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 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.

scholarly journals ELLIPTIC K3 SURFACES ASSOCIATED WITH THE PRODUCT OF TWO ELLIPTIC CURVES: MORDELL–WEIL LATTICES AND THEIR FIELDS OF DEFINITION

2016 ◽  
Vol 228 ◽  
pp. 124-185 ◽  
Author(s):  
ABHINAV KUMAR ◽  
MASATO KUWATA
Keyword(s):  
Elliptic Curves ◽  
Complex Multiplication ◽  
K3 Surfaces ◽  
Double Cover ◽  
Base Change ◽  
Elliptic Surfaces ◽  
Low Degree ◽  
Weil Group ◽  
Finite Index Subgroup ◽  
Rational Elliptic Surfaces

To a pair of elliptic curves, one can naturally attach two K3 surfaces: the Kummer surface of their product and a double cover of it, called the Inose surface. They have prominently featured in many interesting constructions in algebraic geometry and number theory. There are several more associated elliptic K3 surfaces, obtained through base change of the Inose surface; these have been previously studied by Masato Kuwata. We give an explicit description of the geometric Mordell–Weil groups of each of these elliptic surfaces in the generic case (when the elliptic curves are non-isogenous). In the nongeneric case, we describe a method to calculate explicitly a finite index subgroup of the Mordell–Weil group, which may be saturated to give the full group. Our methods rely on several interesting group actions, the use of rational elliptic surfaces, as well as connections to the geometry of low degree curves on cubic and quartic surfaces. We apply our techniques to compute the full Mordell–Weil group in several examples of arithmetic interest, arising from isogenous elliptic curves with complex multiplication, for which these K3 surfaces are singular.

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 Modularity of nearly ordinary 2-adic residually dihedral Galois representations

2014 ◽  
Vol 150 (8) ◽  
pp. 1235-1346 ◽  
Author(s):  
Patrick B. Allen
Keyword(s):  
Elliptic Curves ◽  
Galois Representations ◽  
Real Field ◽  
Two Dimensional ◽  
Hida Families ◽  
Totally Real ◽  
Totally Real Fields ◽  
Totally Real Field

AbstractWe prove modularity of some two-dimensional,$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}2$-adic Galois representations over a totally real field that are nearly ordinary at all places above$2$and that are residually dihedral. We do this by employing the strategy of Skinner and Wiles, using Hida families, together with the$2$-adic patching method of Khare and Wintenberger. As an application we deduce modularity of some elliptic curves over totally real fields that have good ordinary or multiplicative reduction at places above $2$.

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