Overview
The presented research focuses on developing Mellin transform formulas for the logarithm function associated with abelian, uniformizable Anderson modules. These formulas are then utilized to establish connections between distinct mathematical objects: special values of dual Goss $L$-functions of Drinfeld modules and the rigid analytic trivialization of their corresponding $t$-comotives. Further application of these formulas demonstrates a relationship between specific $L$-values and special values of matrix valued modular forms.
Research Context
The work builds upon a recently introduced motivic map by the second author. This map serves as a combinatorial link, enabling the connection between the newly derived Mellin transform formulas and the rigid analytic trivialization of $t$-comotives. The investigation extends prior work on Drinfeld modular forms, specifically referencing the vector valued Drinfeld modular forms developed by Pellarin and by Pellarin and Perkins, particularly in the rank two setting. The current research generalizes these earlier findings by incorporating matrix valued modular forms.
Approach
The core of the approach involves introducing specific formulas for the logarithm function within the context of abelian, uniformizable Anderson modules. Following the establishment of these formulas, the research integrates them with a motivic map. This integration allows for the explicit relating of special values of dual Goss $L$-functions of Drinfeld modules to the rigid analytic trivialization of their corresponding $t$-comotives. Subsequently, these formulas are applied to demonstrate a relationship between certain $L$-values and special values of matrix valued modular forms.
Findings
- Formulas have been introduced for the logarithm function of abelian, uniformizable Anderson modules.
- These formulas, in conjunction with a motivic map, relate special values of dual Goss $L$-functions of Drinfeld modules to the rigid analytic trivialization of their corresponding $t$-comotives.
- The formulas are applied to connect certain $L$-values to special values of matrix valued modular forms.
- The connection to matrix valued modular forms generalizes the vector valued Drinfeld modular forms of Pellarin and Pellarin and Perkins, particularly in the rank two context.