I am currently working on the theory of elliptic curves and trying to understand what the genus a curve is.
Therefore I'm trying to understand the theory behind the Riemann-Roch Theorem. Hence the concept of Devisors.
But already the definition of a devisor makes me struggel.
"Definition 4.100: Let $C/K$ be a curve. The divisor group $Div_C$ of $C$ is the free abelian group over the places of $K(C)/K$. An element $D\in Div_C$ is called a divisor. It is given by $$D = \sum_{\mathfrak{p}_i\in\sum_{K(C)/K}}n_i\mathfrak{p}_i,$$ where $n_i\in\mathbb{Z}$ and $n_i=0$ for almost all $i$." ("Handbook of Elliptic and Hyperelliptic Cryptography" by Cohen and Frey)
$\mathfrak{p}$ is a place = the equivalence class of a valuation $v$ of $K(C)$
$\sum_{K(C)/K}$ is the set of places of $K(C)/K$
$K(C) = Quot(K[x_1,x_2]/f(x_1,x_2))$.
"Definition 4.102 Let $C/K$ be a curve and $f\in K(C)^*$. The divisor $div(f)$ of $f$ is given by $$div:K(C)\to Div_C$$ $$f\mapsto div(f) = \sum_{\mathfrak{p}_i \in \sum_{K(C)/K}}v_{\mathfrak{p}_i}(f)\mathfrak{p}_i$$" ("Handbook of Elliptic and Hyperelliptic Cryptography" by Cohen and Frey)
What does this really mean? I am looking at the definitions but I get no feeling for this theory.
I would be really happy if someone could explain a bit the concept of the divisors, the genus and finally the resulting Riemann Roch Theorem.
Or maybe someone knows a good source to work with, to understand this theory.
"Theorem 11.15 (Riemann Roch) Given and algebraic curve C, there exists an integer $g$ (called the genus of $C$) such that $$l(D)-l(\mathscr{K}-D) = deg(D)-g+1$$ for all divisors $D$." (Washington)
with $l(D) = dim\mathscr{L}(D)$, where $\mathscr{L}(D) = \{fct. f|div(f)+D\geq 0\}\cup\{0\}$
I'm working mainly with the books "Handbook of Elliptic and Hyperelliptic Cryptography", "The Arithmetic of Elliptic Curves" by Silverman and "Elliptic Curves Number Theory and Cryptography" by Washington.