198,283 research outputs found
Leben und Denkmal des Herrn M. Christian Ernst Schmidts, weiland Stiftssuperintendents und Konsistorialrassessors zu Merseburg / [Christian Gottlieb Schmidt]
No full textDie Vorlage enth. insgesamt 2 WerkeErm. Verf. nach Angaben aus dem Alphabet. Katalog der ULB Sachsen-Anhalt: Christian Gottlieb SchmidtAutopsie nach dem Exemplar der ULB Sachsen-AnhaltVorlageform des Erscheinungsvermerks: Leipzig, bey Paul Gotthelf Kummer 178
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
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
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
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
On Computations in Kummer Extensions
No full textAbstractLet k be an algebraic number field containing a primitive m th root of unity. An extension K=k([m ] μ) of k with μ∈k is called a Kummer extension. These extensions have been studied extensively in the past and they play an important role in class field theory. Recently many new algorithms dealing with Kummer extensions emerged. In this paper we will give algorithms to solve two problems, which are of particular interest; the computation of the relative discriminant dK/k and the computation of Hilbert norm symbols
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
Approximation of Analytic Functions by Kummer Functions
No full textWe solve the inhomogeneous Kummer differential equation of the form xy′′+(β-x)y′-αy=∑m=0∞amxm and apply this result to the proof of a local Hyers-Ulam stability of the Kummer differential equation in a special class of analytic functions
Explicit Kummer theory for the rational numbers
No full textpeer reviewedLet G be a finitely generated multiplicative subgroup of Q* having rank r. The ratio between n^r and the Kummer degree [Q(\zeta_m,\sqrt[n]{G}) : Q(\zeta_m)], where n divides m, is bounded independently of n and m. We prove that there exist integers m_0, n_0 such that the above ratio depends only on G, \gcd(m,m_0), and \gcd(n,n_0). Our results are very explicit and they yield an algorithm that provides formulas for all the above Kummer degrees (the formulas involve a finite case distinction)
- …
