By Duke W., Tschinkel Y. (eds.)

ISBN-10: 0821843079

ISBN-13: 9780821843079

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 venerated the a hundred and fiftieth anniversary of the loss of life of C.-F. Gauss and the 2 hundredth anniversary of the beginning of J.-L. Dirichlet. the quantity starts off with a definitive precis of the existence and paintings of Dirichlet and keeps with 13 papers through major specialists on study themes of present curiosity in quantity thought that have been without delay encouraged through Gauss and Dirichlet. one of the issues are the distribution of primes (long mathematics progressions of primes and small gaps among primes), category teams of binary quadratic kinds, numerous facets of the speculation of $L$-functions, the idea of modular varieties, and the examine of rational and crucial recommendations to polynomial equations in numerous variables. Titles during this sequence are co-published with the Clay arithmetic Institute (Cambridge, MA).

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**Sample text**

Elliptic functions according to Eisenstein and Kronecker. : Zetafunktionen und quadratische K¨ orper. : Springer, 1981 [Wa] ¨ t Mu ¨nster, Einsteinstr. 62, 48149 Mathematisches Institut, Westf. D. Browning Abstract. This paper surveys recent progress towards the Manin conjecture for (singular and non-singular) del Pezzo surfaces. To illustrate some of the techniques available, an upper bound of the expected order of magnitude is established for a singular del Pezzo surface of degree four. 1. Introduction A fundamental theme in mathematics is the study of integer or rational points on algebraic varieties.

The law of quadratic reciprocity implies that n → D is a n so-called primitive Dirichlet character mod |D|. It is known that any primitive real Dirichlet character is one of the characters D· for some fundamental discriminant D. In terms of generating functions the last sum formula means, supposing that D < 0, h (fj (x, y))−s = wζ(s)L s, j=1 (x,y)=(0,0) D · with w = 2, 4 or 6 as D < −4, D = −4 or D = −3, respectively. Using geometric considerations, Dirichlet deduces by a limiting process the ﬁrst of his class number formulae w |D| D L 1, if D < 0 , 2π · √ (4) h(D) = D D if D > 0 .

A discussion of these more general versions of the conjecture can be found in the survey of Tschinkel [Tsc03]. The purpose of this note is to give an overview of our progress in the case that V is a suitable Fano variety of dimension 2. Let d 3. A non-singular surface S ⊂ Pd of degree d, with very ample anticanonical divisor −KS , is known as a del Pezzo surface of degree d. Their geometry has been expounded by Manin [Man86], for example. It is well-known that such surfaces S arise either as the quadratic Veronese embedding of a quadric in P3 , which is a del Pezzo surface of degree 8 in P8 (isomorphic to P1 × P1 ), or as the blow-up of P2 at 9 − d points in general position, in which case the degree of S satisﬁes 3 d 9.

### Analytic number theory. A tribute to Gauss and Dirichlet by Duke W., Tschinkel Y. (eds.)

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