On the standard conjecture for a fibre product of three elliptic surfaces with pairwise-disjoint discriminant loci

2019 ◽  
Vol 83 (3) ◽  
pp. 613-653
Author(s):  
S. G. Tankeev
Keyword(s):  
Pairwise Disjoint ◽  
Elliptic Surfaces ◽  
Fibre Product

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 On the Number of Perfect Matchings in Random Lifts

2010 ◽  
Vol 19 (5-6) ◽  
pp. 791-817 ◽  
Author(s):  
CATHERINE GREENHILL ◽  
SVANTE JANSON ◽  
ANDRZEJ RUCIŃSKI
Keyword(s):  
Asymptotic Formula ◽  
Complete Graph ◽  
Pairwise Disjoint ◽  
Perfect Matching ◽  
Perfect Matchings ◽  
Lattice Points ◽  
Laplace’S Method ◽  
Second Moment ◽  
Multiple Dimensions ◽  
Laplace's Method

Let G be a fixed connected multigraph with no loops. A random n-lift of G is obtained by replacing each vertex of G by a set of n vertices (where these sets are pairwise disjoint) and replacing each edge by a randomly chosen perfect matching between the n-sets corresponding to the endpoints of the edge. Let XG be the number of perfect matchings in a random lift of G. We study the distribution of XG in the limit as n tends to infinity, using the small subgraph conditioning method.We present several results including an asymptotic formula for the expectation of XG when G is d-regular, d ≥ 3. The interaction of perfect matchings with short cycles in random lifts of regular multigraphs is also analysed. Partial calculations are performed for the second moment of XG, with full details given for two example multigraphs, including the complete graph K4.To assist in our calculations we provide a theorem for estimating a summation over multiple dimensions using Laplace's method. This result is phrased as a summation over lattice points, and may prove useful in future applications.

scholarly journals The Matching Property of Infinitesimal Isometries on Elliptic Surfaces and Elasticity of Thin Shells

2011 ◽  
Vol 200 (3) ◽  
pp. 1023-1050 ◽  
Author(s):  
Marta Lewicka ◽  
Maria Giovanna Mora ◽  
Mohammad Reza Pakzad
Keyword(s):  
Thin Shells ◽  
Elliptic Surfaces

Extremal rational elliptic surfaces in characteristicp

1991 ◽  
Vol 207 (1) ◽  
pp. 429-437 ◽  
Author(s):  
William E. Lang
Keyword(s):  
Elliptic Surfaces ◽  
Rational Elliptic Surfaces

Generalized reduction theorems for model-theoretic analogs of the class of coanalytic sets

1993 ◽  
Vol 58 (1) ◽  
pp. 81-98
Author(s):  
Shaughan Lavine
Keyword(s):  
Pairwise Disjoint ◽  
Admissible Set ◽  
Effective Version ◽  
Reduction Property ◽  
Uniformization Property ◽  
Weakly Stable

AbstractLet be an admissible set. A sentence of the form is a sentence if φ ∈ (φ is ∨ Φ where Φ is an -r.e. set of sentences from ). A sentence of the form is an , sentence if φ is a sentence. A class of structures is, for example, a ∀1 class if it is the class of models of a ∀1() sentence. Thus ∀1() is a class of classes of structures, and so forth.Let i, be the structure 〈i, <〉, for i > 0. Let Γ be a class of classes of structures. We say that a sequence J1, …, Ji,…, i < ω, of classes of structures is a Γ sequence if Ji ∈ Γ, i < ω, and there is I ∈ Γ such that ∈ Ji, if and only if [],i, where [,] is the disjoint sum. A class Γ of classes of structures has the easy uniformization property if for every Γ sequence J1,…, Ji,…, i < ω, there is a Γ sequence J′t, …, J′i, …, i < ω, such that J′i ⊆ Ji, i < ω, ⋃J′i = ⋃Ji, and the J′i are pairwise disjoint. The easy uniformization property is an effective version of Kuratowski's generalized reduction property that is closely related to Moschovakis's (topological) easy uniformization property.We show over countable structures that ∀1() and ∃2() have the easy uniformization property if is a countable admissible set with an infinite member, that and have the easy uniformization property if α is countable, admissible, and not weakly stable, and that and have the easy uniformization properly. The results proved are more general. The result for answers a question of Vaught(1980).

scholarly journals Rooted $K_4$-Minors

10.37236/3476 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Ruy Fabila-Monroy ◽  
David R. Wood
Keyword(s):  
Planar Graph ◽  
Pairwise Disjoint ◽  
Single Face ◽  
Special Cases ◽  
Connected Subgraphs ◽  
Connected Planar Graph

Let $a,b,c,d$ be four vertices in a graph $G$. A $K_4$ minor rooted at $a,b,c,d$ consists of four pairwise-disjoint pairwise-adjacent connected subgraphs of $G$, respectively containing $a,b,c,d$. We characterise precisely when $G$ contains a $K_4$-minor rooted at $a,b,c,d$ by describing six classes of obstructions, which are the edge-maximal graphs containing no $K_4$-minor rooted at $a,b,c,d$. The following two special cases illustrate the full characterisation: (1) A 4-connected non-planar graph contains a $K_4$-minor rooted at $a,b,c,d$ for every choice of $a,b,c,d$. (2) A 3-connected planar graph contains a $K_4$-minor rooted at $a,b,c,d$ if and only if $a,b,c,d$ are not on a single face.

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