arXiv:1001.1621 The Hilbert-Polya strategy and height pairings from arXiv Front: math.AG by C. Deninger Previously we gave a conjectural cohomological argument for the validity of the Riemann hypotheses for Hasse-Weil zeta functions. In the present note we sketch how the same cohomological formalism would imply the conjectured positivity properties of the height pairings of homologically trivial cycles.
http://mathoverflow.net/questions/61620/beilinsons-height-pairing-vs-neron-tate
Bloch: Height pairings for algebraic cycles (1983). Memo notes from MR:
Let be a -dimensional smooth projective variety over a number field. Let be the -VS of homologically trivial algebraic cycles of codim on mod rational equivalence. Under certain mild assumptions, can construct height pairings
Theses generalize the Neron pairing (Neron 1965) between divisors and zero-cycles. The discriminant of the Neron pairing shows up as a regulator in the BSD conjecture. Bloch shows that the above pairing plays a similar role for the L-function associated to at . For some evidence, see Bloch: Algebraic cycles and values of L-functions. Beilinson (Higher regulators and values of L-functions of curves; English translation exists!) has independently constructed a height pairing and regulators, conjecturally related to special values.
The pairing above is defined as a sum of local factors over the primes of the number field. The general method uses certain homotopy-theoretic methods and properties of the spaces of Quillen. At the finite primes one uses the usual theory of Chern classes in etale cohomology. At the Archimedean primes, one needs a theory of Chern classes in Deligne cohomology, as constructed by Gillet.
arXiv:1103.0570 Néron’s pairing and relative algebraic equivalence from arXiv Front: math.NT by Cédric Pépin Let R be a complete discrete valuation ring with algebraically closed residue field k and fraction field K. Let X_K be a projective smooth and geometrically connected scheme over K. Néron defined a canonical pairing on X_K between 0-cycles of degree zero and divisors which are algebraically equivalent to zero. When X_K is an abelian variety, and if one restricts to those 0-cycles supported by K-rational points, Néron gave an expression of his pairing involving intersection multiplicities on the Néron model A of A_K over R. When X_K is a curve, Gross and Hriljac gave independantly an analogous description of Néron’s pairing, but for arbitrary 0-cycles of degree zero, by means of intersection theory on a proper flat regular R-model X of X_K. In this article, we show that these intersection computations are valid for an arbitrary scheme X_K as above and arbitrary 0-cyles of degree zero, by using a proper flat normal and semi-factorial model X of X_K over R. When X_K=A_K is an abelian variety, and X is a semi-factorial compactification of its Néron model A, these computations can be used to study the algebraic equivalence on X. We then obtain an interpretation of Grothentieck’s duality for the Néron model A, in terms of the Picard functor of X over R.
Muller-Stach: A remark on height pairings. Contains many good references, and three different equivalent definitions!
K. Ku¨nnemann, Height pairings for algebraic cycles on abelian varieties, Annls Scient. E´c. Norm. Sup. 34(4) (2001), 503–523. MR1852008
arXiv:1001.4788 Positivity of heights of codimension 2 cycles over function field of characteristic 0 from arXiv Front: math.AG by Shou-Wu Zhang In this note, we show how the classical Hodge index theorem implies the Hodge index conjecture of Beilinson for height pairing of homologically trivial codimension two cycles over function field of characteristic 0. Such an index conjecture has been used in our paper on Gross-Schoen cycles to deduce the Bogomolov conjecture and a lower bound for Hodge class (or Faltings height) from some conjectures about metrized graphs which have just been recently proved by Zubeyir Cinkir.
2.30pm Chris Wuthrich (Nottingham). “The class group pairing on elliptic curves”. Abstract: There is a pairing on the Mordell-Weil group of an elliptic curve over a number field with values in the class group of the field. It sits somewhere between the monodromy pairing and the Neron-Tate height pairing. I would like to discuss how this pairing shows up in questions on Galois module structures, how one can compute it effectively and how it links to p-descent.
arXiv:1003.0777 A generalisation of Zhang’s local Gross-Zagier formula from arXiv Front: math.NT by Kathrin Maurischat On the background of Zhang’s local Gross-Zagier formulae for GL(2), we study some p-adic problems. The local Gross-Zagier formulae give identities of very special local geometric data (local linking numbers) with certain local Fourier coefficients of a Rankin L-function. The local linking numbers are local coefficients of a geometric (height) pairing. The Fourier coefficients are products of the local Whittaker functions of two automorphic representations of GL(2). We establish a matching of the space of local linking numbers with the space of all those Whittaker products. Further, we construct a universally defined operator on the local linking numbers which reflects the behavior of the analytic Hecke operator. Its suitability is shown by recovering from it an equivalent of the local Gross-Zagier formulae. Our methods are throughout constructive and computational.
nLab page on Height pairing