scholarly journals Bounding Tangencies of Sections on Elliptic Surfaces

2020 ◽  
Author(s):  
Douglas Ulmer ◽  
Giancarlo Urzúa
Keyword(s):  
Upper Bound ◽  
Characteristic Zero ◽  
Infinite Order ◽  
Elliptic Surface ◽  
Zero Section ◽  
Elliptic Surfaces

Abstract Given an elliptic surface $\mathcal{E}\to \mathcal{C}$ over a field $k$ of characteristic zero equipped with zero section $O$ and another section $P$ of infinite order, we give a simple and explicit upper bound on the number of points where $O$ is tangent to a multiple of $P$.

scholarly journals The Nef Cone of the Hilbert Scheme of Points on Rational Elliptic Surfaces and the Cone Conjecture

2020 ◽  
pp. 1-12
Author(s):  
John Kopper
Keyword(s):  
Hilbert Scheme ◽  
Elliptic Surface ◽  
Elliptic Surfaces ◽  
Hilbert Scheme Of Points ◽  
Rational Elliptic Surfaces ◽  
Nef Cone ◽  
Rational Elliptic Surface

Abstract We compute the nef cone of the Hilbert scheme of points on a general rational elliptic surface. As a consequence of our computation, we show that the Morrison–Kawamata cone conjecture holds for these nef cones.

Properties of the Invariants of Solvable Lie Algebras

2000 ◽  
Vol 43 (4) ◽  
pp. 459-471 ◽  
Author(s):  
J. C. Ndogmo
Keyword(s):  
Lie Algebras ◽  
Upper Bound ◽  
Characteristic Zero ◽  
Complex Numbers ◽  
Coadjoint Representation ◽  
Base Field ◽  
Invariant Functions ◽  
Solvable Lie Algebras ◽  
Restricted Type

AbstractWe generalize to a field of characteristic zero certain properties of the invariant functions of the coadjoint representation of solvable Lie algebras with abelian nilradicals, previously obtained over the base field ℂ of complex numbers. In particular we determine their number and the restricted type of variables on which they depend. We also determine an upper bound on the maximal number of functionally independent invariants for certain families of solvable Lie algebras with arbitrary nilradicals.

Automorphism groups of rational elliptic surfaces with section and constant J-map

2014 ◽  
Vol 12 (12) ◽  
Author(s):  
Tolga Karayayla
Keyword(s):  
Automorphism Groups ◽  
Zero Section ◽  
Ground Field ◽  
Elliptic Surfaces ◽  
Singular Fibers ◽  
Weil Group ◽  
Biholomorphic Maps ◽  
Rational Elliptic Surfaces ◽  
Fixed Section

AbstractIn this paper, the automorphism groups of relatively minimal rational elliptic surfaces with section which have constant J-maps are classified. The ground field is ℂ. The automorphism group of such a surface β: B → ℙ1, denoted by Au t(B), consists of all biholomorphic maps on the complex manifold B. The group Au t(B) is isomorphic to the semi-direct product MW(B) ⋊ Aut σ (B) of the Mordell-Weil groupMW(B) (the group of sections of B), and the subgroup Aut σ (B) of the automorphisms preserving a fixed section σ of B which is called the zero section on B. The Mordell-Weil group MW(B) is determined by the configuration of singular fibers on the elliptic surface B due to Oguiso and Shioda [9]. In this work, the subgroup Aut σ (B) is determined with respect to the configuration of singular fibers of B. Together with a previous paper [4] where the case with non-constant J-maps was considered, this completes the classification of automorphism groups of relatively minimal rational elliptic surfaces with section.

scholarly journals On the Modularity of Three Calabi–Yau Threefolds With Bad Reduction at 11

2006 ◽  
Vol 49 (2) ◽  
pp. 296-312 ◽  
Author(s):  
Matthias Schütt
Keyword(s):  
Elliptic Curves ◽  
Galois Representations ◽  
Elliptic Surface ◽  
Two Dimensional ◽  
Cohomology Groups ◽  
Elliptic Surfaces ◽  
Fibre Products ◽  
Fibre Product ◽  
Rational Elliptic Surfaces

