One of the most essential ingredients in the theory of motivic integration are the space of arcs (in german: the "formalen Schleifen": literally that translates as "formal loops") of a given $k$-variety $X$. This is a scheme, whose $k$-rational points are the $k[[t]]$-valued points of $X$.

In german the space of arcs is also called the "Raum der formalen Schleifen": literally that translates as "space of formal loops".

My question is if there any geometrical reason & motivation to call these objects "arcs" or "formal loops"? Does there exist any analogy to the topological loops and loop spaces which motivates the choice of the name "arcs" or "formal loops" for these objects occuring in motivic integration?

if $char \ k =0$ we can also associate to $X$ the topological loop space by taking the space $\Omega X(\mathbb{C}):=Hom(S^1, X(\mathbb{C}))$ where the set of $\mathbb{C}$-valued points $X(\mathbb{C})$ can be endowed with analytic topology if $X$ nice enough. $\Omega X(\mathbb{C})$ is the topological loop space of $X$. Is there any meaningful relation between this topological loop space and the space of arcs as defined above justifying the choice of similar names?

One of the most essential ingredients in the theory of motivic integration are the space of arcs of a given $k$-variety $X$. This is a scheme, whose $k$-rational points are the $k[[t]]$-valued points of $X$.

In german the space of arcs is also called the "Raum der formalen Schleifen": literally that translates as "space of formal loops".

My question is if there any geometrical reason & motivation to call these objects "arcs" or "formal loops"? Does there exist any analogy to the topological loops and loop spaces which motivates the choice of the name "arcs" or "formal loops" for these objects occuring in motivic integration?

if $char \ k =0$ we can also associate to $X$ the topological loop space by taking the space $\Omega X(\mathbb{C}):=Hom(S^1, X(\mathbb{C}))$ where the set of $\mathbb{C}$-valued points $X(\mathbb{C})$ can be endowed with analytic topology if $X$ nice enough. $\Omega X(\mathbb{C})$ is the topological loop space of $X$. Is there any meaningful relation between this topological loop space and the space of arcs as defined above justifying the choice of similar names?

One of the most essential ingredients in the theory of motivic integration are the space of arcs (in german: the "formalen Schleifen": literally that translates as "formal loops") of a given $k$-variety $X$. This is a scheme, whose $k$-rational points are the $k[[t]]$-valued points of $X$.

In german the space of arcs is also called the "Raum der formalen Schleifen": literally that translates as "space of formal loops".

My question is if there any geometrical reason & motivation to call these objects "arcs" or "formal loops"? Does there exist any analogy to the topological loops and loop spaces which motivates the choice of the name "arcs" or "formal loops" for these objects occuring in motivic integration?

One of the most essential ingredients in the theory of motivic integration are the space of arcs (in german: the "formalen Schleifen": literally that translates as "formal loops") of a given $k$-variety $X$. This is a scheme, whose $k$-rational points are the $k[[t]]$-valued points of $X$.

In german the space of arcs is also called the "Raum der formalen Schleifen": literally that translates as "space of formal loops".

My question is if there any geometrical reason & motivation to call these objects "arcs" or "formal loops"? Does there exist any analogy to the topological loops and loop spaces which motivates the choice of the name "arcs" or "formal loops" for these objects occuring in motivic integration?

if $char \ k =0$ we can also associate to $X$ the topological loop space by taking the space $\Omega X(\mathbb{C}):=Hom(S^1, X(\mathbb{C}))$ where the set of $\mathbb{C}$-valued points $X(\mathbb{C})$ can be endowed with analytic topology if $X$ nice enough. $\Omega X(\mathbb{C})$ is the topological loop space of $X$. Is there any meaningful relation between this topological loop space and the space of arcs as defined above justifying the choice of similar names?