers up to 256 bits We prove that if X and S are smooth varieties and f: X → S is an elliptic fibration with singular fibers curves of types I N with N ≥ 1, II, III and IV, then the relative Jacobian f ˆ: M ¯ X / S → S of f, defined as the relative moduli space of semistable pure dimension one sheaves of rank 1 and degree 0 on the fibers of f, is an elliptic fibration such that all its fibers are irreducible Title: Jacobian varieties with many elliptic curves Authors: Ruben A. Hidalgo (Submitted on 28 Jul 2015 ( v1 ), revised 6 Sep 2018 (this version, v3), latest version 16 Oct 2019 ( v5 )  ### Elliptic Curves - Hyperelliptic Curve

p of genus (ℓ−1)/2 whose Jacobian is isogenous to the power of one ordinary elliptic curve. Key words: Elliptic curves of high rank, Jacobians MSC2000: Primary: 11G05; Secondary: 11G20, 14H40, 14H52. 1 Introduction The purpose of this work is to prove the following theorem and its corollary. Theorem Let pand ℓbe odd prime numbers. Then there exists an ordi-nary hyperelliptic curve Hover. Let p be an odd prime number. Let K be the p-th cyclotomic field. We give general formulae for the root numbers of the Jacobian varieties of the Ferm Miller ), the jacobian of a hyperelliptic curve deﬁned over a ﬁnite ﬁeld (Koblitz ), and the class group of an imaginary quadratic number ﬁeld (Buchmann and Williams ). 103. 174 KOBLITZ ET AL. Elliptic curves have been extensively studied for over a hundred years, and there is a vast literature on the topic. Originally pursued mainly for aesthetic reasons, elliptic curves. ### Computing in the Jacobian of a hyper-elliptic curve, (1987

Elliptic curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. The use of elliptic curves in cryptography was suggested independently by Neal Koblitz1 and Victor S. Miller2 in 1985. Elliptic curves are also used in several integer factorization algorithms that have applications in cryptography, such as Lenstra. ii J.S. MILNE The canonical form of the equation The group law for the canonical form 6. Reduction of an Elliptic Curve Modulo p 23 Algebraic groups of dimension 1 Singular cubi EC operations in both affine and Jacobian coordinates. An elliptic curve E over GF(p) in affine coordinates is the set of solutions for an equation such as 2 = 3 + + (1) where x, y, a, b ∈ GF(p) with 4 3 + 27 2 ≠ 0. The coefficients a, ∈ specifying an elliptic curve ( ) are defined by (1). The number of points on elliptic curve E is represented by # ( ). It is defined over as nh, where n. A discussion of alternative coordinate systems, such as projective, Jacobian, and Edwards coordinates, along with related computational issues; A more complete treatment of the Weil and Tate Lichtenbaum pairings ; Doud s analytic method for computing torsion on elliptic curves over Q; An explanation of how to perform calculations with elliptic curves in several popular computer algebra systems. Elliptic Curve. An extensible library of elliptic curves used in cryptography research. Curve representations. An elliptic curve E(K) over a field K is a smooth projective plane algebraic cubic curve with a specified base point O, and the points on E(K) form an algebraic group with identity point O.By the Riemann-Roch theorem, any elliptic curve is isomorphic to a cubic curve of the for

### elliptic curves - Why are Jacobian Coordinates used

Elliptic Curves: Basics. Given an elliptic curve, it is possible to do some operations between its points: for example one can add two points P and Q obtaining the point P + Q that belongs to the curve ; given a point P on the elliptic curve, it is possible to double P, that means find P = P + P (the square brackets are used to indicate [n]P, the point P added n times), and also find. - The generalisation of Jacobian coordinates from the elliptic curve setting to the hyperelliptic curve setting: these coordinates essentially cast aﬃne points into projective space according to the weights of xand yin the deﬁning curve equation. While applying Jacobian coordinates t Jacobian of of a modular curve of level N. Another formulation of modularity property is by using L functions which generalizes the famous Riemann zeta function ζ(s) := X∞ n=1 1 ns Riemann hypothesis claims that all the non-trivial zeros of ζ lies on <(s) = 1 2 and it has strong consequences on the growth of prime number. For the L functions associated to elliptic curves one has the Birch.

### Jacobian of a general hyperelliptic curve — Sage 9

P → P1 is a Jacobian elliptic ﬁbration. We want to show the existence of a saliently ramiﬁed elliptic or rational multisection in order to use the Proposition 2.2. We have to consider several possibilities, depending on the curve R. Deﬁnition 3.2 (Tangent correspondence) Let R be a reduced curve in P2 and denote by R0 the set of smooth. Roughly speaking this means that every elliptic curve over Q appears in the Jacobian of of a modular curve of level N. Another formulation of modularity property is by using L functions which generalizes the famous Riemann zeta function z(s):= ¥ å n=1 1 ns Riemann hypothesis claims that all the non-trivial zeros of z lies on ´(s)= 1 2 and it has strong consequences on the growth of prime. Shimura curves arising from inde nite quaternion algebras over Q. We study these curves and their Jacobians mostly using rigid-analytic techniques. Our main results are: (1) A description of the action of the Hecke operators in terms of the rigid-analytic uniformization. (2) An analytic construction of elliptic curves associated to harmonic Heck

