. Remark, One motivation to consider this form comes from (9.11) and Goncharov's explicit formulas for regulators (cf

M. F. Atiyah, H. Donnelly, and I. M. Singer, Eta invariants, signature defects of cusps, and values of Lfunctions, Ann. of Math, vol.118, issue.2, pp.131-177, 1983.

A. Beilinson, G. Kings, and A. Levin, Topological polylogarithms and p-adic interpolation of L-values of totally real fields, Math. Ann, vol.371, issue.3-4, pp.1449-1495, 2018.

N. Berline, E. Getzler, and M. Vergne, Heat kernels and Dirac operators. Grundlehren Text Editions, 2004.

A. Be?linson and A. Levin, The elliptic polylogarithm, Motives, vol.55, pp.123-190, 1991.

A. A. Be?linson, Higher regulators and values of L-functions, Current problems in mathematics, vol.24, pp.181-238, 1984.

J. Bismut and J. Cheeger, Remarks on the index theorem for families of Dirac operators on manifolds with boundary, Differential geometry, vol.52, pp.59-83, 1991.

J. Bismut, H. Gillet, and C. Soulé, Analytic torsion and holomorphic determinant bundles. I. Bott-Chern forms and analytic torsion, Comm. Math. Phys, vol.115, issue.1, pp.49-78, 1988.

J. Bismut and J. Cheeger, Transgressed Euler classes of SL(2n, Z) vector bundles, adiabatic limits of eta invariants and special values of L-functions, Ann. Sci. École Norm. Sup, vol.25, issue.4, pp.335-391, 1992.

D. Blasius, On the critical values of Hecke L-series, Ann. of Math, vol.124, issue.2, pp.23-63, 1986.

L. A. Borisov and P. E. Gunnells, Toric modular forms and nonvanishing of L-functions, J. Reine Angew. Math, vol.539, pp.149-165, 2001.

L. A. Borisov and P. E. Gunnells, Toric varieties and modular forms, Invent. Math, vol.144, issue.2, pp.297-325, 2001.

R. Bott and L. W. Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics, vol.82, 1982.

F. Brunault, Beilinson-Kato elements in K 2 of modular curves, Acta Arith, vol.134, issue.3, pp.283-298, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00561919

P. Cassou-noguès, Valeurs aux entiers négatifs des fonctions zêta et fonctions zêta p-adiques, Invent. Math, vol.51, issue.1, pp.29-59, 1979.

P. Charollois and S. Dasgupta, Integral Eisenstein cocycles on GLn, I: Sczech's cocycle and p-adic L-functions of totally real fields, Camb. J. Math, vol.2, issue.1, pp.49-90, 2014.

P. Charollois, S. Dasgupta, and M. Greenberg, II: Shintani's method, Integral Eisenstein cocycles on GLn, vol.90, issue.2, pp.435-477, 2015.

P. Charollois and R. Sczech, Elliptic functions according to Eisenstein and Kronecker: an update, Eur. Math. Soc. Newsl, issue.101, pp.8-14, 2016.

. Shiing-shen-chern, A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds, Ann. of Math, vol.45, issue.2, pp.747-752, 1944.

P. Colmez, Algébricité de valeurs spéciales de fonctions L, Invent. Math, vol.95, issue.1, pp.161-205, 1989.

R. M. Damerell, L-functions of elliptic curves with complex multiplication, I. Acta Arith, vol.17, pp.287-301, 1970.

P. Deligne, Automorphic forms, representations and Lfunctions (Proc. Sympos. Pure Math, vol.2, pp.313-346, 1977.

P. Deligne and K. A. Ribet, Values of abelian L-functions at negative integers over totally real fields, Invent. Math, vol.59, issue.3, pp.227-286, 1980.

G. Faltings, Arithmetic Eisenstein classes on the Siegel space: some computations, Number fields and function fields-two parallel worlds, vol.239, pp.133-166, 2005.

J. Flórez, C. Karabulut, and T. Wong, Eisenstein cocycles over imaginary quadratic fields and special values of L-functions, J. Number Theory, vol.204, pp.497-531, 2019.

