t-motives: Hodge structures, transcendence and other motivic aspects (09w5094)

Objectives

Recent years have seen a large number of interesting developments in the arithmetic of functions fields centered around the notion of a t-motive as defined by Greg Anderson. Some of the most important ones are:

(1) New developments in the transcendence theory over function fields: For instance, it has been shown that the period matrix of a t-motive has transcendence degree equal to the dimension of a difference Galois group associated to the t-motive, much in the same way that Grothendieck's conjecture predicts the transcendence degree of the period matrix of an abelian variety to be equal to the dimension of its Mumford-Tate group.

(2) Hodge structures for function fields: Defined by Pink in 1997, they
allow him to formulate a Mumford-Tate conjecture for certain t-motives
and to prove the conjecture for Drinfeld modules.

(3) Period domains: More recently Pink's Hodge structures have been used extensively to lay foundations for period domains over function fields and state an analogue of Fontaine's theory of crystalline Galois
representations.

(4) Galois representations: While the Tate-conjecture for t-motives has
been proved already in the early 90's, only recently results on the
openness of the image (l-adically and adelically) have been obtained.

(5) Tannakian formalisms: Such have recently been described for t-motives in various contexts. They should ultimately link
transcendence, Hodge structures and images of Galois representations.

(6) L-series: There are now cohomological approaches to L-series attached to t-motives. Moreover, recently new results on the zeroes of these L-series have emerged.

The above topics are tightly interwoven. The Tannakian formalism is used in transcendence theory as well as in a formulation of a Mumford-Tate conjecture based on function field Hodge structures (which is proved for Drinfeld modules). This in turn spurs the interest in Galois representations over function fields. All of the above topics have close relations to similar questions in number theory. The function field Hodge structures and analogues of Fontaine's theory have influenced questions on period spaces for number fields. Other developments such as the transcendence theory have gone far beyond comparable results in number theory.

Some aspects of the above developments have played an important role in various conferences which took place in recent years. On the other hand, a large scale international conference centered around the notion of t-motive has not taken place in recent years. The proposers of the present workshop feel that there is large need for such a conference and that the Banff center would be an ideal place to host it, focusing on the following aims:

(a) Present the state of the art in the subject in a single conference including the above and other recent topics in function field arithmetic.

(b) Bring together the most active researchers to discuss further possible developments and the potential that lies in the recently developed methods.

(c) Get a fair number of post-docs and recent PhD students in this and neighboring areas involved, among them in particular some that work in algebraic number theory.

(d) Have two educational lecture series of 3-4 talks each, one on transcendence and another one Hodge structures and periods. We hope that this will benefit younger researchers, as well as bring together researchers with expertise on different aspects of t-motives.

(e) Enhance interaction among the participants on a personal level and in a stimulating environment to foster further joint projects.

The following are further materials to support the application; since we had assembled them before filling out this online form [we had done so since in the guidelines for applicants there were mentioned 2-4 pages for the Proposal And the Objectives] we decided to add them on - but they are not parts of the objectives:

Materials supporting the proposal

Recent achievements and further goals (in some core topics)

I) Transcendence:
New developments in the transcendence theory over function fields
have led to important results on algebraic independence that as yet
are out of reach over number fields. In the 1980's and 1990's work
of Brownawell, Thakur, Yu, and many others demonstrated that there
is a rich transcendence theory of exponentials and logarithms of
Drinfeld modules that parallels ideas and results over number
fields. This culminated with Yu's Sub-t-module Theorem [JYu97],
which characterized completely linear relations among logarithms of
algebraic points on Drinfeld modules and t-modules.

More recently, there have been important new discoveries toward
algebraic independence. Work of Anderson, Brownawell, and
Papanikolas [ABP04] demonstrated that results on linear
relations could be adapted to approach problems of algebraic
independence, and they proved theorems on the algebraic independence
of function field Gamma-values. In [Pa05], Papanikolas showed
that the period matrix of a t-motive has transcendence degree equal
to the dimension of a difference Galois group associated to the
t-motive, much in the same way that Grothendieck's conjecture
predicts the transcendence degree of the period matrix of an abelian
variety to be equal to the dimesion of its Mumford-Tate group. Using
this point of view, Papanikolas proved the algebraic independence of
Carlitz logarithms, and subsequently Chang and Yu [CYu07]
have applied Papanikolas' results to prove the algebraic
independence of Carlitz zeta values. However, as much as the
potential to characterize all algebraic relations among periods and
logarithms of t-motives is clear, these problems require more
detailed study.

