64,176 research outputs found
On a theorem of Kummer
No full textAbstractThe author gives a simple proof of a theorem of Kummer. Let q denote an odd prime, e = (q − 1)2 and let fe(x) denote the polynomial with leading coefficient 1 whose roots are 2 cos (2mπq) with 1 ≤ m ≤ e. Then all prime divisors p of the polynomial fe(x) have the form p ≡ ± 1 (rmmod q), except for p = q
Jahrbuch des öffentlichen Rechts der Gegenwart, vol. 16, p. 204. M. Kummer, Das urheberrechtlich schutzbare Werk
No full textJahrbuch des öffentlichen Rechts der Gegenwart, vol. 16, p. 204. M. Kummer, Das urheberrechtlich schutzbare Werk. In: Revue internationale de droit comparé. Vol. 22 N°1, Janvier-mars 1970. pp. 204-205
Jahrbuch des öffentlichen Rechts der Gegenwart, vol. 16, p. 204. M. Kummer, Das urheberrechtlich schutzbare Werk
No full textJahrbuch des öffentlichen Rechts der Gegenwart, vol. 16, p. 204. M. Kummer, Das urheberrechtlich schutzbare Werk. In: Revue internationale de droit comparé. Vol. 22 N°1, Janvier-mars 1970. pp. 204-205
Syzygies of Kummer varieties
No full textWe study syzygies of Kummer varieties proving that their behavior is half of
the abelian varieties case. Namely, an -th power of an ample line bundle on
a Kummer variety satisfies the Green-Lazarsfeld property , if .Comment: 15 pages, final version. To appear in Trans. Amer. Math. So
Ryhiner-Kartensammlung / 42 Carte von Tibet
No full textnach den neuesten Nachrichten entworfenTitel oben MitteNullmeridian: LondonUrsprungswerk: "Neue Beiträge zur Völker- und Länderkunde", Band 3, hrsg. von M. C. Sprengel und G. Forster (Leipzig : Kummer, 1790
Moduli spaces of abstract and embedded Kummer varieties
No full textIn this paper, we investigate the construction of two moduli stacks of Kummer varieties. The first one is the stack of abstract Kummer varieties and the second one is the stack of embedded Kummer varieties. We will prove that is a Deligne-Mumford stack and its coarse moduli space is isomorphic to , the coarse moduli space of principally polarized abelian varieties of dimension . On the other hand we give a modular family of embedded Kummer varieties embedded in P^{2^g-1}\timesP^{2^g-1}, meaning that every geometric fiber of this family is an embedded Kummer variety and every isomorphic class of such varieties appears at least once as the class of a fiber.
As a consequence, we construct the coarse moduli space of embedded Kummer surfaces and prove that it is obtained from by contracting the locus swept by a particular linear equivalence class of curves. We conjecture that this is a general fact: \boldsymbo{K}^{em}_g could be obtained from via a contraction for all g>1
Normal integral bases and tameness conditions for Kummer extensions
No full textIn this paper we present a detailed analysis of some properties of a general tamely ramified Kummer extension of number fields L/K. Our main achievement is a criterion for the existence of a normal integral basis for a general Kummer extension, which generalizes the existing results (Thm. 11). Our approach also allows us to explicitly describe the Steinitz class of L/K and we get an easy criterion for this class to be trivial. In the second part of the paper we restrict to the particular case of tame Kummer extensions Q(ζm , m√a1 , . . . , m√an )/Q(ζm ), with ai ∈ Z. We prove that these extensions always have trivial Steinitz classes. We also give sufficient condition for the existence of a normal integral basis for such extensions and an example showing that such conditions are sharp in the general case. An accurate study of the ramification produces explicit necessary and sufficient conditions on the elements ai for the extension to be tame
Carlitz-Kummer function fields
No full textAbstractLet Fq be a finite field with q elements where q is a power of a prime p. Also, let M be any polynomial in Fq[x] and let kM be the Mth cyclotomic function field. If K is any finite extension field of kM, then we define Carlitz-Kummer extensions of K as an analog of the Kummer extensions of algebraic number fields. More specifically, let z ∈ K. Then the Carlitz-Kummer extension KM,z is defined as the splitting field over K of the polynomial uM − z. The Carlitz-Kummer extension KM,z is a simple, separable, Abelian extension whose degree is a power of the characteristic. Our main results are on the factorization of primes in Carlitz-Kummer extensions. Let q be any prime divisor of K and let Q be any prime divisor of KM,z that lies over q. We show that q can ramify in KM,z only if q is an infinite prime, q divides M, or q divides the denominator of z. Finally, we show that the factorization of q in KM,z in determined by certain congruence conditions on the polynomial uM − z modulo powers of the prime q
Explicit Serre weights for GL_2 via Kummer theory
No full textWe give an explicit formulation of the weight part of Serre's conjecture for GL_2 using Kummer theory. This avoids any reference to p-adic Hodge theory. The key inputs are a description of the reduction modulo p of crystalline extensions in terms of certain "G_K-Artin-Scheier cocycles" and a result of Abrashkin which describes these cocycles in terms of Kummer theory. An alternative explicit formulation in terms of local class field theory was previously given by Dembele-Diamond-Roberts in the unramified case and by the second author in general. We show that the description of Dembele-Diamond-Roberts can be recovered directly from ours using the explicit reciprocity laws of Brueckner-Shaferevich-Vostokov. These calculations illustrate how our use of Kummer theory eliminates certain combinatorial complications appearing in these two papers
M. Kummer, Anwendungsbereich und Schutzgut der privatrechtlichen Rechtssàtze gcgen unlautern und gegen freiheitsbeschrànkenden Wettbewerb
No full textM. Kummer, Anwendungsbereich und Schutzgut der privatrechtlichen Rechtssàtze gcgen unlautern und gegen freiheitsbeschrànkenden Wettbewerb. In: Revue internationale de droit comparé. Vol. 13 N°4, Octobre-décembre 1961. p. 885
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