The Tate conjecture

This web page contains material for the workshop The Tate conjecture.

Contributions from the workshop participants are available in dvi, postscript or pdf.

Background reading

An introduction to the Tate conjecture by Ramakrishnan.

Bibliography

Participants are invited to contribute to this list of papers. By the end of the workshop we hope to have a database of all the papers dealing, directly or indirectly, with the Tate conjecture, from various angles. (The selection below is a highly *non-complete and subjective* list of papers the organizers have been involved with or looked at.)

Basic references

1) J.Tate, Conjectures on algebraic cycles in $l$-adic cohomology. Motives
(Seattle, WA, 1991), 71--83, Proc. Sympos. Pure Math., 55, Part 1, Amer.
Math. Soc., Providence, RI, 1994.

2) N.Katz, Review of $l$-adic cohomology. Motives (Seattle, WA, 1991),
21--30, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., Providence,
RI, 1994.

Tate's article is connected to two old papers of his, namely

1a) J.Tate, Endomorphisms of abelian varieties over finite fields. Invent.
Math. 2 1966 134--144;

1b) J.Tate, Algebraic cycles and poles of zeta functions, in Arithmetical
Algebraic Geometry (Proc. Conf. Purdue Univ., 1963) pp. 93--110 Harper &
Row, New York (1965).

Curves, K3 Surfaces, Abelian Varieties, and Motives

1) P.Deligne, J.Milne, A.Ogus and K.Shih, Hodge cycles, motives, and Shimura
varieties. Lecture Notes in Mathematics, 900. Springer-Verlag, Berlin-New
York, 1982. ii+414 pp.

2) G.Faltings, Finiteness theorems for abelian varieties over number fields.
Translated from the German original [Invent. Math. 73 (1983), no. 3,
349--366; ibid. 75 (1984), no. 2, 381; MR 85g:11026ab] by Edward Shipz.
Arithmetic geometry (Storrs, Conn., 1984), 9--27, Springer, New York, 1986.

3) S.Lichtenbaum, Behavior of the zeta-function of open surfaces at $s=1$.
Algebraic number theory, 271--287, Adv. Stud. Pure Math., 17, Academic
Press, Boston, MA, 1989.

4) J.Milne, Lefschetz motives and the Tate conjecture. Compositio Math. 117
(1999), no. 1, 45--76.

5) J.Milne, The Tate conjecture for certain abelian varieties over finite
fields. Acta Arith. 100 (2001), no. 2, 135--166.

6) Y.Andre, Cycles de Tate et cycles motiv�s sur les vari�t�s ab�liennes en
caract�ristique $p>0$. (French) [Tate cycles and motivated cycles on abelian
varieties of characteristic $p>0$] J. Inst. Math. Jussieu 5 (2006), no. 4,
605--627. 

7) N.Nygard and A.Ogus, Tate's conjecture for $K3$ surfaces of finite
height. Ann. of Math. (2) 122 (1985), no. 3, 461--507.

8) Y.Zarhin, The Tate conjecture for powers of ordinary $K3$ surfaces over
finite fields. J. Algebraic Geom. 5 (1996), no. 1, 151--172.

9) T.Geisser, Tate's conjecture, algebraic cycles and rational $K$-theory in
characteristic $p$. $K$-Theory 13 (1998), no. 2, 109--122.

10) D.Ulmer, Elliptic curves with large rank over function fields. Ann. of
Math. (2) 155 (2002), no. 1, 295--315.

11) W.Raskind, A generalized Hodge-Tate conjecture for algebraic varieties
with totally degenerate reduction over $p$-adic fields. Algebra and number
theory, 99--115, Hindustan Book Agency, Delhi, 2005.

Modular varieties

0) Gerard Van der Geer, Hilbert modular surfaces, Springer 1988 ( the part
near the end deals with the Tate conjecture and exposes material from 1), 2)
below)

1) G.Harder, R.P.Langlands and M.Rapoport, Algebraische Zyklen auf
Hilbert-Blumenthal-Fl�chen. (German) [Algebraic cycles on Hilbert-Blumenthal
surfaces] J. Reine Angew. Math. 366 (1986), 53--120.

2) V.K.Murty and D.Ramakrishnan, Period relations and the Tate conjecture
for Hilbert modular surfaces. Invent. Math. 89 (1987), no. 2, 319--345.

3) K.Klingenberg, Die Tate-Vermutungen f�r Hilbert-Blumenthal-Fl�chen.
(German) [The Tate conjectures for Hilbert-Blumenthal surfaces] Invent.
Math. 89 (1987), no. 2, 291--317.

4) D.Blasius and J.Rogawski, Tate classes and arithmetic quotients of the
two-ball. The zeta functions of Picard modular surfaces, 421--444, Univ.
Montr�al, Montreal, QC, 1992.

5) D.Prasad and V.K.Murty, Tate cycles on a product of two Hilbert modular
surfaces. J. Number Theory 80 (2000), no. 1, 25--43.

6) D.Ramakrishnan, Algebraic cycles on Hilbert modular fourfolds and poles
of $L$-functions. Algebraic groups and arithmetic, 221--274, Tata Inst.
Fund. Res., Mumbai, 2004; the case of fourfold modular curves was considered
earlier in
(sec.4.5 of) Modularity of the Rankin-Selberg L-series, and multiplicity one
for ${\rm SL}(2)$,
Ann. of Math. (2) 152 (2000), no. 1, 45--111.

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A list of registered participants is available.

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