I became interested in the history of the Riemann-Roch theorem, so I searched various materials.
So, I read Über die Wechselwirkungen zwischen der französischen Schule, Riemann und Weierstraß. Eine Übersicht mit zwei Quellenstudien by E. Neuenschwander. If you read pages 3 to 5 of this paper, you will see that Riemann became interested in the conformal map, and also function theory, through a conversation with his teacher Gauss. You can see that it was from. The original text was in German, so I translated the parts I found into English.
A letter from Riemann to his father from Berlin on March 30, 1849 stated the following:
“I looked for a long time in the library's data for another work by Gauss that won a prize in Copenhagen, and I finally found it, and I'm still studying it.” Gauss' paper mentioned in the last part here is about conformal mapping.
And I was intrigued, so I searched other papers to find information about the conversation between Riemann and Gauss. So I found this paper, Riemann, Betti and the birth of topology by Andre Weil. If you read this paper, you will know that after Riemann heard the concept of cuts from Gauss, he became interested in the study of Analysis Situs, now called topology, and began to study topological contents.
What gave Riemann the idea of the cuts was that Gauss defined them to him, talking about other matters, in a private conversation. In his writings one finds that analysis situs, that is, this consideration of quantities independently from their measure, is "wichtig"; in the last years of his life he has been much concerned with a problem in analysis situs, namely: given a winding thread and knowing, at every one of its selfintersections, which part is above and which below, to find whether it can be unwound without making knots; this problem he did not succeed in solving except in special cases ...
However, since the current Riemann-Roch theorem is also known as a combination of complex analysis and topology, it is possible that these contents were the basis for Riemann to think about Riemann-inequality, the first prototype of the Riemann-Roch theorem, which appeared in the above papers. I started to think that it might have been a conversation with Gauss. Is this idea right?
Edit: Felix Klein's Development of Mathematics in the 19th Century
After researching further, I found a part in the above book by Felix Klein that seems to describe the Riemann-Roch theorem, so I referred to it.
Generally speaking, for $\zeta = f(z)$ not only will be many-valued in $z$, but $z$ will also be many-valued in $\zeta$. To any portion, no matter how it is bounded, of the surface over the $z$-plane there then corresponds biuniquely, and in general conformally, a portion of the surface over the $\zeta$-plane. And this is where the new geometric subject Analysis Situs (=topology) intervenes. The biunique conformal mapping is a special case of the biunique continuous mapping. The question arises: When can two surfaces be mapped to each other biuniquely and continuously? I only report historically: The surface has two characteristic numbers: $p$, the maximal number of simultaneously possible closed cuts that do not disconnect the surface and do not cross each other, and $\mu$, the number of bounding curves. I can briefly answer the question posed above by saying: Two surfaces are biuniquely and continuously related to each other if and only if their numbers $\mu$ and $p$ coincide. Riemann does not state this fundamental theorem explicitly, but he uses it again and again. (page 241)
