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According to some authors, it is built in A.A.Beilinson "Higher regulator of modular curves" a class $\mathbf{Eis}_{\phi}$ in the motivic cohomology of the modular curve where $\phi$ is a Schwartz function over the finite adeles. Since the modular curve is only quasi-projective, I assume it is mixed-motivic cohomology? I am right ?

I say "according to some authors" because I haven't read this article by Beilinson (I can't find it anywhere). This document is a bit old and sometimes Beilinson is hard to read so my question is

Does anyone know of a reference where this construction is done with the technical details? If not how can I find this article of Beilinson?

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The Eisenstein classes $\mathrm{Eis}^k_\phi$ live in the motivic cohomology $H^{k+1}_{\mathcal{M}}(E^k, \mathbf{Q}(k+1))$, where $E \to Y(N)$ is the universal elliptic curve over the open modular curve $Y(N)$. For example the classes $\mathrm{Eis}^0_\phi$ are the Siegel modular units. At the time, Beilinson defined motivic cohomology as a graded piece of algebraic $K$-theory.

Beilinson's construction is hard to read. Scholl gave an account of the Eisenstein symbol in An introduction to Kato's Euler systems, in the survey The Beilinson conjectures (with Deninger), and in his book in preparation $L$-functions of modular forms and higher regulators. Doran and Kerr also explained Beilinson's construction in Algebraic $K$-theory of toric hypersurfaces, Section 5.

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  • $\begingroup$ +1. Scholl's account in the "Introduction to Kato's ES" article is, I think, the most readable I have come across. $\endgroup$ Feb 18, 2023 at 21:23

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