II) Hodge-Pink structures:
In [Pi97a] Pink has clarified the concept of Hodge structures in equal characteristic, which today are called Hodge-Pink structures. This enabled him to prove his celebrated analogue of the Mumford-Tate Conjecture for Drinfeld modules~[Pi97b]. It also allowed to transfer the notion of Fontaine's filtered isocrystals to the arithmetic of function fields, and the analogy of p-adic Hodge theory to emerge (e.g. [Ha06]). The latter is called Hodge-Pink theory. By a correspondence strongly paralleling the situation for abelian varieties, one can, in particular, associate to a t-motive with good reduction over a local field an isocrystal with Hodge-Pink structure. The latter structures possess period domains and the picture (see [FGL07,GHKR07]) is much the same as the one in mixed characteristic developed by Rapoport, Zink, Faltings and others, which had great impact on the understanding of Shimura varieties and the Langlands program. Besides being a beautiful theory in its own right, the insights gained from Hodge-Pink theory have inspired recent work of Faltings and Hartl on classical p-adic Hodge theory, which has attracted much attention from researchers in number theory.

Despite this progress, many questions remain open. The Mumford-Tate conjecture for higher dimensional t-motives deserves further study, as well as its possible relations to transcendence theory. Semi-stable reduction of t-motives and their Hodge-Pink theory needs to be investigated. The cohomology of period domains and related spaces has not been considered much.

III) L-functions:
As in number theory, a fascinating topic are L-functions for function fields, and especially such which take values in positive characteristic [Go92]. By work of Taguchi and Wan [TW96] with methods in the spirit of Dwork, these L-functions are known to have an entire continuation to an analogue of the complex plane in positive characteristic. A cohomological proof was later given by Boeckle and Pink in 2001. But no functional equation is known. The special values of these L-functions at "even" positive integers are related to Bernoulli numbers over function fields. Beyond that there are few interpretations of special values. There is no analogue of the Birch and Swinnerton-Dyer conjecture, although the analogous objects, Drinfeld modules and Drinfeld cusp forms are fairly well understood. A starting point for such conjectures is [An96]. But even more elementary questions such as the number of zeros at the negative integers remain mysterious, cf.~[Th95].

References

[An96] G. W. Anderson, Log-algebraicity of twisted A-harmonic series and special values of L-series in characteristic p, J. Number Theory 60(1996), no. 1, 165--209.

[ABP04] G. W. Anderson, W. D. Brownawell, and M. A. Papanikolas, Determination of the algebraic relations among special Gamma-values in positive characteristic, Ann. of Math. (2) 160 (2004), 237--313.

[CYu07] C.~Y. Chang and J. Yu, Determination of algebraic relations among special zeta values in positive characteristic, Adv. Math. (to appear).

[FGL07] L. Fargues, A. Genestier, V. Lafforgue: L'isomorphisme entre les tours de Lubin-Tate et de Drinfield, Progress in Mathematics, Vol. 262, Birkhaeuser Verlag, Basel 2007.

[Go92] D. Goss, L-series of t-motives and Drinfeld modules, in "The arithmetic of function fields (Columbus, OH, 1991)", 313--402, de Gruyter, Berlin, 1992.

[GHKR06] U. G"ortz, T. Haines, R. Kottwitz, D. Reuman, Dimensions of some affine Deligne-Lusztig varieties, Ann. Sci. 'Ecole Norm. Sup. (4) 39 (2006), no. 3, 467--511.

[Ha06] U. Hartl, Period Spaces for Hodge Structures in Equal Characteristic, Preprint on arXiv:math.NT/0511686.

[Pa07] M. A. Papanikolas, Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms, Inv. Math. (to appear).

[Pi97a] R. Pink, Hodge Structures over Function Fields, Preprint 1997, available at http://www.math.ethz.ch/~pink.

[Pi97b] R. Pink, The Mumford-Tate conjecture for Drinfeld-modules, Publ. Res. Inst. Math. Sci. 33 (1997), no. 3, 393--425.

[TW96] Y. Taguchi, D. Wan, L-functions of phi-sheaves and Drinfeld modules, J. Amer. Math. Soc. 9 (1996), no. 3, 755--781.

[Th95] D. Thakur, On characteristic p zeta functions, Compositio Math. 99 (1995), no. 3, 231--247.

[JYu97] J. Yu, Analytic homomorphisms into Drinfeld modules, Ann. of Math. (2) 145 (1997), 215--233.

Recent conferences related to the proposal

a) NCTS Conference on "Galois Representations and Function Field Arithmetic", Taiwan 2007, organized by Wen-Ching Winnie Li and Jing Yu.

b) AMS Special Session on "Arithmetic of Function Fields", New Orleans, 2007, organized by A. M. Pacelli and M. J. Rosen.

c) 4th Texel conference on "The analogy between number fields and function fields", Texel 2004, organized by Gerard van der Geer, Ben Moonen and René Schoof.

d) Conference on "Function Fields and L-functions", Amherst 2002, organized by G. Call, M. Robinson, and S. Wong.

e) Workshop on "Drinfeld Modules, t-Motives and Rigid Geometry", Gent, Belgium, 2002, organized by G. Cornelissen, F. Gardeyn, M. van der Put, J. Top, J. Van Geel.