7,380 research outputs found
The explicit Mordell Conjecture for families of curves (with an appendix by M. Stoll)
In this article we prove the explicit Mordell Conjecture for large families
of curves. In addition, we introduce a method, of easy application, to compute
all rational points on curves of quite general shape and increasing genus. The
method bases on some explicit and sharp estimates for the height of such
rational points, and the bounds are small enough to successfully implement a
computer search. As an evidence of the simplicity of its application, we
present a variety of explicit examples and explain how to produce many others.
In the appendix our method is compared in detail to the classical method of
Manin-Demjanenko and the analysis of our explicit examples is carried to
conclusion.Comment: 42 pages, 1 figure, 1 tabl
On the Diophantine equation
By finding all integral points on certain elliptic and hyperelliptic curves
we completely solve the Diophantine equation for
and Moreover, we present some other
observations of computational and theoretical nature concerning the title
equation
The second moment of the number of integral points on elliptic curves is bounded
In this paper, we show that the second moment of the number of integral
points on elliptic curves over is bounded. In particular, we prove
that, for any , the -th moment of the
number of integral points is bounded for many families of elliptic curves ---
e.g., for the family of all integral short Weierstrass curves ordered by naive
height, for the family of only minimal such Weierstrass curves, for the family
of semistable curves, or for subfamilies thereof defined by finitely many
congruence conditions. For certain other families of elliptic curves, such as
those with a marked point or a marked -torsion point, the same methods show
that for , the -th moment of the number of
integral points is bounded.
The main new ingredient in our proof is an upper bound on the number of
integral points on an affine integral Weierstrass model of an elliptic curve
depending only on the rank of the curve and the number of square divisors of
the discriminant. We obtain the bound by studying a bijection first observed by
Mordell between integral points on these curves and certain types of binary
quartic forms. The theorems on moments then follow from H\"older's inequality,
analytic techniques, and results on bounds on the average sizes of Selmer
groups in the families.Comment: 14 pages, comments welcome
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