Let $k$ be an arbitrary field and $C \subset \mathbb{P}^2(k)$ an elliptic curve. In order to define the group law on $C$ we need to establish some geometric facts first, e.g.
- Any line intersects $C$ at exactly three points (counting multiplicity).
- For any point $p \in C$ there is a unique line intersecting $C$ at $p$ with multiplicity 2 (tangent line at $p$).
If $C$ is given by a (homogeneous) Weierstrass equation $Y^2Z = X^3+aXZ^2+bZ^3$ these facts could be shown by elementary calculations without many difficulties. However, if $\operatorname{char} k \in \{2,3\}$ the general equation for $C$ is more complicated, and the calculations will become quite cumbersome.
I am looking for a reference, where the group law on elliptic curves as well as the proof of associativity is treated in full generality, and which avoids cumbersome calculations. Instead, such geometric facts should be derived by methods of algebraic geometry. Furthermore, it would be nice if this could be done within classical algebraic geometry (i.e. by examining the coordinate ring of $C$) since I am not quite familiar with the theory of schemes yet.
If you haven't already, you might have a look at the description of the group law in terms of divisors on the curve. This makes use of more advanced (but still classical) machinery to preclude painful verifications of associativity, etc. One reference would be Shafarevich's Basic Algebraic Geometry I. The relevant information begins in III.3.1, and III.3.2 features the group law. (You might want to read/skim from the beginning of III, depending on how familiar you are with divisors.)
After introducing divisors and the class group, Shafarevich explains how to identify points of the elliptic curve with a subgroup of the class group of the curve. You can therefore "transfer" the group structure from the class group onto the curve itself. This construction is in full generality, and the explanations are fairly complete. Shafarevich also goes through the explicit formulas for a cubic in Weierstrass normal form (this part shows that the group law is given by rational formulas, which is significant); for this he does assume $\operatorname{char} k \neq 2,3$ "to simplify formulas".
I'm not sure if there is a way to construct the group law using the coordinate ring of the curve alone. It seems to me that any such method would result in the same type of tedious algebraic manipulations that you're seeking to avoid, but this is just speculation.