Year: 2025
Type: article
Abstract: Let X be a smooth projective and geometrically irreducible curve over the finite field $$\mathbb {F}_q$$ with q elements and K be its function field. Let $$\infty $$ be a fixed closed point of X and A be the ring of functions regular away from $$\infty $$ . In the present paper, by generalizing the previous work of Chen and the first author, we introduce the notion of nearly holomorphic Drinfeld modular forms for congruence subgroups of $${{\,\textrm{GL}\,}}_2(K)$$ as continuous but non-holomorphic functions on a certain subdomain of the Drinfeld upper half plane. By extending the de Rham sheaf to a compactification $$\overline{M_I^2}$$ of the Drinfeld moduli space $$M_I^2$$ parametrizing rank 2 Drinfeld A-modules with level I-structure over K-schemes, we also describe such forms algebraically as global sections of an explicitly described sheaf on $$\overline{M_I^2}$$ as well as construct a comparison isomorphism between analytic and algebraic description of them. Furthermore, we show the transcendence of special values of nearly holomorphic Drinfeld modular forms at CM points and relate them to the periods of CM Drinfeld A-modules.
Source: Research in Number Theory
Authors: Oğuz Gezmiş, Sriram Chinthalagiri Venkata
Institutions: National Tsing Hua University, Heidelberg University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,590
OpenAlex ID: W4414167120