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Gauss sums and algebraic cycles. (English) Zbl 0635.14014

Let L be a function field over a finite field k, C a complete smooth curve over k with function field L, X a complete smooth curve over L, A the Jacobi variety of X, and \({\mathcal X}^ a \)minimal model of X over C. Under assumptions which implies the Tate-Birch-Swinnerton-Dyer conjecture, the author introduces an invariant \(\Delta(\ell)\) in terms of Gauss sums arising from étale and crystalline cohomology of \({\mathcal X}\) and \(\Delta\) provides a factorization of the square root of the determinant in local terms. This leads to the formula \[ e^{2\pi i(- r/8)}(2m)^ r\lim_{s\to 1}\sqrt{\frac{L(A,s)}{(s-1)^ r}}=\prod_{\ell}\frac{[\text{Russian{Sh}}(\ell)]^{1/2}\Delta (\ell)}{[A(L)_{tor}(\ell)]}. \]
Reviewer: K.Lai

MSC:

14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14C99 Cycles and subschemes
14H05 Algebraic functions and function fields in algebraic geometry
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14F20 Étale and other Grothendieck topologies and (co)homologies
11R58 Arithmetic theory of algebraic function fields
14F30 \(p\)-adic cohomology, crystalline cohomology
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References:

[1] S. Abhyankar : Resolution of singularities of arithmetical surfaces . In: Arithmetical Algebraic Geometry , ed. O. F. G. Schilling. Harper and Row, New York (1965). · Zbl 0147.20503
[2] P. Deligne (notes by J.S. Milne ): Hodge cycles on abelian varieties . In: Hodge Cycles, Motives, and Shimura Varieties . Lecture Notes in Math. 900. Springer-Verlag, Berlin, Heidelberg, New York (1982). · Zbl 0537.14006
[3] W.J. Gordon : Linking the conjectures of Artin-Tate and Birch-Swinnerton-Dyer . Compositio Math. 38, 2 (1979) 163-199. · Zbl 0425.14003
[4] A. Grothendieck : Le groupe de Brauer III . In: Dix exposés sur la cohomologie des schémas . North-Holland, Amsterdam (1968). · Zbl 0198.25901
[5] M. Gros : Classes de Chern et classes de cycles en cohomologie logarithmique , These 3^\circ cycle. Universite de Paris-Sud, Centre d’Orsay (1983). · Zbl 0615.14011 · doi:10.24033/msmf.322
[6] P. Hriljac : Heights and Arakelov’s intersection theory . Amer. J. Math. 107, 1 (1986) 23-38. · Zbl 0593.14004 · doi:10.2307/2374455
[7] M. Knebush and W. Scharlau : Quadratische Formen und quadratische Reziprozitätsgesetze über algebraischen Zahlkörpern . Math. Z. 121 (1971) 346-368. · Zbl 0216.04602 · doi:10.1007/BF01109981
[8] S. Lang : Fundamentals of Diophantine Geometry . Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, (1983). · Zbl 0528.14013
[9] S. Lang and A. Néron : Rational points of abelian varieties over function fields . Amer. J. Math. 81 (1959) 95-118. · Zbl 0099.16103 · doi:10.2307/2372851
[10] S. Lichtenbaum : Curves over discrete valuation rings . Amer. J. Math. 90 (1968) 380-405. · Zbl 0194.22101 · doi:10.2307/2373535
[11] J.S. Milne : On a conjecture of Artin and Tate . Ann. Math. (2) 102 (1975) 517-533. · Zbl 0343.14005 · doi:10.2307/1971042
[12] J.S. Milne : Étale Cohomology . Princeton University Press, Princeton (1980). · Zbl 0433.14012
[13] A. Néron : Modèles minimaux des variétés abéliennes sur les corps locaux et globaux . Inst. Hautes Études Sci. Publ. Math. 21 (1964) 361-484. · Zbl 0132.41403 · doi:10.1007/BF02684271
[14] A. Néron : Quasi-fonctions et hauteurs sur les variétés abéliennes . Ann. Math. 82 (1965) 249-331. · Zbl 0163.15205 · doi:10.2307/1970644
[15] H. Reiter : Über den Satz von Weil-Cartier . Monatsh. Math. 86 (1978) 13-62. · Zbl 0396.43015 · doi:10.1007/BF01300054
[16] I.R. Shafarevich : Lectures on Minimal Models and Birational Transformations of Two Dimensional Schemes . Tata Institute of Fundamental Research, Bombay (1966). · Zbl 0164.51704
[17] J. Tate : Fourier analysis in number fields and Hecke’s zeta-functions . In: Algebraic Number Theory , eds. J.W.S. Cassels and A. Fröhlich. Thompson Book Company Inc., Washington, DC (1967).
[18] J. Tate : Algebraic cycles and poles of zeta functions . In: Arithmetical Algebraic Geometry , ed. O.F.G. Schilling. Harper and Row, New York (1965). · Zbl 0213.22804
[19] J. Tate : On the conjectures of Birch and Swinnerton-Dyer and a geometric analog . In: Dix exposés sur la cohomologie des schémas . North-Holland, Amsterdam (1968). · Zbl 0199.55604
[20] A. Weil : Sur certains groupes d’opérateurs unitaires . Acta Math. 111 (1964) 143-211. · Zbl 0203.03305 · doi:10.1007/BF02391012
[21] A. Weil : Basic Number Theory , third edition. Springer-Verlag, New York, Heidelberg, Berlin (1974). · Zbl 0326.12001
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