2,481 research outputs found
Multiplicative excellent families of elliptic surfaces of type E_7 or E_8
We describe explicit multiplicative excellent families of rational elliptic
surfaces with Galois group isomorphic to the Weyl group of the root lattices
E_7 or E_8. The Weierstrass coefficients of each family are related by an
invertible polynomial transformation to the generators of the multiplicative
invariant ring of the associated Weyl group, given by the fundamental
characters of the corresponding Lie group. As an application, we give examples
of elliptic surfaces with multiplicative reduction and all sections defined
over Q for most of the entries of fiber configurations and Mordell-Weil
lattices in [Oguiso-Shioda '91], as well as examples of explicit polynomials
with Galois group W(E_7) or W(E_8).Comment: 23 pages. Final versio
Points of Low Height on Elliptic Curves and Surfaces, I: Elliptic surfaces over P^1 with small d
For each of n=1,2,3 we find the minimal height h^(P) of a nontorsion point P
of an elliptic curve E over C(T) of discriminant degree d=12n (equivalently, of
arithmetic genus n), and exhibit all (E,P) attaining this minimum. The minimal
h^(P) was known to equal 1/30 for n=1 (Oguiso-Shioda) and 11/420 for n=2
(Nishiyama), but the formulas for the general (E,P) were not known, nor was the
fact that these are also the minima for an elliptic curve of discriminant
degree 12n over a function field of any genus. For n=3 both the minimal height
(23/840) and the explicit curves are new. These (E,P) also have the property
that that mP is an integral point (a point of naive height zero) for each
m=1,2,...,M, where M=6,8,9 for n=1,2,3; this, too, is maximal in each of the
three cases.Comment: 15 pages; some lines in the TeX source are commented out with "%" to
meet the 15-page limit for ANTS proceeding
Lines on Fermat surfaces
We prove that the Neron-Severi groups of several complex Fermat surfaces are
generated by lines. Specifically, we obtain these new results for all degrees
up to 100 that are relatively prime to 6. The proof uses reduction modulo a
supersingular prime. The techniques are developed in detail. They can be
applied to other surfaces and varieties as well.Comment: 29 pages; v3: major extension thanks to RvL who joined as third
author; results and techniques strengthened, paper reorganise
Constructing Elliptic Curves over with Moderate Rank
We give several new constructions for moderate rank elliptic curves over
. In particular we construct infinitely many rational elliptic
surfaces (not in Weierstrass form) of rank 6 over using
polynomials of degree two in . While our method generates linearly
independent points, we are able to show the rank is exactly 6 \emph{without}
having to verify the points are independent. The method generalizes; however,
the higher rank surfaces are not rational, and we need to check that the
constructed points are linearly independent.Comment: 11 page
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