Common divisors of elliptic divisibility sequences over function fields

Let E/k(T) be an elliptic curve defined over a rational function field of characteristic zero. Fix a Weierstrass equation for E. For points RE(k(T)), write x_R=A_R/D_R² with relatively prime polynomials A_R(T),D_R(T)k[T]. The sequence {D_nR}_{n 1} is called the elliptic divisibility sequence of R. Let P,QE(k(T)) be independent points. We conjecture that deg (gcd(D_nP, D_mQ)) is bounded for m, n 1, and that gcd(D_nP, D_nQ) = gcdD_P, D_Q) for infinitely many n 1. We prove these conjectures in the case that j(E)k. More generally, we prove analogous statements with k(T) replaced by the function field of any curve and with P and Q allowed to lie on different elliptic curves. If instead k is a finite field of characteristic p and again assuming that j(E)k, we show that deg (gcd(D_nP, D_nQ)) is as large as for infinitely many n0 (mod p).Mathematics Subject Classification (2000): Primary: 11D61; Secondary: 11G35Acknowledgements. I would like to thank Gary Walsh for rekindling my interest in the arithmetic properties of divisibility sequences and for bringing to my attention the articles [1] and [3], and David McKinnon for showing me his article [14]. I also want to thank Zeev Rudnick for his helpful comments concerning the first draft of this paper, especially for Remark 5, for pointing out [7], and for letting me know that he described conjectures similar to those made in this paper at CNTA 7 in 2002.     

Elliptic Tate curves over local Γ-extensions

The group A(K)/N is computed, where A(K) is the group of points of a Tate curve over a local field while N is the group of universal norms from the group of points over a -extension. As an application, the Mazurl-modulus of modular elliptic curves is computed for values ofl dividing the denominator of the absolute invariant.Translated from Matematicheskie Zametki, Vol. 13, No. 4, pp. 531–539, April, 1973.In conclusion, I wish to thank Yu. I. Manin for having guided this work.     

Perfect powers in elliptic divisibility sequences

It is shown that there are finitely many perfect powers in an elliptic divisibility sequence whose first term is divisible by 2 or 3. For Mordell curves the same conclusion is shown to hold if the first term is greater than 1. Examples of Mordell curves and families of congruent number curves are given with corresponding elliptic divisibility sequences having no perfect power terms. The proofs combine primitive divisor results with modular methods for Diophantine equations.     

Elliptic and hyperelliptic curves over supersimple fields in characteristic 2

In this paper, we extend a previous result of A. Pillay and the author regarding existence of rational points over elliptic and hyperelliptic curves with generic moduli defined over supersimple fields to the even characteristic case. We give a detailed exposition of the affine models of these families of curves in characteristic 2 and the transformations between members in the same rational isomorphism class.     

Primitive divisors of elliptic divisibility sequences

Silverman proved the analogue of Zsigmondy's Theorem for elliptic divisibility sequences. For elliptic curves in global minimal form, it seems likely this result is true in a uniform manner. We present such a result for certain infinite families of curves and points. Our methods allow the first explicit examples of the elliptic Zsigmondy Theorem to be exhibited. As an application, we show that every term beyond the fourth of the Somos-4 sequence has a primitive divisor.     

Prime powers in elliptic divisibility sequences

Certain elliptic divisibility sequences are shown to contain only finitely many prime power terms. In some cases the methods prove that only finitely many terms are divisible by a bounded number of distinct primes.

     

Reduction of elliptic curves over certain real quadratic number fields

The main result of this paper is that an elliptic curve having good reduction everywhere over a real quadratic field has a -rational point under certain hypotheses (primarily on class numbers of related fields). It extends the earlier case in which no ramification at is allowed. Small fields satisfying the hypotheses are then found, and in four cases the non-existence of such elliptic curves can be shown, while in three others all such curves have been classified.

     

Number of points on certain hyperelliptic curves defined over finite fields

For odd primes p and l such that the order of p modulo l is even, we determine explicitly the Jacobsthal sums _l(v), ψ_l(v), and ψ_2l(v), and the Jacobsthal–Whiteman sums and , over finite fields F_q such that . These results are obtained only in terms of q and l. We apply these results pertaining to the Jacobsthal sums, to determine, for each integer n1, the exact number of F_qⁿ-rational points on the projective hyperelliptic curves aY²Z^e−2=bX^e+cZ^e (abc≠0) (for e=l,2l), and aY²Z^l−1=X(bX^l+cZ^l) (abc≠0), defined over such finite fields F_q. As a consequence, we obtain the exact form of the ζ-functions for these three classes of curves defined over F_q, as rational functions in the variable t, for all distinct cases that arise for the coefficients a,b,c. Further, we determine the exact cases for the coefficients a,b,c, for each class of curves, for which the corresponding non-singular models are maximal (or minimal) over F_q.     

On divisibility properties of certain multinomial coefficients—II

A recent conjecture of Myerson and Sander concerns divisibility properties of certain multinomial coefficients. We obtain results in this direction by further pursuing a line of attack developed earlier by the first author.     

Elliptic subcovers of hyperelliptic curves

The main result of this paper states that if C is a hyperelliptic curve of even genus over an arbitrary field K , then there is a natural bijection between the set of equivalence classes of elliptic subcovers of and the set of elliptic subgroups of its Jacobian .     

Elliptic curves on spinor varieties

We prove irreducibility of the scheme of morphisms, of degree large enough, from a smooth elliptic curve to spinor varieties. We give an explicit bound on the degree.     

Nonexistence of elliptic curves having good reduction everywhere over certain quadratic fields

The purpose of this paper is to show the nonexistence of elliptic curves having good reduction everywhere over certain quadratic fields.     

Elliptic curves of odd modular degree

The modular degree m _E of an elliptic curve E/Q is the minimal degree of any surjective morphism X ₀(N) → E, where N is the conductor of E. We give a necessary set of criteria for m _E to be odd. In the case when N is prime our results imply a conjecture of Mark Watkins. As a technical tool, we prove a certain multiplicity one result at the prime p = 2, which may be of independent interest. Supported in part by the American Institute of Mathematics. Supported in part by NSF grant DMS-0401545.