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?**



References for the construction of Beilinson's motivic Eisenstein classes
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](https://www.dpmms.cam.ac.uk/%7Eajs1005/preprints/euler.pdf), in the survey [The Beilinson conjectures](https://www.dpmms.cam.ac.uk/%7Eajs1005/preprints/d-s.pdf) (with Deninger), and in his book in preparation [$L$-functions of modular forms and higher regulators](https://www.dpmms.cam.ac.uk/%7Eajs1005/mono/index.html). Doran and Kerr also explained Beilinson's construction in [Algebraic $K$-theory of toric hypersurfaces](https://arxiv.org/abs/0809.4669), Section 5.


