scholarly journals On the arithmetic of a family of degree - two K3 surfaces

2018 ◽  
Vol 166 (3) ◽  
pp. 523-542 ◽  
Author(s):  
FLORIAN BOUYER ◽  
EDGAR COSTA ◽  
DINO FESTI ◽  
CHRISTOPHER NICHOLLS ◽  
MCKENZIE WEST
Keyword(s):  
Projective Space ◽  
Function Field ◽  
Weighted Projective Space ◽  
Module Structure ◽  
K3 Surface ◽  
Galois Module ◽  
Generic Element ◽  
The Family ◽  
Formula Group ◽  
Image Position

AbstractLet ℙ denote the weighted projective space with weights (1, 1, 1, 3) over the rationals, with coordinates x, y, z and w; let $\mathcal{X}$ be the generic element of the family of surfaces in ℙ given by \begin{equation*} X\colon w^2=x^6+y^6+z^6+tx^2y^2z^2. \end{equation*} The surface $\mathcal{X}$ is a K3 surface over the function field ℚ(t). In this paper, we explicitly compute the geometric Picard lattice of $\mathcal{X}$, together with its Galois module structure, as well as derive more results on the arithmetic of $\mathcal{X}$ and other elements of the family X.

Module invariants and root numbers for quaternion fields of degree 41r

1974 ◽  
Vol 76 (2) ◽  
pp. 393-399 ◽  
Author(s):  
A. Fröhlich
Keyword(s):  
Module Structure ◽  
Integral Group ◽  
Galois Module ◽  
Integral Basis ◽  
Generators And Relations ◽  
Root Numbers ◽  
Normal Number ◽  
Image Position ◽  
Normal Integral Basis ◽  
Free Modules

1. The results. Let l be an odd prime, r ≥ 1, and letbe the quaternion group of order 4lr, as given by generators and relations. Throughout N is a tamely ramified normal number field with Galois group Gal (N/Q) = H (a ‘quaternion field’), and its ring of integers. We are interested in the structure of as a module over the integral group ring ZH. Deriving, first, certain classgroup invariants for locally free ZH-modules, we shall then determine those for the module in terms of the arithmetic invariants of N/Q. When 1 ≡ – 1 (mod 4), this yields again a Galois module interpretation of Artin root numbers quite analogous to that in (2). On the other hand for l ≡ 1 (mode 4), we shall get a weak ‘normal integral basis theorem’. The original impetus for this work came from computations of J. Queyrut, who – in different language – obtained these results in the case l = 3, r = 1 (cf. (7)). The tools, we are using, come from the general theory developed in recent years with such concrete applications in mind, and it is perhaps of interest to see how the various ‘strands’, on root numbers (cf. (3), (4)), on locally free modules (cf. (5)), and on Galois module structure (cf. (6)) are here pulled together. For technical reasons, we shall impose on N the slight further restriction, that l be non-ramified, although our results would remain true without this. Both the statements and the proofs of the theorem depend on ideas contained in (5) and (6). The reader who is prepared to take them for granted should, however, be able to read the present paper independently of those papers.

scholarly journals Galois module structure of holomorphic differentials

2000 ◽  
Vol 62 (3) ◽  
pp. 493-509 ◽  
Author(s):  
Martha Rzedowski-Calderón ◽  
Gabriel Villa-Salvador ◽  
Manohar L. Madan
Keyword(s):  
Rational Function ◽  
Prime Divisor ◽  
Function Field ◽  
Module Structure ◽  
Closed Field ◽  
Galois Module ◽  
Indecomposable Modules ◽  
Algebraically Closed Field ◽  
Rational Function Field ◽  
Galois Module Structure

For a finite cyclic P–extension L/K of a rational function field K = κ(x) over an algebraically closed field κ of characteristic P > 0 such that every ramified prime divisor is fully ramified, we find a basis of the κ[G]-module structure of ωL(0) in terms of indecomposable modules.

Vanishing sums in function fields

1986 ◽  
Vol 100 (3) ◽  
pp. 427-434 ◽  
Author(s):  
W. D. Brownawell ◽  
D. W. Masser
Keyword(s):  
Projective Space ◽  
Function Field ◽  
Function Fields ◽  
Order Group ◽  
Zero Characteristic ◽  
Sum Formula ◽  
Image Position

Let k be a field of zero characteristic, and let F be a function field over k of genus g. We normalize each valuation v on F so that its order group consists of all rational integers, and for elements u1, …, un of F, not all zero, we define the (projective) height asThe sum formula on F shows that this is really a height on the projective space .

scholarly journals Embedding Feynman integral (Calabi-Yau) geometries in weighted projective space

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Jacob L. Bourjaily ◽  
Andrew J. McLeod ◽  
Cristian Vergu ◽  
Matthias Volk ◽  
Matt von Hippel ◽  
...  
Keyword(s):  
Projective Space ◽  
Feynman Integral ◽  
Weighted Projective Space

scholarly journals Singular perturbations of the unicritical polynomials with two parameters

