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arXiv · 1709.00459

A description of a Drinfeld module with class number $h=1$ and rank $1$

V. Bautista-Ancona·J. Diaz-Vargas·J. A. Lara Rodriguez·F. X. Portillo-Bobadilla
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Abstract

We work with detail the Drinfeld module over the ring $$A=F_2[x,y]/(y^2+y=x^3+x+1).$$ The example in question is one of the four examples that come from quadratic imaginary fields with class number $h = 1$ and rank one. We develop specific formulas for the coefficients $d_k$ and $\ell_k$ of the exponential and logarithmic functions and relate them with the product $D_k$ of all monic elements of $A$ of degree $k$. On the Carlitz module, $D_k$ and $d_k$ coincide, but this is not true in general Drinfeld modules. On this example, we obtain a formula relating both invariants. We prove also using elementary methods a theorem due to Thakur that relate two different combinatorial symbols important in the analysis of solitons.

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V. Bautista-Ancona, J. Diaz-Vargas, J. A. Lara Rodriguez, F. X. Portillo-Bobadilla. 2017-09-01. A description of a Drinfeld module with class number $h=1$ and rank $1$. https://arxiv.org/abs/1709.00459

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