A function field analogue of Romanoff's theorem

作者：Kuan Yen Liang^*

来源：International Journal of Number Theory, 2015, 11(7): 2161-2173.

DOI：10.1142/S1793042115500955

• Function fields
• Drinfeld modules
• Romanoff's theorem

We prove an analogue for Drinfeld modules of a theorem of Romanoff. Specifically, let phi be a Drinfeld A-module over a global function field L and denote by phi(L) the A-module structure on L coming from phi. Let T subset of phi(L) be a free A-submodule of finite rank. For each effective divisor D = Sigma(v)n(v)v of L, let f(Gamma)(D) be the cardinality of the image of the reduction map Gamma -> Pi(v) O-v/m(v)(nv) if all elements of Gamma are relatively prime to the divisor D; otherwise, just set f(Gamma)(D) = infinity. We give explicit upper bounds for the series Sigma(D) 1/NDf(Gamma)(D)(epsilon) and Sigma(v) log N-v/N(v)f(Gamma)(v)(epsilon).

• 出版日期2015-11