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AbstractIn this note we study the existence of primes and of primitive divisors in function field analogues of classical divisibility sequences. Under various hypotheses, we prove that Lucas sequences and elliptic divisibility sequences over function fields defined over number fields contain infinitely many irreducible elements. We also prove that an elliptic divisibility sequence over a function field has only finitely many terms lacking a primitive divisor.

Abstract. We apply Vojta’s conjecture to blowups and deduce a number of deep statements regarding (generalized) greatest common divisors on varieties, in particular on projective space and on abelian varieties. Special cases of these statements generalize earlier results and conjectures. We also discuss the relationship between generalized greatest common divisors and the divisibility sequences attached to algebraic groups, and we apply Vojta’s conjecture to obtain a strong bound on the divisibility sequences attached to abelian varieties of dimension at least two.

Abstract. For a fixed rational point P ? E (K) on an elliptic curve, we consider the sequence of values (Fn (P))n=1 of the division polynomials of E at P. For a finite field we prove that the sequence is periodic. For a local field we prove (under certain hypotheses) that there is a power q=pe so that for all m=1, the limit of exists in K and is algebraic over We apply this result to prove an analogous p-adic limit and algebraicity result for elliptic divisibility sequences.

Let E ? C be an elliptic surface defined over a number field K. For a finite covering C' ? C defined over K, let E' = E ×CC' be the corresponding elliptic surface over C'. In this paper we give a strong upper bound for the rank of E' (C'/K) in the case of geometrically abelian unramified coverings C' ? C and under the assumption that the Tate conjecture is true for E'/K. In the case that C is an elliptic curve and the map C' = C ? C is the multiplication-by-n map, the bound for rank(E'(C'/K)) takes the form O(nk/log logn), which may be compared with the elementary bound of O(n2).

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 R ∈  E( k( T)), write x R= A R/ D R 2 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, Q ∈  E( 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) = gcd D 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 n≢0 (mod p).

The complexity of the multiplication operation in finite fields is of interest for both theoretical and practical reasons. For example, an optimal normal basis for F2Nhas complexity 2N−1. A construction described in J. H. Silverman, (“Cryptographic Hardware and Embedded Systems,” Lecture Notes in Computer Science, Vol. 1717, pp. 122–134, Springer–Verlag, Berlin, 1999.) allows multiplication of complexity N+1 to be performed in F2Nby working in a larger ring Rof dimension N+1 over F2. In this paper we give a complete classification of all such rings and show that this construction is the only one which also has a certain useful permutability property.

Let Kbe a number field and E/Kan elliptic curve. As is well known [3, 4,[ if Khas class number 1, then there exists a global minimal Weierstrass equation for E. Our main goal in this paper is to prove the following converse to this statement.

Abstract.: Nagao has recently given a conjectural limit formula for the rank of an elliptic surface E in terms of a weighted average of fibral Frobenius trace values. We show that Tate's conjecture on the order of vanishing of L 2(E,s) essentially implies Nagao's formula; in particular, we prove Nagao's formula for rational elliptic surfaces. In the case that E is a twist, we reduce Nagao's and Tate's conjectures to the case of products of curves, and we verify the conjectures for many new classes of elliptic surfaces of Kodaira dimension 0 and 1.

Let φ(z)∈C(z) be a rational function of degreeD≥2. A well known theorem in dynamical systems says that if some iterate φn(z)=φ∘φ∘···∘φ(z) is a polynomial, then already φ2(z) is a polynomial. More generally, this is true for φ(z)∈K(z) for any fieldKprovided that φ is separable (i.e., φ′(z)≠0). However, it need not be true for inseparable φ. In this note we give a complete description, in all characteristics and with no separability assumptions, of the rational functions which have a polynomial iterate.

The set of Weierstrass points for pluricanonical linear systems on an algebraic curve C have been extensively studied from a geometric viewpoint. If the curve is defined over a number fieldk, then thesen th order Weierstrass points are defined over an algebraic extensionkn ofk, and it is an interesting question to ask for the arithmetic properties of the points and the extension that they generate. In this paper we begin the study of the arithmetic properties of higher order Weierstrass points in the special case of hyperelliptic curves. We give an upper bound for the average height of these points, and we show that for sufficiently large primesp, the first order Weierstrass points and then th order Weierstrass points remain distinct modulop. This limits to some extent the ramification that can occur in the extensionkn/k. We also present two numerical examples which indicate that a complete description of the ramification is likely to be complicated.

Estimates for the difference of the Weil height and the canonical height of points on elliptic curves are used for many purposes, both theoretical and computational. In this note we give an explicit estimate for this difference in terms of the j-invariant and discriminant of the elliptic curve. The method of proof, suggested by Serge Lang, is to use the decomposition of the canonical height into a sum of local heights. We illustrate one use for our estimate by computing generators for the Mordell-Weil group in three examples.

We describe how to compute the canonical height of points on elliptic curves. Tate has given a rapidly converging series for Archimedean local heights over R. We describe a modified version of Tate's series which also converges over C, and give an efficient procedure for calculating local heights at non-Archimedean places. In this way we can calculate heights over number fields having complex embeddings. We also give explicit estimates for the tail of our series, and present several examples.

Let $ K$ $ a,b,c \in {K^ \ast }$ $ \max \{ m,n\} $ $ a{x^m} + b{y^n} = c$.

We show that there are only finitely many elliptic curves of bounded degree and height, provided that one takes a naive height defined in terms of minimal Weierstrass equations. We show that the corresponding statement is false if instead one uses the Faltings-Parshin modular height.