9,054 research outputs found
On group structures realized by elliptic curves over a fixed finite field
We obtain explicit formulas for the number of non-isomorphic elliptic curves
with a given group structure (considered as an abstract abelian group).
Moreover, we give explicit formulas for the number of distinct group structures
of all elliptic curves over a finite field. We use these formulas to derive
some asymptotic estimates and tight upper and lower bounds for various counting
functions related to classification of elliptic curves accordingly to their
group structure. Finally, we present results of some numerical tests which
exhibit several interesting phenomena in the distribution of group structures.
We pose getting an explanation to these as an open problem
On group structures realized by elliptic curves over arbitrary finite fields
We study the collection of group structures that can be realized as a group
of rational points on an elliptic curve over a finite field (such groups are
well known to be of rank at most two). We also study various subsets of this
collection which correspond to curves over prime fields or to curves with a
prescribed torsion. Some of our results are rigorous and are based on recent
advances in analytic number theory, some are conditional under certain widely
believed conjectures, and others are purely heuristic in nature
Monodromy of Kodaira Fibrations of Genus
A Kodaira fibration is a non-isotrivial fibration
from a smooth algebraic surface to a smooth algebraic curve such that
all fibers are smooth algebraic curves of genus . Such fibrations arise as
complete curves inside the moduli space of genus algebraic
curves. We investigate here the possible connected monodromy groups of a
Kodaira fibration in the case and classify which such groups can arise
from a Kodaira fibration obtained as a general complete intersection curve
inside a subvariety of parametrizing curves whose Jacobians
have extra endomorphisms.Comment: 17 page
Enhanced gauge symmetry in 6D F-theory models and tuned elliptic Calabi-Yau threefolds
We systematically analyze the local combinations of gauge groups and matter
that can arise in 6D F-theory models over a fixed base. We compare the
low-energy constraints of anomaly cancellation to explicit F-theory
constructions using Weierstrass and Tate forms, and identify some new local
structures in the "swampland' of 6D supergravity and SCFT models that appear
consistent from low-energy considerations but do not have known F-theory
realizations. In particular, we classify and carry out a local analysis of all
enhancements of the irreducible gauge and matter contributions from
"non-Higgsable clusters," and on isolated curves and pairs of intersecting
rational curves of arbitrary self-intersection. Such enhancements correspond
physically to unHiggsings, and mathematically to tunings of the Weierstrass
model of an elliptic CY threefold. We determine the shift in Hodge numbers of
the elliptic threefold associated with each enhancement. We also consider local
tunings on curves that have higher genus or intersect multiple other curves,
codimension two tunings that give transitions in the F-theory matter content,
tunings of abelian factors in the gauge group, and generalizations of the
"" rule to include tunings and curves of self-intersection zero. These
tools can be combined into an algorithm that in principle enables a finite and
systematic classification of all elliptic CY threefolds and corresponding 6D
F-theory SUGRA models over a given compact base (modulo some technical caveats
in various special circumstances), and are also relevant to the classification
of 6D SCFT's. To illustrate the utility of these results, we identify some
large example classes of known CY threefolds in the Kreuzer-Skarke database as
Weierstrass models over complex surface bases with specific simple tunings, and
we survey the range of tunings possible over one specific base.Comment: 101 pages, 3 figures, 25 tables; v2: references added, some technical
corrections, minor typos correcte
Group structures of elliptic curves over finite fields
It is well-known that if is an elliptic curve over the finite field
, then
for
some positive integers . Let denote the set of pairs
with and such that there exists an elliptic curve over some
prime finite field whose group of points is isomorphic to
. Banks, Pappalardi and
Shparlinski recently conjectured that if , then a
density zero proportion of the groups in question actually arise as the group
of points on some elliptic curve over some prime finite field. On the other
hand, if , they conjectured that a density one
proportion of the groups in question arise as the group of points on some
elliptic curve over some prime finite field. We prove that the first part of
their conjecture holds in the full range , and we
prove that the second part of their conjecture holds in the limited range . In the wider range , we show that a positive density
of the groups in question actually occur.Comment: 15 pages. Final version, published in IMRN. Some minor change
Preperiodic points for quadratic polynomials over cyclotomic quadratic fields
Given a number field and a polynomial of degree at least
2, one can construct a finite directed graph whose vertices are the
-rational preperiodic points for , with an edge if and
only if . Restricting to quadratic polynomials, the
dynamical uniform boundedness conjecture of Morton and Silverman suggests that
for a given number field , there should only be finitely many isomorphism
classes of directed graphs that arise in this way. Poonen has given a
conjecturally complete classification of all such directed graphs over
, while recent work of the author, Faber, and Krumm has provided a
detailed study of this question for all quadratic extensions of .
