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Einleitung. Eine klassische Konstruktion aus der algebraischen Zahlentheorie ist folgende: Zu jedem algebraischen Zahlkörper K kann man ein sogenanntes System idealer Zahlen S zuordnen, welches eine Untergruppe der multiplikativen Gruppe ℂ* der komplexen Zahlen ist derart, daß die Faktorgruppe S/K* in kanonischer Weise isomorph zu der Klassengruppe von K ist. Diese Konstruktion geht auf Hecke [5] zurück und hat folgende wichtige Eigenschaft, die auch bei dem Hilbertschen Klassenkörper zu K vorkommt:...
We give a geometric interpretation of an arithmetic rule to generate explicit formulas for the Fourier coefficients of elliptic modular forms and their associated Jacobi forms. We discuss applications of these formulas and derive as an example a criterion similar to Tunnel's criterion for a number to be a congruent number.
A result on the orders of the reductions of an element of the group of S-units of a number field is obtained by investigating three height functions for groups of S-units of number fields which are analogous to local, global and canonical height functions for elliptic curves.
We compare general inequalities between invariants of number fields and invariants of elliptic curves over number fields. On the number field side, we remark that there is only a finite number of non-CM number fields with bounded regulator. On the elliptic curve side, assuming the height conjecture of Lang and Silverman, we obtain a Northcott property for the regulator on the set of elliptic curves with dense rational points over a number field. This amounts to say that the arithmetic of CM fields...
We study the behavior of canonical height functions , associated to rational maps f, on totally p-adic fields. In particular, we prove that there is a gap between zero and the next smallest value of on the maximal totally p-adic field if the map f has at least one periodic point not contained in this field. As an application we prove that there is no infinite subset X in the compositum of all number fields of degree at most d such that f(X) = X for some non-linear polynomial f. This answers a...
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