Jacobians of Genus One Curves

An, Sang Yook; Kim, Seog Young; Marshall, David C.; Marshall, Susan; McCallum, William G.; Perlis, Alexander R.

doi:10.1006/jnth.2000.2632

Cited by 62 publications

(117 citation statements)

References 4 publications

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“…In order to obtain the WSF of X F 1 , given the absence of sections of its fibration, we have to calculate the associated Jacobian fibration J(X F 1 ). The algorithm for computing J(X F 1 ) is well known in the mathematics literature, see for example [76], from where we calculate f and g, given explicitly in (B.1) and (B.2), and subsequently the discriminant ∆. The discriminant does not factorize, which shows the absence of codimension one singularities…”

Section: Polyhedronmentioning

confidence: 99%

“…it is a genusone fibration. We obtain its WSF by computing its associated Jacobian fibration J(X F 2 ), employing again the straightforward algorithms from the mathematics literature [76]. The results for the functions f and g can be found in (B.3) and (B.4), from which the discriminant can be readily computed.…”

Section: Polyhedronmentioning

confidence: 99%

“…Since this fibration does not have a section one has to utilize its associated Jacobian fibration J(X F 4 ) in order to find its WSF. We readily compute the functions f and g in (2.1) using the algorithm in [76]. From the discriminant of this WSF, we find one I 2 -singularity over the divisors S b SU(2) = {d 9 = 0} ∩ B in B.…”

Section: Polyhedronmentioning

confidence: 99%

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F-theory on all toric hypersurface fibrations and its Higgs branches

Klevers

Pena²,

Oehlmann³

et al. 2015

J. High Energ. Phys.

139

570

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We consider F-theory compactifications on genus-one fibered Calabi-Yau manifolds with their fibers realized as hypersurfaces in the toric varieties associated to the 16 reflexive 2D polyhedra. We present a base-independent analysis of the codimension one, two and three singularities of these fibrations. We use these geometric results to determine the gauge groups, matter representations, 6D matter multiplicities and 4D Yukawa couplings of the corresponding effective theories. All these theories have a non-trivial gauge group and matter content. We explore the network of Higgsings relating these theories. Such Higgsings geometrically correspond to extremal transitions induced by blow-ups in the 2D toric varieties. We recover the 6D effective theories of all 16 toric hypersurface fibrations by repeatedly Higgsing the theories that exhibit Mordell-Weil torsion. We find that the three Calabi-Yau manifolds without section, whose fibers are given by the toric hypersurfaces in P 2 , P 1 × P 1 and the recently studied P 2 (1, 1, 2), yield F-theory realizations of SUGRA theories with discrete gauge groups Z 3 , Z 2 and Z 4 . This opens up a whole new arena for model building with discrete global symmetries in F-theory. In these three manifolds, we also find codimension two I 2 -fibers supporting matter charged only under these discrete gauge groups. Their 6D matter multiplicities are computed employing ideal techniques and the associated Jacobian fibrations. We also show that the Jacobian of the biquadric fibration has one rational section, yielding one U(1)-gauge field in F-theory. Furthermore, the elliptically fibered Calabi-Yau manifold based on dP 1 has a U(1)-gauge field JHEP01 (2015)142 induced by a non-toric rational section. In this model, we find the first F-theory realization of matter with U(1)-charge q = 3.

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Section: Polyhedronmentioning

confidence: 99%

Section: Polyhedronmentioning

confidence: 99%

See 1 more Smart Citation

F-theory on all toric hypersurface fibrations and its Higgs branches

Klevers

Pena²,

Oehlmann³

et al. 2015

J. High Energ. Phys.

139

570

View full text Add to dashboard Cite

show abstract

“…, 10. We can use [58] to obtain the Jacobian corresponding to (129). By using the corresponding maps of theŝ i to f and g or by noting that f and g are sections of K −4…”

Section: Scps In Toric Modelsmentioning

confidence: 99%

Global Tensor‐Matter Transitions in F‐Theory

Dierigl

Oehlmann

Ruehle

2018

Fortschritte der Physik

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We use F-theory to study gauge algebra preserving transitions of 6d supergravity theories that are connected by superconformal points. While the vector multiplets remain unchanged, the hyper-and tensor multiplet sectors are modified. In 6d F-theory models, these transitions are realized by tuning the intersection points of two curves, one of them carrying a non-Abelian gauge algebra, to a (4, 6, 12) singularity, followed by a resolution in the base. The six-dimensional supergravity anomaly constraints are strong enough to completely fix the possible non-Abelian representations and to restrict the Abelian charges in the hypermultiplet sector affected by the transition, as we demonstrate for all Lie algebras and their representations. Furthermore, we present several examples of such transitions in torically resolved fibrations. In these smooth models, superconformal points lead to non-flat fibers which correspond to non-toric Kähler deformations of the torus-fibered Calabi-Yau 3-fold geometry.

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“…Consider, for example, the quadratic number fields K such that the curve Y 1 (18) has a K-rational point 1 . As shown in [16,Thm.…”

Section: Introductionmentioning

confidence: 99%

Squarefree parts of polynomial values

Krumm

2016

J. Théor. Nombres Bordeaux

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Given a separable nonconstant polynomial f pxq with integer coefficients, we consider the set S consisting of the squarefree parts of all the rational values of f pxq, and study its behavior modulo primes. Fixing a prime p, we determine necessary and sufficient conditions for S to contain an element divisible by p. Furthermore, we conjecture that if p is large enough, then S contains infinitely many representatives from every nonzero residue class modulo p. The conjecture is proved by elementary means assuming f pxq has degree 1 or 2. If f pxq has degree 3, or if it has degree 4 and has a rational root, the conjecture is shown to follow from the Parity Conjecture for elliptic curves. For polynomials of arbitrary degree, a local analogue of the conjecture is proved using standard results from class field theory, and empirical evidence is given to support the global version of the conjecture.

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Jacobians of Genus One Curves

Cited by 62 publications

References 4 publications

F-theory on all toric hypersurface fibrations and its Higgs branches

F-theory on all toric hypersurface fibrations and its Higgs branches

Global Tensor‐Matter Transitions in F‐Theory

Squarefree parts of polynomial values

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