In this paper we inspect from closer the local and global points of the twists of the Klein quartic. For the local ones we use geometric arguments, while for the global ones we strongly use the ...modular interpretation of the twists. The main result is providing families with (conjecturally infinitely many) twists of the Klein quartic that at counter-examples to the Hasse principle.
The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many ...important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students.
We set up a general framework to study Tate cohomology groups of Galois modules along $\mathbb{Z}_p$-extensions of number fields. Under suitable assumptions on the Galois modules, we establish the ...existence of a five-term exact sequence in a certain quotient category whose objects are simultaneously direct and inverse systems, subject to some compatibility. The exact sequence allows one, in particular, to control the behaviour of the Tate cohomology groups of the units along $\mathbb{Z}_p$-extensions. As an application, we study the growth of class numbers along what we call "fake $\mathbb{Z}_p$-extensions of dihedral type". This study relies on a previous work, where we established a class number formula for dihedral extensions in terms of the cohomology groups of the units.
Here we approach the problem of FLT using the Binomial Theorem and two cases: n even or odd. In this paper we will show that the polynomial (a + b − c)^n can be factored non-trivially iff a^n + b^n = ...c^n . Using this factored form we attain a general solution for FLT.
Building on Bosca’s method, we extend to tame ray class groups the results on capitulation of ideals of a number field by composition with abelian extensions of a subfield first studied by Gras. More ...precisely, for every extension of number fields K/k, where at least one infinite place splits completely, and every squarefree divisor m of K, we prove that there exist infinitely many abelian extensions F/k such that the ray class group mod m of K capitulates in KF. As a consequence we generalize to tame ray class groups the results of Kurihara on triviality of class groups for maximal abelian pro-extensions of totally real number fields.
Nous montrons que la méthode utilisée par Bosca pour faire capituler les classes d’idéaux d’un corps de nombres s’étend aux classes de rayons dans le cas modéré. Plus précisément, nous prouvons que pour toute extension K/k de corps de nombres dans laquelle une au moins des places à l’infini se décompose complètement et tout diviseur m sans facteur carré, il existe une infinité d’extensions abéliennes F/k telles que les classes de rayons modulo m de K capitulent dans KF. Il suit de là que les résultats de Kurihara sur la trivialité des groupes de classes des pro-extensions abéliennes maximales des corps de nombres totalement réels valent encore pour les groupes de classes de rayons dans le cas modéré.
SummaryFrom the fact that the lattice points on the diagonal of an m by n rectangle R are equally spaced, we have deduced that the finest subdivision of R into a regular k by k checkerboard of ...integer-sided tiles gives the greatest common factor (k) and the least common multiple (the total area of k tiles), with the product being the area mn of R.
In this paper, we obtain two analogues of the Sturm bound for modular forms in the function field setting. In the case of mixed characteristic, we prove that any harmonic cochain is uniquely ...determined by an explicit finite number of its first Fourier coefficients where our bound is much smaller than the ones in the literature. A similar bound is derived for generators of the Hecke algebra on harmonic cochains. As an application, we present a computational criterion for checking whether two elliptic curves over the rational function field $\mathbb{F}_{q}(\theta)$ with same conductor are isogenous. In the case of equal characteristic, we also prove that any Drinfeld modular form is uniquely determined by an explicit finite number of its first coefficients in the $t$-expansion.
New results on Bernoulli numbers of higher order Khattou, Chouaib; Bayad, Abdelmejid; Hernane, Mohand Ouamar
The Rocky Mountain journal of mathematics,
2/2022, Letnik:
52, Številka:
1
Journal Article
PDF
