Proving that a genus 2 curve has complex multiplication

Recently examples of genus 2 curves defined over the rationals were found which, conjecturally, should have complex multiplication. We prove this conjecture. This involves computing an explicit representation of a rational map defining complex multiplication.

     

On a generalization of Deuringʼs results

Using the Dieudonné theory we will study a reduction of an abelian variety with complex multiplication at a prime. Our results may be regarded as generalization of the classical theorem due to Deuring for CM-elliptic curves. We will also discuss a sufficient condition for a prime at which the reduction of a CM-curve is maximal.     

Generalised Kummer constructions and Weil restrictions

We use a generalised Kummer construction to realise all but one known weight four newforms with complex multiplication and rational Fourier coefficients in Calabi-Yau threefolds defined over Q. The Calabi-Yau manifolds are smooth models of quotients of the Weil restrictions of elliptic curves with CM of class number three.     

The double-base number system and its application to elliptic curve cryptography

We describe an algorithm for point multiplication on generic elliptic curves, based on a representation of the scalar as a sum of mixed powers of and . The sparseness of this so-called double-base number system, combined with some efficient point tripling formulae, lead to efficient point multiplication algorithms for curves defined over both prime and binary fields. Side-channel resistance is provided thanks to side-channel atomicity.

     

Hasse invariants and anomalous primes for elliptic curves with complex multiplication

Let C be an elliptic curve defined over Q. Let p be a prime where C has good reduction. By definition, p is anomalous for C if the Hasse invariant at p is congruent to 1 modulo p. The phenomenon of anomalous primes has been shown by Mazur to be of great interest in the study of rational points in towers of number fields. This paper is devoted to discussing the Hasse invariants and the anomalous primes of elliptic curves admitting complex multiplication. The two special cases Y² = X³ + a₄X and Y² = X³ + a₆ are studied at considerable length. As corollaries, some results in elementary number theory concerning the residue classes of the binomial coefficients (_n²ⁿ) (Resp. (_n³ⁿ)) modulo a prime p = 4n + 1 (resp. p = 6n + 1) are obtained. It is shown that certain classes of elliptic curves admitting complex multiplication do not have any anomalous primes and that others admit only very few anomalous primes.     

ON ALMOST PRIME SUBMODULES

In this article,we define almost prime submodules as a new generalization of prime and weakly prime submodules of unitary modules over a commutative ring with identity.We study some basic properties of...     

On strongly prime rings and ideals

Strongly prime rings may be defined as prime rings with simple central closure. This paper is concerned with further investigation of such rings. Various characterizations, particularly in terms of symmetric zero divisors, are given. We prove that the central closure of a strongly (semi-)prime ring may be obtained by a certain symmetric perfect one sided localization. Complements of strongly prime ideals are described in terms of strongly multiplicative sets of rings. Moreover, some relations between a ring and its multiplication ring are examined.     

On Galois representations for abelian varieties with complex and real multiplications

In this paper, we study the image of l-adic representations coming from Tate module of an abelian variety defined over a number field. We treat abelian varieties with complex and real multiplications. We verify the Mumford-Tate conjecture for a new class of abelian varieties with real multiplication.     

Drinfeld Modules of Rank 1 and Algebraic Curves with Many Rational Points

 We construct algebraic curves C defined over a finite prime field such that the number of -rational points of C is large relative to the genus of C. The methods of construction are based on the relationship between algebraic curves and their function fields, as well as on narrow ray class extensions obtained from Drinfeld modules of rank 1.     

Special automorphisms on Shimura curves and non-triviality of Heegner points

We define the notion of special automorphisms on Shimura curves. Using this notion, for a wild class of elliptic curves defined over Q, we get rank one quadratic twists by discriminants having any prescribed number of prime factors. Finally, as an application, we obtain some new results on Birch and Swinnerton-Dyer (BSD) conjecture for the rank one quadratic twists of the elliptic curve X₀(49).     

