Printfusion E-books

Number Theory

Get Frobenius Distributions in GL2-Extensions: Distribution of PDF

By Serge Lang, Hale Trotter

ISBN-10: 354007550X

ISBN-13: 9783540075509

ISBN-10: 3540380949

ISBN-13: 9783540380948

Show description

Read or Download Frobenius Distributions in GL2-Extensions: Distribution of Frobenius Automorphisms in GL2-Extensions of the Rational Numbers PDF

Similar number theory books

Download PDF by Kristina Reiss: Basiswissen Zahlentheorie: Eine Einfuhrung in Zahlen und

Kenntnisse uber den Aufbau des Zahlsystems und uber elementare zahlentheoretische Prinzipien gehoren zum unverzichtbaren Grundwissen in der Mathematik. Das vorliegende Buch spannt den Bogen vom Rechnen mit naturlichen Zahlen uber Teilbarkeitseigenschaften und Kongruenzbetrachtungen bis hin zu zahlentheoretischen Funktionen und Anwendungen wie der Kryptographie und Zahlencodierung.

Read e-book online Selected Chapters of Geometry, Analysis and Number Theory: PDF

This booklet specializes in a few very important classical elements of Geometry, research and quantity conception. the fabric is split into ten chapters, together with new advances on triangle or tetrahedral inequalities; precise sequences and sequence of genuine numbers; numerous algebraic or analytic inequalities with purposes; designated functions(as Euler gamma and beta services) and certain capacity( because the logarithmic, identric, or Seiffert's mean); mathematics services and mathematics inequalities with connections to excellent numbers or similar fields; and plenty of extra.

Extra info for Frobenius Distributions in GL2-Extensions: Distribution of Frobenius Automorphisms in GL2-Extensions of the Rational Numbers

Example text

There are others of a slightly more abelian nature as follows. Cf. Shimura [Sh]. Let Mg = Mat2(Z~), and since g is fixed, write M = M I. Let V n = I + gnMg be the group of matrices --- I mod l n. The map X I " l+lnx induces an isomorphism M/{~! ' Vn/Vn+ 1 9 Consider the case when ~ ~ 3, The map raising to the g-th power gives an is omor phis m Vn/Vn+ I ~ Vn+i/Vn+ 2 9 Let U 1 be a closed subgroup of V 1 and let U n = U I fIV n. We have an embedding U1/U 2 9 Vl/V 2 ~ M/L~ . We may refer to U1/U 2 as the a s s o c i a t e d vector space to U 1 at level 1, or as the tangent space to U 1.

We have HL = H2q • H GL2(Zg) e~2,s, i i and ( S 2 q : H 2 q ) = l or S. Proof. The group Gs/H 5 is the Galois group of a cyclic extension of the rationals which is ramified only at 11 or possibly 5, and of degree 1 or 5. On the other hand, GL/H L is the Galois group of an abelian extension which is unramified at 5. Hence the common subfield K 5 n K L must be the 5-part of the field of 11-th roots of unity, Q(g-11) , or it must be trivial. This proves the 1emma, while also providing additional information.

To see this, it suffices to observe that there is at least one element in the stable group G(M) with trace O. Such an element is given by complex conjugation, for instance, so that FM(O) ~ 0 for a stabilizing M. On the other hand, for other values of t o ~ 0 it may happen that the constant C(to, p) is equal to O. A typical reason for this is the presence of rational points on the elliptic curve, which may impose congruence conditions on the traces of Frobenius. Remark 2. Let A, B be two elliptic curves over the rationals, without complex multiplication.

Download PDF sample

Frobenius Distributions in GL2-Extensions: Distribution of Frobenius Automorphisms in GL2-Extensions of the Rational Numbers by Serge Lang, Hale Trotter


by Donald
4.0

Rated 4.05 of 5 – based on 35 votes