1 results
PERIODS OF DRINFELD MODULES AND LOCAL SHTUKAS WITH COMPLEX MULTIPLICATION
- Part of
-
- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 19 / Issue 1 / January 2020
- Published online by Cambridge University Press:
- 20 March 2018, pp. 175-208
- Print publication:
- January 2020
-
- Article
- Export citation
-
Colmez [Périodes des variétés abéliennes a multiplication complexe, Ann. of Math. (2)138(3) (1993), 625–683; available at http://www.math.jussieu.fr/∼colmez] conjectured a product formula for periods of abelian varieties over number fields with complex multiplication and proved it in some cases. His conjecture is equivalent to a formula for the Faltings height of CM abelian varieties in terms of the logarithmic derivatives at
$s=0$ of certain Artin 
$L$-functions. In a series of articles we investigate the analog of Colmez’s theory in the arithmetic of function fields. There abelian varieties are replaced by Drinfeld modules and their higher-dimensional generalizations, so-called 
$A$-motives. In the present article we prove the product formula for the Carlitz module and we compute the valuations of the periods of a CM 
$A$-motive at all finite places in terms of Artin 
$L$-series. The latter is achieved by investigating the local shtukas associated with the 
$A$-motive.