2016 ◽  
Vol 37 (6) ◽  
pp. 1997-2016 ◽  
Author(s):  
YINGQING XIAO ◽  
FEI YANG
Keyword(s):  
Julia Set ◽  
Dynamical Behavior ◽  
Rational Maps ◽  
Topological Properties ◽  
Fatou Set ◽  
The Family ◽  
Image Position ◽  
Two Parameters ◽  
Unicritical Polynomials

In this paper, we study the dynamics of the family of rational maps with two parameters $$\begin{eqnarray}f_{a,b}(z)=z^{n}+\frac{a^{2}}{z^{n}-b}+\frac{a^{2}}{b},\end{eqnarray}$$ where $n\geq 2$ and $a,b\in \mathbb{C}^{\ast }$. We give a characterization of the topological properties of the Julia set and the Fatou set of $f_{a,b}$ according to the dynamical behavior of the orbits of the free critical points.

Resolution of T. Ward's Question and the Israel–Finch Conjecture: Precise Analysis of an Integer Sequence Arising in Dynamics

2014 ◽  
Vol 24 (1) ◽  
pp. 195-215
Author(s):  
JEFFREY GAITHER ◽  
GUY LOUCHARD ◽  
STEPHAN WAGNER ◽  
MARK DANIEL WARD
Keyword(s):  
Weighted Average ◽  
Analytic Number Theory ◽  
Combinatorial Structure ◽  
Asymptotic Growth ◽  
First Order ◽  
Integer Sequence ◽  
Precise Analysis ◽  
Formula Group ◽  
Fourth Author ◽  
Image Position

We analyse the first-order asymptotic growth of \[ a_{n}=\int_{0}^{1}\prod_{j=1}^{n}4\sin^{2}(\pi jx)\, dx. \] The integer an appears as the main term in a weighted average of the number of orbits in a particular quasihyperbolic automorphism of a 2n-torus, which has applications to ergodic and analytic number theory. The combinatorial structure of an is also of interest, as the ‘signed’ number of ways in which 0 can be represented as the sum of ϵjj for −n ≤ j ≤ n (with j ≠ 0), with ϵj ∈ {0, 1}. Our result answers a question of Thomas Ward (no relation to the fourth author) and confirms a conjecture of Robert Israel and Steven Finch.

SHARP WEIGHTED BOUNDS FOR GEOMETRIC MAXIMAL OPERATORS

2016 ◽  
Vol 59 (3) ◽  
pp. 533-547 ◽  
Author(s):  
ADAM OSȨKOWSKI
Keyword(s):  
Maximal Operator ◽  
General Setting ◽  
Maximal Operators ◽  
List Type ◽  
Type Number ◽  
Type Estimate ◽  
Formula Group ◽  
Probability Spaces ◽  
Image Position

AbstractLet $\mathcal{M}$ and G denote, respectively, the maximal operator and the geometric maximal operator associated with the dyadic lattice on $\mathbb{R}^d$. (i)We prove that for any 0 < p < ∞, any weight w on $\mathbb{R}^d$ and any measurable f on $\mathbb{R}^d$, we have Fefferman–Stein-type estimate $$\begin{equation*} ||G(f)||_{L^p(w)}\leq e^{1/p}||f||_{L^p(\mathcal{M}w)}. \end{equation*} $$ For each p, the constant e1/p is the best possible.(ii)We show that for any weight w on $\mathbb{R}^d$ and any measurable f on $\mathbb{R}^d$, $$\begin{equation*} \int_{\mathbb{R}^d} G(f)^{1/\mathcal{M}w}w\mbox{d}x\leq e\int_{\mathbb{R}^d} |f|^{1/w}w\mbox{d}x \end{equation*} $$ and prove that the constant e is optimal. Actually, we establish the above estimates in a more general setting of maximal operators on probability spaces equipped with a tree-like structure.

scholarly journals Minimum Number of Monotone Subsequences of Length 4 in Permutations

2014 ◽  
Vol 24 (4) ◽  
pp. 658-679 ◽  
Author(s):  
JÓZSEF BALOGH ◽  
PING HU ◽  
BERNARD LIDICKÝ ◽  
OLEG PIKHURKO ◽  
BALÁZS UDVARI ◽  
...  
Keyword(s):  
Lower Bound ◽  
Complete Graphs ◽  
Formula Group ◽  
Minimum Number ◽  
Additional Stability ◽  
Edge Colourings ◽  
Image Position

We show that for every sufficiently largen, the number of monotone subsequences of length four in a permutation onnpoints is at least\begin{equation*} \binom{\lfloor{n/3}\rfloor}{4} + \binom{\lfloor{(n+1)/3}\rfloor}{4} + \binom{\lfloor{(n+2)/3}\rfloor}{4}. \end{equation*}Furthermore, we characterize all permutations on [n] that attain this lower bound. The proof uses the flag algebra framework together with some additional stability arguments. This problem is equivalent to some specific type of edge colourings of complete graphs with two colours, where the number of monochromaticK4is minimized. We show that all the extremal colourings must contain monochromaticK4only in one of the two colours. This translates back to permutations, where all the monotone subsequences of length four are all either increasing, or decreasing only.

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