AbstractThis paper investigates the modularity of three non-rigid Calabi–Yau threefolds with bad reduction at 11. They are constructed as fibre products of rational elliptic surfaces, involving the modular elliptic surface of level 5. Their middle ℓ-adic cohomology groups are shown to split into two-dimensional pieces, all but one of which can be interpreted in terms of elliptic curves. The remaining pieces are associated to newforms of weight 4 and level 22 or 55, respectively. For this purpose, we develop a method by Serre to compare the corresponding two-dimensional 2-adic Galois representations with uneven trace. Eventually this method is also applied to a self fibre product of the Hesse-pencil, relating it to a newform of weight 4 and level 27.

scholarly journals Every ternary quintic is a sum of ten fifth powers

2015 ◽  
Vol 25 (04) ◽  
pp. 607-631 ◽  
Author(s):  
Alessandro De Paris
Keyword(s):  
Upper Bound ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Waring Rank

To our knowledge at the time of writing, the maximum Waring rank for the set of all ternary forms of degree d (with coefficients in an algebraically closed field of characteristic zero) is known only for d ≤ 4, and the best upper bound that is known for d = 5 is 12. In this work, we lower the upper bound to 10 (and there are evidences that it is likely sharp).

Pseudofree ℤ/3-actions on elliptic surfaces E (4)

2008 ◽  
Vol 45 (2) ◽  
pp. 273-284 ◽  
Author(s):  
Hongxia Li ◽  
Ximin Liu
Keyword(s):  
Cyclic Group ◽  
Elliptic Surface ◽  
Elliptic Surfaces ◽  
Locally Linear

In this paper we give a weak classification of locally linear pseudofree actions of the cyclic group of order 3 on an elliptic surface E (4), and prove the existence of such actions which can not be realized as smooth actions on the standard smooth E (4).

scholarly journals CLASS NUMBERS VIA 3-ISOGENIES AND ELLIPTIC SURFACES

2012 ◽  
Vol 09 (01) ◽  
pp. 125-137
Author(s):  
CAM McLEMAN ◽  
DUSTIN MOODY
Keyword(s):  
Elliptic Curves ◽  
Number Field ◽  
Class Number ◽  
Elliptic Surface ◽  
Class Numbers ◽  
Character Sum ◽  
Elliptic Surfaces ◽  
Higher Dimensional ◽  
Imaginary Number

We show that a character sum attached to a family of 3-isogenies defined on the fibers of a certain elliptic surface over 𝔽p relates to the class number of the quadratic imaginary number field [Formula: see text]. In this sense, this provides a higher-dimensional analog of some recent class number formulas associated to 2-isogenies of elliptic curves.

scholarly journals DONALDSON–THOMAS INVARIANTS OF LOCAL ELLIPTIC SURFACES VIA THE TOPOLOGICAL VERTEX

2019 ◽  
Vol 7 ◽  
Author(s):  
JIM BRYAN ◽  
MARTIJN KOOL
Keyword(s):  
Partition Function ◽  
Elliptic Surface ◽  
Computational Technique ◽  
Partition Functions ◽  
K3 Surface ◽  
Jacobi Forms ◽  
Topological Vertex ◽  
Elliptic Surfaces ◽  
Special Case ◽  
Product Formulas

We compute the Donaldson–Thomas invariants of a local elliptic surface with section. We introduce a new computational technique which is a mixture of motivic and toric methods. This allows us to write the partition function for the invariants in terms of the topological vertex. Utilizing identities for the topological vertex proved in Bryan et al. [‘Trace identities for the topological vertex’, Selecta Math. (N.S.)24 (2) (2018), 1527–1548, arXiv:math/1603.05271], we derive product formulas for the partition functions. The connected version of the partition function is written in terms of Jacobi forms. In the special case where the elliptic surface is a K3 surface, we get a derivation of the Katz–Klemm–Vafa formula for primitive curve classes which is independent of the computation of Kawai–Yoshioka.

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