R202, Astronomy-Mathematics Building, NTU
(台灣大學天文數學館 202室)
The Mordell-Weil Theorem for t-modules
Yen-Liang Kuan (NCTS)
Abstract:
For each positive characteristic multiple zeta value (defined by Thakur)
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5Czeta_A(%5Cmathfrak%7Bs%7D)%24&chf=bg,s,333333&chco=ffffff)
, Chang-Papanikolas-Yu constructed the
![](https://chart.googleapis.com/chart?cht=tx&chl=%24t%24&chf=bg,s,333333&chco=ffffff)
-module
![](https://chart.googleapis.com/chart?cht=tx&chl=%24E_%7B%5Cmathfrak%7Bs%7D%7D%24&chf=bg,s,333333&chco=ffffff)
defined over
![](https://chart.googleapis.com/chart?cht=tx&chl=%24A%24&chf=bg,s,333333&chco=ffffff)
and integral points
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5Cmathbf%7Bv%7D_%7B%5Cmathfrak%7Bs%7D%7D%24&chf=bg,s,333333&chco=ffffff)
,
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5Cmathbf%7Bu%7D_%7B%5Cmathfrak%7Bs%7D%7D%20%5Cin%20E_%7B%5Cmathfrak%7Bs%7D%7D(A)%24&chf=bg,s,333333&chco=ffffff)
. They proved that
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5Czeta_A(%5Cmathfrak%7Bs%7D)%24&chf=bg,s,333333&chco=ffffff)
is Eulerian (resp. zeta-like) if and only if
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5Cmathbf%7Bv%7D_%7B%5Cmathfrak%7Bs%7D%7D%24&chf=bg,s,333333&chco=ffffff)
is an
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5Cmathbb%7BF%7D_q%5Bt%5D%24&chf=bg,s,333333&chco=ffffff)
-torsion point in
![](https://chart.googleapis.com/chart?cht=tx&chl=%24E_%7B%5Cmathfrak%7Bs%7D%7D(A)%24&chf=bg,s,333333&chco=ffffff)
(resp.
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5Cmathbf%7Bv%7D_%7B%5Cmathfrak%7Bs%7D%7D%24&chf=bg,s,333333&chco=ffffff)
,
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5Cmathbf%7Bu%7D_%7B%5Cmathfrak%7Bs%7D%7D%24&chf=bg,s,333333&chco=ffffff)
are
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5Cmathbb%7BF%7D_q%5Bt%5D%24&chf=bg,s,333333&chco=ffffff)
-linearly dependent in
![](https://chart.googleapis.com/chart?cht=tx&chl=%24E_%7B%5Cmathfrak%7Bs%7D%7D(A)%24&chf=bg,s,333333&chco=ffffff)
).
In this talk, we are interested in the structure theory of the
![](https://chart.googleapis.com/chart?cht=tx&chl=%24t%24&chf=bg,s,333333&chco=ffffff)
-module
![](https://chart.googleapis.com/chart?cht=tx&chl=%24E_%7B%5Cmathfrak%7Bs%7D%7D(A)%24&chf=bg,s,333333&chco=ffffff)
. Poonen proved an analogue for Drinfeld modules of the Mordell-Weil theorem. We shall generalize his results to the case of specific families of
![](https://chart.googleapis.com/chart?cht=tx&chl=%24t%24&chf=bg,s,333333&chco=ffffff)
-modules. In particular, we prove that the
![](https://chart.googleapis.com/chart?cht=tx&chl=%24t%24&chf=bg,s,333333&chco=ffffff)
-module
![](https://chart.googleapis.com/chart?cht=tx&chl=%24E_%7B%5Cmathfrak%7Bs%7D%7D(A)%24&chf=bg,s,333333&chco=ffffff)
is the direct sum of its torsion submodule, which is finite, with a free
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5Cmathbb%7BF%7D_q%5Bt%5D%24&chf=bg,s,333333&chco=ffffff)
-module of rank
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5Caleph_0%24&chf=bg,s,333333&chco=ffffff)
.