94 research outputs found
Computing the Arithmetic Genus of Hilbert Modular Fourfolds
The Hilbert modular fourfold determined by the totally real quartic number field k is a desingularization of a natural compactification of the quotient space Gamma(k)\H-4, where Gamma(k) = PSL2(O-k) acts on H-4 by fractional linear transformations via the four embeddings of k into R. The arithmetic genus, equal to one plus the dimension of the space of Hilbert modular cusp forms of weight (2, 2, 2, 2), is a birational invariant useful in the classification of these varieties. In this work, we describe an algorithm allowing for the automated computation of the arithmetic genus and give sample results
Computing the Arithmetic Genus of Hilbert Modular Fourfolds
The Hilbert modular fourfold determined by the totally real quartic number field k is a desingularization of a natural compactification of the quotient space Gamma(k)\H-4, where Gamma(k) = PSL2(O-k) acts on H-4 by fractional linear transformations via the four embeddings of k into R. The arithmetic genus, equal to one plus the dimension of the space of Hilbert modular cusp forms of weight (2, 2, 2, 2), is a birational invariant useful in the classification of these varieties. In this work, we describe an algorithm allowing for the automated computation of the arithmetic genus and give sample results
An intriguing hyperelliptic Shimura curve quotient of genus 16
Let be the maximal totally real subfield of , the cyclotomic field of nd roots of unity. Let be the quaternion algebra over ramified exactly at the unique prime above and 7 of the real places of . Let be a maximal order in , and the Shimura curve attached to . Let , where is the unique Atkin-Lehner involution on . We show that the curve has several striking features. First, it is a hyperelliptic curve of genus , whose hyperelliptic involution is exceptional. Second, there are Weierstrass points on , and exactly half of these points are CM points; they are defined over the Hilbert class field of the unique CM extension of class number contained in , the cyclotomic field of th roots of unity. Third, the normal closure of the field of -torsion of the Jacobian of is the Harbater field , the unique Galois number field unramified outside and , with Galois group . In fact, the Jacobian has the remarkable property that each of its simple factors has a -torsion field whose normal closure is the field . Finally, and perhaps the most striking fact about is that it is also hyperelliptic over
Landscaping with fluxes and the E8 Yukawa Point in F-theory
Integrality in the Hodge theory of Calabi-Yau fourfolds is essential to find
the vacuum structure and the anomaly cancellation mechanism of four dimensional
F-theory compactifications. We use the Griffiths-Frobenius geometry and
homological mirror symmetry to fix the integral monodromy basis in the
primitive horizontal subspace of Calabi-Yau fourfolds. The Gamma class and
supersymmetric localization calculations in the 2d gauged linear sigma model on
the hemisphere are used to check and extend this method. The result allows us
to study the superpotential and the Weil-Petersson metric and an associated tt*
structure over the full complex moduli space of compact fourfolds for the first
time. We show that integral fluxes can drive the theory to N=1 supersymmetric
vacua at orbifold points and argue that fluxes can be chosen that fix the
complex moduli of F-theory compactifications at gauge enhancements including
such with U(1) factors. Given the mechanism it is natural to start with the
most generic complex structure families of elliptic Calabi-Yau 4-fold
fibrations over a given base. We classify these families in toric ambient
spaces and among them the ones with heterotic duals. The method also applies to
the creating of matter and Yukawa structures in F-theory. We construct two
SU(5) models in F-theory with a Yukawa point that have a point on the base with
an -type singularity on the fiber and explore their embeddings in the
global models. The explicit resolution of the singularity introduce a higher
dimensional fiber and leads to novel features.Comment: 150 page
Moduli spaces and Modular forms (hybrid meeting)
The relation between moduli spaces and modular forms goes back
to the theory of elliptic curves. On the one hand both topics
experience their own growth and development, but from time to
time new unexpected links show up and usually these lead to progress on both
sides. One subject where there has been a lot of progress concerns
the moduli of abelian varieties and K3 surfaces and especially
on the Kodaira dimension of these spaces. The idea of the workshop
was to bring together the experts of the two areas in the hope that
discussion, interaction and lectures would spur the development
of new ideas. The lectures of the workshop gave ample evidence
of the interaction and provided opportunities for further interaction.
Besides the lectures participants interacted via zoom in smaller groups
Intersection theory on Shimura surfaces
Kudla has proposed a general program to relate arithmetic intersection
multiplicities of special cycles on Shimura varieties to Fourier coefficients
of Eisenstein series. The lowest dimensional case, in which one intersects two
codimension one cycles on the integral model of a Shimura curve, has been
completed by Kudla-Rapoport-Yang. In the present paper we prove results in a
higher dimensional setting. On the integral model of a Shimura surface we
consider the intersection of a Shimura curve with a codimension two cycle of
complex multiplication points, and relate the intersection to certain cycles
classes constructed by Kudla-Rapoport-Yang. As a corollary we deduce that our
intersection multiplicities appear as Fourier coefficients of a Hilbert modular
form of half-integral weight
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