@all: thank you for the references! This all looks good.

Thank you very much. I just glanced over your article and the one by Taylor to which you included links; later I will read them more thoroughly. They do look like what I have been looking for.

@Tom: this is quite true, and of course that sentence is usually expanded over fifteen minutes or so. But really I have never heard nor read an expository talk or article about something Langlands-related that would not very soon delve into that formalism.

I am not sure if the thing is well-defined at all. For instance, let $a,b\in R$. Do you then identify $a+b$ (addition in $R$) with the formal infinite sum $\sum_{i=1}^{\infty }a_i$, where $a_1=a$, $a_2=b$ and $a_i=0$ for $i>2$? If you want this and other "obvious" relations to hold, then I think you always have $\hat{R}=0$.

Also note that you have the equality $L(s,D)=\zeta_K(s)/\zeta_{\mathbb{Q}}(s)$, where $K=\mathbb{Q}(\sqrt{D})$. Hence at least for even $s$, you may equally well ask for values of the Dedekind zeta function. Siegel has shown long ago that $\zeta_K(s)$ is a rational number for negative odd $s$, which should give you strong arithmetic information on $L(s,D)$ for positive even $s$, via the functional equation.

In the last two sentences, you probably mean $s$ instead of $n$.

You can learn already a great deal of tropical geometry with the knowledge you have. It is a very "visual" and "constructive" area of mathematics, and mainly works with hands-on definitions. Of course, there are some applications of tropical geometry to Grothendieck-style algebraic geometry, but if you do not know schemes and sheaves, you very probably will not be interested in these applications in the first place. So my suggestion is just: start reading some introductory paper right now, e.g. arxiv.org/abs/math/0601322

In nonstandard analysis, what is classically called a differential equation becomes a difference equation.

@Mahdi: Well, let's just wait for him to explain what he actually means, because at least for my taste the current formulation can be interpreted in many different ways, yielding many different answers.

@Mahdi: well, that exactly depends on what you understand by "variety". If you work in scheme theory (as your notation $\mathbb{A}_{\mathbb{R}}^2$ suggests), then they are not the same. Their sets of $\mathbb{R}$-valued points are.