In this article, we give a conjecturally complete classification like Poonen's,
but over the cyclotomic quadratic fields and
. The main tools we use are dynamical modular curves and
results concerning quadratic points on curves.Comment: v4 includes a few more details, especially toward the end of Section
3 and in Appendix A. Other minor changes have been made. An additional Magma
file (main.txt), containing calculations for the main body of the article,
have been added as an ancillary fil
Modularity of Calabi--Yau varieties: 2011 and beyond
This paper presents the current status on modularity of Calabi-Yau varieties
since the last update in 2003. We will focus on Calabi-Yau varieties of
dimension at most three. Here modularity refers to at least two different
types: arithmetic modularity and geometric modularity. These will include: (1)
the modularity (automorphy) of Galois representations of Calabi-Yau varieties
(or motives) defined over Q or number fields, (2) the modularity of solutions
of Picard--Fuchs differential equations of families of Calabi-Yau varieties,
and mirror maps (mirror moonshine), (3) the modularity of generating functions
of invariants counting certain quantities on Calabi-Yau varieties, and (4) the
modularity of moduli for families of Calabi-Yau varieties.Comment: 33 page
Torsion subgroups of rational elliptic curves over the compositum of all extensions of the rational numbers
Let be an elliptic curve and let be
the compositum of all extensions of whose Galois closure has
Galois group isomorphic to a quotient of a subdirect product of a finite number
of transitive subgroups of . In this article we first show that
is in fact the compositum of all extensions of
and then we prove that the torsion subgroup of
is finite and determine the 24 possibilities for
its structure. We also give a complete classification of the elliptic curves
that have each possible torsion structure in terms of their -invariants
Revisiting arithmetic solutions to the condition
The gravitino mass is expected not to be much smaller than the Planck scale
for a large fraction of vacua in flux compactifications. There is no continuous
parameter to tune even by hand, and it seems that the gravitino mass can be
small only as a result of accidental cancellation among period integrals
weighted by integer-valued flux quanta. DeWolfe et.al. (2005) proposed to pay
close attention to vacua where the Hodge decomposition is possible within a
number field, so that the precise cancellation takes place as a result of
algebra. We focus on a subclass of those vacua---those with complex
multiplications---and explore more on the idea in this article. It turns out,
in Type IIB compactifications, that those vacua admit non-trivial
supersymmetric flux configurations if and only if the reflex field of the Weil
intermediate Jacobian is isomorphic to the quadratic imaginary field generated
by the axidilaton vacuum expectation value. We also found that flux statistics
is highly enriched on such vacua, as F-term conditions become linearly
dependent.Comment: 28+25 pages, a few references added and a few typos correcte
Curves of genus 2 with group of automorphisms isomorphic to D_8 or D_12
In this paper we classify curves of genus 2 with group of automorphisms
isomorphic to D_8 or D_12 over an arbitrary field k (of characteristic
different from 2 in the D_8 case and from 2 and 3 in the D_{12} case) up to
k-isomorphism.
As an application of the classification of curves of genus 2 obtained, we get
precise arithmetic information on their elliptic quotients and on their
jacobians. Over the field k=Q, we show that the elliptic quotients of the
curves with automorphisms D_8 and D_{12} are precisely the Q-curves of degrees
2 and 3, respectively, and we determine which curves have jacobians of
GL_2-type.Comment: 26 page
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