Lecture Room B, 4th Floor, The 3rd General Building, NTHU
(清華大學綜合三館 4樓B演講室)
The Rational Torsion Subgroups of the Drinfeld Modular Jacobians for Prime-Power Levels
Sheng-Yang Ho (Pennsylvania State University)
Abstract
Fix a non-zero ideal
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5Cmathfrak%7Bn%7D%24&chf=bg,s,333333&chco=ffffff)
of
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5Cmathbb%7BF%7D_q%5BT%5D%24&chf=bg,s,333333&chco=ffffff)
. Let
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5Cmathcal%7BT%7D(%5Cmathfrak%7Bn%7D)%24&chf=bg,s,333333&chco=ffffff)
be the rational torsion subgroup of the Drinfeld modular Jacobian
![](https://chart.googleapis.com/chart?cht=tx&chl=%24J_0(%5Cmathfrak%7Bn%7D)%24&chf=bg,s,333333&chco=ffffff)
. A generalized Ogg's conjecture states that
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5Cmathcal%7BT%7D(%5Cmathfrak%7Bn%7D)%24&chf=bg,s,333333&chco=ffffff)
coincides with the rational cuspidal divisor class group
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5Cmathcal%7BC%7D(%5Cmathfrak%7Bn%7D)%24&chf=bg,s,333333&chco=ffffff)
of the Drinfeld modular curve
![](https://chart.googleapis.com/chart?cht=tx&chl=%24X_0(%5Cmathfrak%7Bn%7D)%24&chf=bg,s,333333&chco=ffffff)
. Recently, we proved that for any prime-power ideal
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5Cmathfrak%7Bp%7D%5Er%24&chf=bg,s,333333&chco=ffffff)
of
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5Cmathbb%7BF%7D_q%5BT%5D%24&chf=bg,s,333333&chco=ffffff)
, the prime-to-
![](https://chart.googleapis.com/chart?cht=tx&chl=%24q(q-1)%24&chf=bg,s,333333&chco=ffffff)
part of
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5Cmathcal%7BT%7D(%5Cmathfrak%7Bp%7D%5Er)%24&chf=bg,s,333333&chco=ffffff)
is equal to that of
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5Cmathcal%7BC%7D(%5Cmathfrak%7Bp%7D%5Er)%24&chf=bg,s,333333&chco=ffffff)
by studying the Hecke operators and the Eisenstein ideal of level
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5Cmathfrak%7Bp%7D%5Er%24&chf=bg,s,333333&chco=ffffff)
. Moreover, by relating the rational cuspidal divisors of degree
![](https://chart.googleapis.com/chart?cht=tx&chl=%240%24&chf=bg,s,333333&chco=ffffff)
on
![](https://chart.googleapis.com/chart?cht=tx&chl=%24X_0(%5Cmathfrak%7Bp%7D%5Er)%24&chf=bg,s,333333&chco=ffffff)
with
![](https://chart.googleapis.com/chart?cht=tx&chl=%20%24%5CDelta%24&chf=bg,s,333333&chco=ffffff)
-quotients, where
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5CDelta%24&chf=bg,s,333333&chco=ffffff)
is the Drinfeld discriminant function, we are able to compute explicitly the structure of
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5Cmathcal%7BC%7D(%5Cmathfrak%7Bp%7D%5Er)%24&chf=bg,s,333333&chco=ffffff)
. As a result, the structure of the prime-to-
![](https://chart.googleapis.com/chart?cht=tx&chl=%24q(q-1)%24&chf=bg,s,333333&chco=ffffff)
part of
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5Cmathcal%7BT%7D(%5Cmathfrak%7Bp%7D%5Er)%24&chf=bg,s,333333&chco=ffffff)
is completely determined.
Organizer: Chieh-Yu Chang (NTHU)