34 research outputs found
Curves of every genus with many points, II: Asymptotically good families
We resolve a 1983 question of Serre by constructing curves with many points
of every genus over every finite field. More precisely, we show that for every
prime power q there is a positive constant c_q with the following property: for
every non-negative integer g, there is a genus-g curve over F_q with at least
c_q * g rational points over F_q. Moreover, we show that there exists a
positive constant d such that for every q we can choose c_q = d * (log q). We
show also that there is a constant c > 0 such that for every q and every n > 0,
and for every sufficiently large g, there is a genus-g curve over F_q that has
at least c*g/n rational points and whose Jacobian contains a subgroup of
rational points isomorphic to (Z/nZ)^r for some r > c*g/n.Comment: LaTeX, 18 page
Linear independence in linear systems on elliptic curves
Let be an elliptic curve, with identity , and let be a cyclic
subgroup of odd order , over an algebraically closed field with
. For , let be a rational
function with divisor . We ask whether the functions
are linearly independent. For generic , we prove that the answer
is yes. We bound the number of exceptional when is a prime by using
the geometry of the universal generalized elliptic curve over . The
problem can be recast in terms of sections of an arbitrary degree line
bundle on .Comment: 10 page
Curves of every genus with many points, I: Abelian and toric families
Let N_q(g) denote the maximal number of F_q-rational points on any curve of
genus g over the finite field F_q. Ihara (for square q) and Serre (for general
q) proved that limsup_{g-->infinity} N_q(g)/g > 0 for any fixed q. In their
proofs they constructed curves with many points in infinitely many genera;
however, their sequences of genera are somewhat sparse. In this paper, we prove
that lim_{g-->infinity} N_q(g) = infinity. More precisely, we use abelian
covers of P^1 to prove that liminf_{g-->infinity} N_q(g)/(g/log g) > 0, and we
use curves on toric surfaces to prove that liminf_{g-->infty} N_q(g)/g^{1/3} >
0; we also show that these results are the best possible that can be proved
with these families of curves.Comment: LaTeX, 20 page
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Subjectivity in a context of environmental change: opening new dialogues in mental health research
In a period of unstable experimentation with challenges of globalization of associated risks, and disenchantment with ‘enduring injustice’, we bring forward a consideration of subjectivity to the study of environmental change and mental health. We begin by identifying how mainstream climate change and mental health studies are unable to explain the emergent and co-evolutionary pathways of agency. As a means of freeing these studies of their objective dimensions of linear-causation, we argue in favour of a re-positioning of subjectivity within an appreciation of recognition conflicts and beyond the over-deterministic interpretations of power centres—state, market or religion. We draw on one example of scientific research that was conducted in a region undergoing strong environmental, social and cultural changes, in the state of São Paulo/Brazil, with the aim to open mental health research to new dialogues, to which we contribute with the notion of the ‘pluriversal subject’