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K3 and Enriques Surfaces Read chapter Read first chapter

2013 | OriginalPaper | Chapter

K 1 ind of Elliptically Fibered K3 Surfaces: A Tale of Two Cycles

Author : Matt Kerr

Published in: Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds

Publisher: Springer New York

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Abstract

We discuss two approaches to the computation of transcendental invariants of indecomposable algebraic K 1 classes. Both the construction of the classes and the evaluation of the regulator map are based on the elliptic fibration structure on the family of K3 surfaces. The first computation involves a Tauberian lemma, while the second produces a “Maass form with two poles”.

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previous chapter On a Family of K3 Surfaces with Symmetry
next chapter A Note About Special Cycles on Moduli Spaces of K3 Surfaces
Footnotes
1
Though presented on Jacobians of genus 2 curves, these can be transferred (using a correspondence) to the corresponding family of Kummer K3 surfaces.
 
2
In the setup of [2], π is induced by slicing Δ horizontally. This suggests a significant generalization of the computation carried out in this section. Also note that this particular π has Mordell–Weil rank 1.
 
3
This degeneration is not semistable, which can be fixed by blowing up the components of X 0 at a few points; this need not trouble us.
 
4
Cf. (for example) [13, (6.15)ff].
 
5
Where G is incorrectly identified as a transcendental number; that is the conjecture, but its irrationality is still unproven. This has no bearing on nontriviality of 16iG modulo \(\mathbb{Q}(2)\).
 
6
We will usually drop the subscript \(\underline{\lambda }\).
 
7
Since \(\left \vert \log \left \vert \frac{\mathsf{z}+i} {\mathsf{z}-i}\right \vert \right \vert < C\left \vert {\mathsf{z}}^{2} - 1\right \vert \) for z near ± 1, this clearly converges.
 
8
It is possible, but tedious, to instead check the asymptotics for Ψ at 0, ± 1, 2, ∞ directly from the formula (7), cf. the appendix to Sect. 6 of [9].
 
9
To see this, examine the function (n − 1) − nβ + β n .
 
Literature
1.
M. Asakura, On dlog image of K 2 of elliptic surface minus singular fibers, Preprint, Available at arXiv:math/0511190v4
2.
A. Avram, M. Kreuzer, M. Mendelberg, H. Skarke, Searching for K3 fibrations. Nucl. Phys. B494, 567–589 (1997)CrossRef
3.
A. Beilinson, Higher regulators and values of L-functions. J. Sov. Math. 30, 2036–2070 (1985)MATHCrossRef
4.
A. Beilinson, Higher regulators of modular curves, in Applications of Algebraic K-Theory to Algebraic Geometry and Number Theory, Boulder, CO, 1983. Contemporary Mathematics, vol. 55 (AMS, Providence, 1986), pp. 1–34
5.
F. Beukers, C. Peters, A family of K3 surfaces and ζ(3). J. Reine Angew. Math. 351, 42–54 (1984)MathSciNetMATH
6.
S. Bloch, The moving lemma for higher Chow groups. J. Algebr. Geom. 3(3), 537–568 (1993)MathSciNet
7.
P. Brosnan, G. Pearlstein, C. Schnell, The locus of Hodge classes in an admissible variation of mixed Hodge structure. C. R. Math. Acad. Sci. Paris 348(11–12), 657–660 (2010)MathSciNetMATHCrossRef
8.
9.
X. Chen, C. Doran, M. Kerr, J. Lewis, Normal functions, Picard-Fuchs equations, and elliptic fibrations on K3 surfaces, Preprint, Available at arXiv:1108.2223v2
10.
11.
A. Collino, Griffiths’s infinitesimal invariant and higher K-theory on hyperelliptic Jacobians. J. Algebr. Geom. 6(3), 393–415 (1997)MathSciNetMATH
12.
P. del Angel, S. Müller-Stach, Differential equations associated to families of algebraic cycles. Ann. l’institut Fourier 58(6), 2075–2085 (2008)MATHCrossRef
13.
14.
15.
M. Kerr, A regulator formula for Milnor K-groups. K-Theory 29(3), 175–210 (2003)
16.
17.
J. Lewis, Transcendental Methods in the Study of Algebraic Cycles with a Special Emphasis on Calabi-Yau Varieties, in Arithmetic and Geometry of K3 surfaces and Calabi-Yau threefolds (this volume), (Springer, 2013)
18.
A. Mellit, Higher Green’s functions for modular forms. The University of Bonn, Ph.D. Thesis, 2008, Available at arXiv:0804.3184
19.
J. Neukirch, The Beilinson conjecture for algebraic number fields, in Beilinson’s Conjectures on Special Values of L-Functions, ed. by Rapoport et al. (Academic, New York, 1988)
20.
C. Peters, Monodromy and Picard-Fuchs equations for families of K3 surfaces and elliptic curves. Ann. ENS 19, 583–607 (1986)MATH
21.
C. Peters, J. Stienstra, A pencil of K3 surfaces related to Apéry’s recurrence for ζ(3) and Fermi surfaces for potential zero, in Arithmetic of Complex Manifolds, Erlangen, 1988. Springer Lecture Notes in Mathematics, vol. 1399 (1989), pp. 110–127MathSciNetCrossRef
22.
F. Rohsiepe, Fibration structure in toric Calabi-Yau fourfolds [aXiv:hep-th/0502138]
23.
S. Shokurov, Holomorphic forms of highest degree on Kuga’s modular varieties (Russian) Mat. Sb. (N.S.) 101 (143), (1), 131–157, 160 (1976)
24.
Metadata
Title
K 1 ind of Elliptically Fibered K3 Surfaces: A Tale of Two Cycles
Author
Matt Kerr
Copyright Year
2013
Publisher
Springer New York
Book
Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds

Print ISBN: 978-1-4614-6402-0

Electronic ISBN: 978-1-4614-6403-7

Copyright Year: 2013

DOI
https://doi.org/10.1007/978-1-4614-6403-7_13

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