### Cover attacks for elliptic curves with prime order

intermediate Jacobian: formal Brauer group: K3 cohomology: n = 3 n = 3: Calabi-Yau 3-fold: line 3-bundle: intermediate Jacobian: CY3 cohomology: 7d Chern-Simons theory/M5-brane: n n: intermediate Jacobian : References. Elliptic cohomology General. The concept of elliptic cohomology originates around: Peter Landweber, Elliptic Cohomology and Modular Forms, in Elliptic Curves and Modular Forms. Remark 2. The curve C is not always absolutely irreducible. Here is an example q= n= 3; E: y2 = x3 x; 3 = 1: Then h 1 = h 2 and C (IF 3) = fOg, so some condition on is needed. Using an elliptic curve of the same equation but now q= n= p>3, pprime, p = c, where cis a non-square in IF p, we get an example where C is absolutely irreducible and, yet, we still have In particular I work on elliptic curve descent calculations, and the construction of explicit elements in the Tate-Shafarevich group. I am compiling a list of genus one curves that are counter-examples to the Hasse principle and have Jacobian of small conductor. The list so far covers elements of Sha of order 3 and order 5

### Cover attacks for elliptic curves with cofactor two

They are called elliptic curves. If the equation is non-singular, one can use the following procedure: Suppose we know a rational solution (x,y). Compute the tangent line of the curve at this point. Compute the intersection with the curve. The point you obtain is also a rational solution involution and so we are able to get more information about the Jacobian of this curve. Theorem 1 produces: J C ×J2 C/ha,ci ∼ J C/ha i×J c ×J ac. (4.3) Considering ﬁxed points and using Theorem 3, we may conclude that each quotient on the right has genus one and so J C ∼ E 1 ×E 2 ×E 3 for three elliptic curves. 4.1.3 C 4 ×C 2 Both U 2 and H 2 are isomorphic to C 4×C elliptic curve arithmetic is cheaper in Jacobian coordinate system and it is harmonized with Figure 2.2 as expected. 4. Conclusions & Contributions The speed problem of ECC implementation comes from the highly computational cost of elliptic curve arithmetic. The inversion has the highest cost and i 5.2 Hyper Elliptic Curve of genus 3 and Jacobian variety curve . . . . . . . . . 61 5.3 Hyper Elliptic Curve of genus 4 and Jacobian variety curve . . . . . . . . . 71 5.4 Hyper Elliptic Curve of genus 2 as it touch the Jacobian variety curve. . . . 83 5.5 Hyper Elliptic Curve of genus 3 as it touch the Jacobian variety curve. . . . 9     Jacobian curve: lt;table class=metadata plainlinks ambox ambox-content ambox-multiple_issues compact-ambox role=p... World Heritage Encyclopedia, the aggregation. Example 1.2 (Elliptic curves) . An elliptic curve is a genus 1 curve with a speci ed k-point O. If we assume char (k) 6= 2 ;3, then any elliptic curve Ehas a Weierstrass quatione of the shape E: y2 = x3 + ax+ b; a;b2k such that the discriminant E = 16(4a3 + 27b2) is 6= 0 (this is equivalent to the equation being smooth). Strictly speaking this equation de nes an a ne curve, and we should. I'm supposed to give a talk on this subject for one of my courses, so I consider this post as a pre-exposition. I learned from and heavily used the great exposition Vector bundles on curves by Montserrat Teixidor I Bigas in this post.I wrote up the pre-requisites here. In 1957, Atiyah in this famous paper Vector bundles over an elliptic curve classified indecomposable. Abstract: A genus 2 curve $C$ has an elliptic subcover if there exists a degree $n$ maximal covering $\psi: C \to E$ to an elliptic curve $E$. Degree $n$ elliptic subcovers occur in pairs $(E, E')$. The Jacobian $J_C$ of $C$ is isogenous of degree $n^2$ to the product $E \times E'$. We say that $J_C$ is $(n, n)$-split. The locus of $C$, denoted by $Ł_n$, is an algebraic subvariety of the moduli space $\M_2$. The space $Ł_2$ was studied in Shaska/Völklein and Gaudry/Schost. The. Jacobian have cost of 16M+3S and 5M+5S respectively. Mixed addition using affine and Jacobian coordinates has cost of 11M+3S. Dimitrov et al  proposed efficient tripling formula in Jacobian with cost of 15M+7S. He also proposed tripling formula in affine for elliptic curve over binary field . The tripling operation is usin For a general genus two curve , the Jacobian will be a simple abelian surface: it will not admit any nontrivial abelian subvarieties. However, for some (precisely, for a union of countably many divisors in ), the Jacobian will be non-simple, or equivalently there will exist an isogeny. for two elliptic curves . The curves are determined uniquely up to isogeny by the corollary of the Poincaré.

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