A note on meromorphic functions on a compact Riemann surface having poles at a single point

TL;DR

The paper proves the Weierstrass gap theorem for a compact Riemann surface of genus $g\ge1$, showing that at any point $P$ there are exactly $g$ gaps $1=n_1< n_2<\cdots< n_g<2g$ where no meromorphic function can have a pole of order $n_k$ at $P$. It derives the result by a cohomological RR framework using divisors $D_n$ supported at $P$ and tracking the变化 of $\dim H^0(X,\mathcal{O}_{D_n})$ through $\dim H^0(X,\Omega_{-D_n})$, which decreases exactly $g$ times. The note also develops the gap/non-gap structure, defines the Weierstrass weight $\omega(P)$, and counts Weierstrass points with $\#W(X)=g^3-g$, while discussing hyperelliptic and exceptional cases and connections to automorphism finiteness. Overall, it links pole orders to cohomological dimensions, provides a combinatorial perspective on gaps, and highlights implications for the geometry of $X$ and its automorphism group.

Abstract

The Riemann -Rock theorem plays a central role in the theory of Riemann surfaces with applications to several branches in Mathematics and Physics. Suppose $X$ ia a compact Riemann surface of genus $g$ and $P \in X$. By the Riemann-Roch theorem there exists a meromorphic function on $X$ having a pole at $P$ and is holomorphic in $X \setminus \{P\}$. The Weierstrass gap theorem gives more information on the order of the pole at $P$. It determines a sequence of $g$ distinct numbers $1 < n_k < 2g$, $1 \leq k \leq g$ for which a meromorphic function with the order $n_k$, fails to exist at $P$ and it can be obtained again as an application of Riemann-Roch theorem. In this note, we give proof of the Weierstrass gap theorem, using the dimensions of the cohomology groups and find an interesting combinatorial problem, which may be seen as a byproduct from the statement of the Weierstrass gap theorem. A short note is given at the end on Weierstrass points where a meromorphic function with lower order pole $\leq g$ exists and obtain some consequences of Weierstrass gap theorem.

Theorems & Definitions (11)

• Theorem 1: Weierstrass gap theorem
• Definition 2
• Theorem 3
• Remark 4
• Lemma 5
• Example 6
• Definition 8
• Definition 9
• Theorem 10
• Remark 11