5,943 research outputs found
Parametrizing quartic algebras over an arbitrary base
We parametrize quartic commutative algebras over any base ring or scheme
(equivalently finite, flat degree four -schemes), with their cubic
resolvents, by pairs of ternary quadratic forms over the base. This generalizes
Bhargava's parametrization of quartic rings with their cubic resolvent rings
over by pairs of integral ternary quadratic forms, as well as
Casnati and Ekedahl's construction of Gorenstein quartic covers by certain rank
2 families of ternary quadratic forms. We give a geometric construction of a
quartic algebra from any pair of ternary quadratic forms, and prove this
construction commutes with base change and also agrees with Bhargava's explicit
construction over .Comment: submitte
Examples of CM curves of genus two defined over the reflex field
In "Proving that a genus 2 curve has complex multiplication", van Wamelen
lists 19 curves of genus two over with complex multiplication
(CM). For each of the 19 curves, the CM-field turns out to be cyclic Galois
over . The generic case of non-Galois quartic CM-fields did not
feature in this list, as the field of definition in that case always contains a
real quadratic field, known as the real quadratic subfield of the reflex field.
We extend van Wamelen's list to include curves of genus two defined over this
real quadratic field. Our list therefore contains the smallest "generic"
examples of CM curves of genus two.
We explain our methods for obtaining this list, including a new
height-reduction algorithm for arbitrary hyperelliptic curves over totally real
number fields. Unlike Van Wamelen, we also give a proof of our list, which is
made possible by our implementation of denominator bounds of Lauter and Viray
for Igusa class polynomials.Comment: 31 pages; Updated some reference
On number fields with equivalent integral trace forms
Let be a number field. The \textit{integral trace form} is the integral
quadratic form given by In this
article we study the existence of non-conjugated number fields with equivalent
integral trace forms. As a corollary of one of the main results of this paper,
we show that any two non-totally real number fields with the same signature and
same prime discriminant have equivalent integral trace forms. Additionally,
based on previous results obtained by the author and the evidence presented
here, we conjecture that any two totally real quartic fields of fundamental
discriminant have equivalent trace zero forms if and only if they are
conjugated
Curves of genus 3 over small finite fields
We present a table containing the maximal number of rational points on a
genus 3 curve over a field of cardinality q, for all q<100. Also, some remarks
on Frobenius non-classical quartics over finite fields are given.Comment: 9 page
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