I am a beginning graduate student. I have the following basic question I am very confused about:

Suppose $C$ is a smooth geometrically irreducible curve over a finite field $\mathbb{F}_q$, $q=p^m$, $p$ prime. Now take the ring $A$ of functions on $C$ regular away from a rational point, say $\infty$. Let $K$ be its function field and $K_\infty$ be the completed algebraic closure of $K$ wrt a valuation at $\infty$. Now define a rigid analytic function, say $f$, over $K_\infty$.

On the other hand suppose $B= \mathbb{F}_q[\theta]$, where $\theta$ is an indeterminate, and let $B_\infty$ be defined same way as $K_\infty$. Then define the Tate algebra to be $T_B = B_\infty[[T_1,\cdots, T_l]]$, where $T_i$ are indeterminates different from $\theta$. Again define the rigid analytic function, say $g$, over this Tate algebra $T_B$.

What is the difference between theory developed around the functions $f$ and $g$?

I am a beginning graduate student. I have the following basic question I am very confused about:

Suppose $C$ is a smooth geometrically irreducible curve over a finite field $\mathbb{F}_q$, $q=p^m$, $p$ prime. Now take the ring $A$ of functions on $C$ regular away from a rational point, say $\infty$. Let $K$ be its function field and $K_\infty$ be the completed algebraic closure of $K$ wrt a valuation at $\infty$. Now define a rigid analytic function, say $f$, over $K_\infty$.

On the other hand suppose $B= \mathbb{F}_q[\theta]$, where $\theta$ is an indeterminate, and let $B_\infty$ be defined same way as $K_\infty$. Then define the Tate algebra to be $T_B = B_\infty[[T_1,\cdots, T_l]]$, where $T_i$ are indeterminates different from $\theta$. Again define the rigid analytic function, say $g$, over this Tate algebra $T_B$.

What is the difference (which one is more general) between theory developed around the functions $f$ and $g$?

I am a beginning graduate student. I have the following basic question I am very confused about:

Suppose C be a smooth geometrically irreducible curve over finite field F_q, q=p^m, p prime. Now take the ring A of functions on C regular away from a rational point say, \infty. And K is its function field and C_\infty be the completed algebraic closure of K wrt a valuation at \infty. now define a rigid analytic function say, f defined over C_\infty.

On the other hand suppose B= F_q[\theta], \theta indeterminate. and B_\infty defined same way as C_\infty. And then define the Tate algebra to be T_B = B_\infty[[T_1,•••••, T_l]], T_i are indeterminate different from \theta. Again define the rigid analytic function say, g over this Tate algebra T_B.

What is the difference between theory developed around the functions $f$ and $g$?

I am a beginning graduate student. I have the following basic question I am very confused about:

Suppose $C$ is a smooth geometrically irreducible curve over a finite field $\mathbb{F}_q$, $q=p^m$, $p$ prime. Now take the ring $A$ of functions on $C$ regular away from a rational point, say $\infty$. Let $K$ be its function field and $K_\infty$ be the completed algebraic closure of $K$ wrt a valuation at $\infty$. Now define a rigid analytic function, say $f$, over $K_\infty$.

On the other hand suppose $B= \mathbb{F}_q[\theta]$, where $\theta$ is an indeterminate, and let $B_\infty$ be defined same way as $K_\infty$. Then define the Tate algebra to be $T_B = B_\infty[[T_1,\cdots, T_l]]$, where $T_i$ are indeterminates different from $\theta$. Again define the rigid analytic function, say $g$, over this Tate algebra $T_B$.

What is the difference between theory developed around the functions $f$ and $g$?