By William Duke, Yuri Tschinkel

Articles during this quantity are in accordance with talks given on the Gauss-Dirichlet convention held in GÃ¶ttingen on June 20-24, 2005. The convention honored the a hundred and fiftieth anniversary of the loss of life of C.-F. Gauss and the 2 hundredth anniversary of the delivery of J.-L. Dirichlet. the amount starts with a definitive precis of the existence and paintings of Dirichlet and keeps with 13 papers through prime specialists on examine subject matters of present curiosity in quantity conception that have been at once inspired by way of Gauss and Dirichlet. one of the themes are the distribution of primes (long mathematics progressions of primes and small gaps among primes), classification teams of binary quadratic kinds, quite a few elements of the speculation of $L$-functions, the speculation of modular kinds, and the examine of rational and imperative options to polynomial equations in numerous variables. Titles during this sequence are co-published with the Clay arithmetic Institute (Cambridge, MA).

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2], [H], [He], No. 4], [MC], [Si], [Wei]). D. Dirichlet’s Unit Theorem. An algebraic integer is, by deﬁnition, a zero of a monic polynomial with integral coeﬃcients. 1], pp. 619–623), but his notion of what Hilbert later called the ring of algebraic integers in a number ﬁeld remained somewhat imperfect, since for an algebraic integer ϑ he considered only the set Z[ϑ] as the ring of integers of Q(ϑ). 1], pp. 639–644). 2]). In the more familiar modern notation, the unit theorem describes the structure of the group of units as follows: Let K be an algebraic number ﬁeld with r1 real and 2r2 complex (non-real) embeddings and ring of integers oK .

By C. Schilling and I. G. Jacob Jacobi. : Gauss, Dirichlet, and the law of biquadratic reciprocity. Math. Intell. 10, No. : Gauß zum Ged¨ achtnis. R. Wohlwend, Schaan/Liechtenstein, 1981 [Sch] Scharlau, W. ): Richard Dedekind, 1831/1981. Eine W¨ urdigung zu seinem 150. Geburtstag. G. Lejeune Dirichlet. Biographische Mitteilungen zum Werdegang Dirichlets. NTM, Schriftenr. Gesch. Naturwiss. Tech. Med. : Die Erinnerungen von Karl Emil Gruhl (1833–1917) an sein Studium der ¨ Mathematik und Physik in Berlin (1853–1856).

Throughout this work we shall work with the height metrized by the choice of norm |x| := max0 i n |xi |. Given a suitable Zariski open subset U ⊆ V , the goal is then to study the quantity (1) NU,H (B) := #{x ∈ U (Q) : H(x) B}, as B → ∞. It is natural to question whether the asymptotic behaviour of NU,H (B) can be related to the geometry of V , for suitable open subsets U ⊆ V . Around 1989 Manin initiated a program to do exactly this for varieties with ample anticanonical divisor [FMT89]. Suppose for simplicity that V ⊂ Pn is a non-singular complete intersection, with V = W1 ∩ · · · ∩ Wt for hypersurfaces Wi ⊂ Pn of degree di .