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, but, under certain conditions, there is a canonical choice of elliptic curve E' and algebraic f:X—...

‎In a (t,n)-threshold secret sharing scheme‎, ‎a secret s is distributed among n participants such that any group of t or more participants can reconstruct the secret together‎, ‎but no group of fewer than t participants can do‎. In this paper, we propose a verifiable (t,n)-threshold multi-secret sharing scheme based on Shao and Cao‎, ‎and the intractability of the elliptic curve discrete logar...

In this paper the family of elliptic curves over Q given by the equation Ep :Y2 = (X - p)3 + X3 + (X + p)3 where p is a prime number, is studied. Itis shown that the maximal rank of the elliptic curves is at most 3 and someconditions under which we have rank(Ep(Q)) = 0 or rank(Ep(Q)) = 1 orrank(Ep(Q))≥2 are given.

In this paper, we make a study of the Iwasawa theory of an elliptic curve at a supersingular prime p along an arbitrary Zp-extension of a number field K in the case when p splits completely in K. Generalizing work of Kobayashi [8] and Perrin-Riou [16], we define restricted Selmer groups and λ ± , µ ±-invariants; we then derive asymptotic formulas describing the growth of the Selmer group in ter...

The first purpose of this paper is to examine the relationship between equivariant elliptic genera and orbifold elliptic genera. We apply the character theory of [HKR00] to the Borel-equivariant genus associated to the sigma orientation of [AHS01] to define an orbifold genus for certain total quotient orbifolds and supersingular elliptic curves. We show that our orbifold genus is given by the s...

In this paper, we make a study of the Iwasawa theory of an elliptic curve at a supersingular prime p along an arbitrary Zp-extension of a number field K in the case when p splits completely in K. Generalizing work of Kobayashi [9] and Perrin-Riou [17], we define restricted Selmer groups and λ±, μ±-invariants; we then derive asymptotic formulas describing the growth of the Selmer group in terms ...

We investigate the E2-term of Adams spectral sequence based on elliptic homology. The main results describe this E2-term from a ‘chromatic’ perspective. At a prime p > 3, the Bousfield class of Ell is the same as that of K(0)∨K(1)∨ K(2). Using delicate facts due to Katz (which also play a major rôle in work on the structure Ell∗Ell by Clarke & Johnson, the author and Laures) as well as our desc...

We show that the Davenport-Stothers inequality from characteristic 0 fails in any characteristic p > 3. The proof uses elliptic surfaces over P and inseparable base change. We then present adjusted inequalities. These follow from results of PesentiSzpiro. For characteristic 2 and 3, we achieve a similar result in terms of the maximal singular fibres of elliptic surfaces over P. Our ideas are al...