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The title says it. I am looking for a good exposition on the K-theory of the curves $X_{i}(N)$, $Y_{i}(N)$, where $i\in\{0,1\}$. I have some background in $K$-theory and also some background in modular curves.

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  • $\begingroup$ Maybe you should specify a little more explicitly what you are looking for. Beilinson's conjectures give a fairly general picture of what is expected of the K-theory (in particular of modular curves). Besides Beilinson's papers on higher regulators for modular curves, there are a couple of expository papers on Beilinson's conjectures by Nekovar, Deninger-Scholl, Schappacher-Scholl, etc. There is quite a follow-up literature on construction of special elements in K-groups, but essentially Beilinson's conjectures remain unproved even in the curve case. $\endgroup$ – Matthias Wendt Dec 21 '15 at 15:31
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    $\begingroup$ Beilinson's "Higher regulators and values of L-functions", J. Sov. Math. 30 (1985) is supposed to be a good (general) reference here. $\endgroup$ – jvo Dec 21 '15 at 20:56
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I wouldn't recommend Beilinson's 1985 paper as a general reference -- it's terrifyingly compressed, developing an entire new subject in a single short paper, and crashes through the necessary material on modular curves in a couple of sentences.

A much gentler reference would be Flach's 1992 Inventiones paper "A finiteness theorem for the symmetric square of an elliptic curve", which gives a much fuller introduction to K-groups, Gersten complexes, regulator maps etc. You could also try looking at parts of my paper with Lei and Zerbes "Euler systems for Rankin--Selberg convolutions" for more on this.

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