Rational torsion of J0(N) for hyperelliptic modular curves and families of Jacobians of genus 2 and genus 3 curves with a rational point of order 5,7 or 10

We describe a way of constructing Jacobians of hyperelliptic curves of genus g ≥ 2, defined over a number field, whose Jacobians have a rational point of order of some (well chosen) integer l ≥ g + 1; the method is based on a polynomial identity. Using this approach we construct new families of genus 2 curves defined over — which contain the modular curves X₀(31) (and X₀(22) as a by-product) and X₀(29), the Jacobians of which have a rational point of order 5 and 7 respectively. We also construct a new family of hyperelliptic genus 3 curves defined over —, which contains the modular curve X₀(41), the Jacobians of which have a rational point of order 10. Finally we show that all hyperelliptic modular curves X₀(N) with N a prime number fit into the described strategy, except for N = 37 in which case we give another explanation. The authors thank the FNR (project FNR/04/MA6/11) for their support.     

Drinfeld Modules of Rank 1 and Algebraic Curves with Many Rational Points

 We construct algebraic curves C defined over a finite prime field such that the number of -rational points of C is large relative to the genus of C. The methods of construction are based on the relationship between algebraic curves and their function fields, as well as on narrow ray class extensions obtained from Drinfeld modules of rank 1. Received 21 July 1997; in revised form 5 February 1998     

Krull dimension for differential graded algebras

We introduce a naive notion of a system of parameters for a homologically finite complex over a commutative noetherian local ring and compare it to the system of parameters defined by Christensen. We show that these notions differ in general but that they agree when the complex in question is a DG R-algebra. In this case we also show that the Krull dimension defined in terms of the lengths of such systems of parameters agrees with Krull dimensions defined in terms of certain chains of prime ideals.     

Associative Cones and Integrable Systems

We identify R7 as the pure imaginary part of octonions. Then the multiplication in octonions gives a natural almost complex structure for the unit sphere S6. It is known that a cone over a surface M in S6 is an associative submanifold of R7 if and only if M is almost complex in S6. In this paper, we show that the Gauss-Codazzi equation for almost complex curves in S6 are the equation for primitive maps associated to the 6-symmetric space G2/T2, and use this to explain some of the known results. Moreover, the equation for S1-symmetric almost complex curves in S6 is the periodic Toda lattice, and a discussion of periodic solutions is given.     

On basic concepts of tropical geometry

We introduce a binary operation over complex numbers that is a tropical analog of addition. This operation, together with the ordinary multiplication of complex numbers, satisfies axioms that generalize the standard field axioms. The algebraic geometry over a complex tropical hyperfield thus defined occupies an intermediate position between the classical complex algebraic geometry and tropical geometry. A deformation similar to the Litvinov-Maslov dequantization of real numbers leads to the degeneration of complex algebraic varieties into complex tropical varieties, whereas the amoeba of a complex tropical variety turns out to be the corresponding tropical variety. Similar tropical modifications with multivalued additions are constructed for other fields as well: for real numbers, p-adic numbers, and quaternions.     

A Hyperelliptic Smoothness Test, II

This series of papers presents and rigorously analyzes a probabilisticalgorithm for finding small prime factors of an integer. Thealgorithm uses the Jacobian varieties of curves of genus 2 inthe same way that the elliptic curve method uses elliptic curves.This second paper in the series is concerned with the orderof the group of rational points on the Jacobian of a curve ofgenus 2 defined over a finite field. We prove a result on thedistribution of these orders. 2000 Mathematical Subject Classification:11Y05, 11G10, 11M20, 11N25.     

Efficient Arithmetic on Koblitz Curves

It has become increasingly common to implement discrete-logarithm based public-key protocols on elliptic curves over finite fields. The basic operation is scalar multiplication: taking a given integer multiple of a given point on the curve. The cost of the protocols depends on that of the elliptic scalar multiplication operation.Koblitz introduced a family of curves which admit especially fast elliptic scalar multiplication. His algorithm was later modified by Meier and Staffelbach. We give an improved version of the algorithm which runs 50 than any previous version. It is based on a new kind of representation of an integer, analogous to certain kinds of binary expansions. We also outline further speedups using precomputation and storage.     

Isogenies of polyquadratic $ \mathbb{Q} $-curves to their Galois conjugates

In this paper, we determine the field of definition of the isogenies of a \mathbbQ \mathbb{Q} -curve without complex multiplication defined over a polyquadratic field to its Galois conjugates.