490 research outputs found
Explicit classification of isogeny graphs of rational elliptic curves
Let be an integer such that has genus ,
and let be a field of characteristic or relatively prime to . In
this article, we explicitly classify the isogeny graphs of all rational
elliptic curves that admit a non-trivial isogeny over . We achieve
this by introducing parameterized families of elliptic curves
defined over , which have the following two
properties for a fixed : the elliptic curves are
isogenous over , and there are integers and such that
the -invariants of and
are given by the Fricke parameterizations. As a
consequence, we show that if is an elliptic curve over a number field
with isogeny class degree divisible by , then there
is a quadratic twist of that is semistable at all primes of
such that .Comment: 22 pages; incorporates referee's suggestions; final version to appear
in International Journal of Number Theor
Lower bounds for the modified Szpiro ratio
Let be an elliptic curve. The modified Szpiro ratio of is
the quantity where and are the invariants
associated to a global minimal model of , and denotes the conductor
of . In this article, we show that for each of the fifteen torsion subgroups
allowed by Mazur's Torsion Theorem, there is a rational number such
that if , then . We also show that this bound is sharp.Comment: 15 pages; incorporates referee's suggestions; sharpness of lower
bounds is no longer conditional on the abc conjecture; final version to
appear in Acta Arithmetic
Reduced minimal models and torsion
Let be an elliptic curve. The reduced minimal model of is
a global minimal model
which satisfies the additional conditions that and
. The reduced minimal model of is unique, and in this
article, we explicitly classify the reduced minimal model of an elliptic curve
with a non-trivial torsion point. We obtain this classification
by first showing that the reduced minimal model of is uniquely determined
by a congruence on modulo . We then apply this result to
parameterized families of elliptic curves to deduce our main result. We also
show that the reduction at and of affects the reduced minimal model
of .Comment: 14 page
Minimal Models of Rational Elliptic Curves with non-Trivial Torsio
This dissertation concerns the formulation of an explicit modified Szpirobconjecture and the classification of minimal discriminants of rational elliptic curves with non-trivial torsion subgroup. The Frey curve y2=x( x+a) ( x-b) is a two-parameter family of elliptic curves which comes equipped with an easily computable minimal discriminant which helped pave the mathematical bridge that led to the proof of Fermat\u27s Last Theorem. In this dissertation, we extend the ideas of the Frey curve by considering two- and three- parameter families of elliptic curves which parameterize all rational elliptic curves with non-trivial torsion. First, we use these families to give a new proof of a classic result of Frey, Flexor, and Oesterlé which pertains to the primes at which an elliptic curve over a number field can have additive reduction. While our proof gives a weaker variant of the original statement, it is explicit and does not require the Néron model of an elliptic curve. As a consequence of this new proof, we attain our classification of minimal discriminants of rational elliptic curves with non-trivial torsion. In addition, we give necessary and sufficient conditions for when a rational elliptic curve with non-trivial torsion has additive reduction at a given prime. We also study the connection between torsion structure of a rational elliptic curve and the possible reduced minimal models The second theme of this dissertation concerns the modified Szpiro conjecture, which is equivalent to the ABC Conjecture. Roughly speaking, the modified Szpiro conjecture states that certain elliptic curves, known as good elliptic curves, are rare in nature. Masser gave a non-constructive proof which showed that there were infinitely many good Frey curves. In this dissertation, we give a constructive proof of Masser\u27s assertion. We then extend this result by proving that for each of the fifteen torsion subgroups T allowed by Mazur\u27s Torsion Theorem, there are infinitely many good elliptic curves E with torsion subgroup isomorphic to T. This proof is also constructive and allows for the construction of a database which consists of 13870964 good elliptic curves. We provide an analysis of these good elliptic curves to parallel the work done by the ABC@Home project concerning the ABC Conjecture and good ABC triples. The data obtained is then used to formulate an explicit version of the modified Szpiro conjecture. We then show that this explicit formulation allows for the construction of databases of elliptic curves which are exhaustive up to a given conductor. Lastly, we use the classification of minimal discriminants to study the local data of rational elliptic curves at a given prime via Tate\u27s Algorithm. These results and a study of the naive height of an elliptic curve allow us to prove that there is a lower bound on the modified Szpiro ratio which depends only on the torsion structure of an elliptic curve
Representations attached to elliptic curves with a non-trivial odd torsion point
We give a classification of the cuspidal automorphic representations attached
to rational elliptic curves with a non-trivial torsion point of odd order. Such
elliptic curves are parameterizable, and in this paper, we find the necessary
and sufficient conditions on the parameters to determine when split or
non-split multiplicative reduction occurs. Using this and the known results on
when additive reduction occurs for these parametrized curves, we classify the
automorphic representations in terms of the parameters.Comment: 16 page
Local data of rational elliptic curves with non-trivial torsion
By Mazur's Torsion Theorem, there are fourteen possibilities for the
non-trivial torsion subgroup of a rational elliptic curve. For each ,
such that may have additive reduction at a prime , we consider a
parameterized family of elliptic curves with the property that they
parameterize all elliptic curves which contain in their
torsion subgroup. Using these parameterized families, we explicitly classify
the Kodaira-N\'{e}ron type, the conductor exponent, and the local Tamagawa
number at each prime where has additive reduction. As a
consequence, we find all rational elliptic curves with a -torsion or a
-torsion point that have global Tamagawa number .Comment: 36 pages; incorporates referee's suggestions; final version to appear
in Pacific Journal of Mathematic
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Diseña un sistema web adaptativo a cualquier tipo de dispositivo (computadores, tabletas, móviles) que facilita la gestión de evaluaciones de riesgos en sedes regionales de Visión Mundial
Probing structure and energetics of proton-bound complexes N2…HCO+ and N2H+…OC using computational chemistry methods
N2…HCO+ and N2H+…OC are predicted to exist in interstellar clouds. These complexes involve HCO+ and N2H+ fragments that are bound to N2 and CO, respectively using hydrogen-bonded interaction. The reason these molecules are important is that the existence of nitrogen can be measured indirectly through ion-molecular complexes studied in this work. The measured vibrational spectra of molecules is an excellent way to characterize and detect molecules. We used B3LYP, MP2, and CCSD(T) computational methods to predict the structure and vibrational frequencies of N2…HCO+ and N2H+…OC and their fragments. The aug-cc-pVDZ and aug-cc-pVTZ basis sets were used. The stability of the complexes was described in terms of dissociations energies De and their zero-point energy (ZPE) corrected values, Do. Both molecular complexes exhibit a linear geometry. Vibrational frequencies were obtained using normal mode analysis. The N2H+…OC proton transfer vibrations occur at around 1800 – 1900 cm-1. H+ bound within HCO+ exhibit C-H vibration at ~2500-2700 cm-1. The N2…HCO+. complex is more stable than N2H+…OC by ~7000 cm-1. The ZPE corrected values for dissociation energies, Do for N2…HCO+ --\u3e N2 + HCO+ and N2H+…OC --\u3e N2H+ + OC are ~3500 cm-1 and ~5000 cm-1, respectively
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