L. E. Garcia, Superconnections, theta series, and period domains, Adv. Math, vol.329, pp.555-589, 2018.

E. Getzler, The Thom class of Mathai and Quillen and probability theory, Stochastic analysis and applications, vol.26, pp.111-122, 1989.

B. Alexander, . Goncharov, and . Regulators, Handbook of K-theory, vol.1, pp.295-349, 2005.

P. Graf, Polylogarithms for GL 2 over totally real fields, 2016.

G. Harder and N. Schappacher, Special values of Hecke L-functions and abelian integrals, Workshop Bonn, vol.1111, pp.17-49, 1984.

G. Harder, Some results on the Eisenstein cohomology of arithmetic subgroups of GLn, Cohomology of arithmetic groups and automorphic forms, vol.1447, pp.85-153, 1989.

G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 2008.

R. Howe, ?-series and invariant theory, Automorphic forms, representations and L-functions (Proc. Sympos, pp.275-285, 1977.

T. Ishii and T. Oda, A short history on investigation of the special values of zeta and L-functions of totally real number fields, Automorphic forms and zeta functions, pp.198-233, 2006.

H. Ito, A function on the upper half space which is analogous to the imaginary part of log ?(z), J. Reine Angew. Math, vol.373, pp.148-165, 1987.

K. Kato, p-adic Hodge theory and values of zeta functions of modular forms, Cohomologies p-adiques et applications arithmétiques, vol.295, pp.117-290, 2004.

H. Klingen, über die Werte der Dedekindschen Zetafunktion, Math. Ann, vol.145, pp.265-272, 1961.

S. S. Kudla, Seesaw dual reductive pairs, Automorphic forms of several variables, vol.46, pp.244-268, 1983.

S. Stephen, J. J. Kudla, and . Millson, Intersection numbers of cycles on locally symmetric spaces and Fourier coefficients of holomorphic modular forms in several complex variables, Inst. Hautes Études Sci. Publ. Math, issue.71, pp.121-172, 1990.

V. Mathai and D. Quillen, Superconnections, Thom classes, and equivariant differential forms, Topology, vol.25, issue.1, pp.85-110, 1986.

W. Müller, Signature defects of cusps of Hilbert modular varieties and values of L-series at s = 1, J. Differential Geom, vol.20, issue.1, pp.55-119, 1984.

V. Madhav and . Nori, Some Eisenstein cohomology classes for the integral unimodular group, Proceedings of the International Congress of Mathematicians, vol.1, pp.690-696, 1994.

R. Sczech, Dedekindsummen mit elliptischen Funktionen, Invent. Math, vol.76, issue.3, pp.523-551, 1984.

R. Sczech, Eisenstein cocycles for GL 2 Q and values of L-functions in real quadratic fields, Comment. Math. Helv, vol.67, issue.3, pp.363-382, 1992.

R. Sczech, Eisenstein group cocycles for GLn and values of L-functions, Invent. Math, vol.113, issue.3, pp.581-616, 1993.

T. Shintani, On evaluation of zeta functions of totally real algebraic number fields at non-positive integers, J. Fac. Sci. Univ. Tokyo Sect. IA Math, vol.23, issue.2, pp.393-417, 1976.

C. Siegel, Über die analytische Theorie der quadratischen Formen, III. Ann. of Math, vol.38, issue.2, pp.212-291, 1937.

C. Siegel, Advanced analytic number theory, Tata Institute of Fundamental Research Studies in Mathematics. Tata Institute of Fundamental Research, vol.9, 1980.

C. Siegel, Gesammelte Abhandlungen. IV. Springer Collected Works in Mathematics, 2015.

D. Sullivan, La classe d'Euler réelle d'un fibré vectoriel à groupe structural SLn(Z) est nulle, C. R. Acad. Sci. Paris Sér. A-B, vol.281, issue.1, pp.17-18, 1975.

A. Weil, Elliptic functions according to Eisenstein and Kronecker, Classics in Mathematics, 1999.

F. Wielonsky, Séries d'Eisenstein, intégrales toroïdales et une formule de Hecke, Enseign. Math, issue.2, pp.93-135, 1985.

. Ens-/-psl, D. University, . De, F. Et-applications, P. et al., nicolas.bergeron@ens.fr URL