The Hodge realization of the polylogarithm and the Shintani generating class for totally real fields. (English) Zbl 07857681
The Beilinson conjecture for algebraic varieties and more general motives defined over algebraic number fields is a conjecture relating the special values of Hasse-Weil \(L\)-functions
to the determinant of the regulator map from motivic cohomology to Deligne-Beilinson cohomology.
This conjecture is a vast generalization of the Gross conjecture for special values of Artin \(L\)-functions B. H. Gross [Pure Appl. Math. Q. 1, No. 1, 1–13 (2005; Zbl 1169.11050)], based on the work by A. Borel [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 4, 613–636 (1977; Zbl 0382.57027)] on special values of Dedeking zeta functions. The general strategy for the proof of this conjecture is to construct explicit elements in motivic cohomology whose image by the regulator map is amenable to calculation. Beilinson originally proved the Beilinson conjecture for the noncritical values of Dirichlet \(L\)-functions by constructing the cyclotomic elements in the motivic cohomology, or equivalently \(K\)-groups, associated to the spectrum of cyclotomic fields [A. A. Beilinson, J. Sov. Math. 30, 2036–2070 (1985; Zbl 0588.14013); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 24, 181–238 (1984); J. Neukirch, Perspect. Math. 4, 193–247 (1988; Zbl 0651.12009)]. Beilinson and Deligne provided a universal construction of the cyclotomic elements in terms of the motivic polylogarithm, a class constructed in the motivic cohomology of the projective line minus three points. This allows for a simplified and conceptual proof of the Beilinson conjecture for the Dirichlet \(L\)-function, by reducing the complicated calculation of the regulator to the calculation of the Hodge realization of the polylogarithm class [A. Beilinson and P. Deligne, Proc. Symp. Pure Math. 55, 97–121 (1994; Zbl 0799.19004); P. Deligne, Publ., Math. Sci. Res. Inst. 16, . 79–297 (1989; Zbl 0742.14022); A. Huber and J. Wildeshaus, Doc. Math. 3, 27–133 (1998; Zbl 0906.19004)]. Although perhaps not written explicitly in literature, similar strategy may be applied to the proof of the Beilinson conjecture for Hecke character associated to imaginary quadratic fields [C. Deninger, Symp. Math. 37, 99–137 (1997; Zbl 0888.14002)], originally proved by C. Deninger [Invent. Math. 96, No. 1, 1–69 (1989; Zbl 0721.14004); Ann. Math. (2) 132, No. 1, 131–158 (1990; Zbl 0721.14005)], using the elliptic polylogarithm constructed by A. Beilinson and A. Levin [Proc. Symp. Pure Math. 55, 123–190 (1994; Zbl 0817.14014)].
In this article, we prospect a similar strategy for Hecke \(L\)-functions of totally real fields, using an equivariant version of the polylogarithm for certain algebraic tori associated to totally real fields.
Let \(F\) be a totally real field and \(g=[F\colon\mathbb Q]\). We denote by \(\mathfrak{I}\) the set of nonzero fractional ideals of \(F\), and by the multiplicative group of totally positive elements of \(F\). For any \(\mathfrak{a}\in\mathfrak{I}\), we let \(\mathbb T^{\mathfrak{a}}=\mathrm{Hom}(\mathfrak{a},\mathbb G_m),\) which is a \(g\)-dimensional affine group scheme over \(\mathbb Q\). If we let \begin{align*} U&=\mathrm{coprod}_{\mathfrak{a}\in\mathfrak{I}}U^{\mathfrak{a}} \end{align*} for \(U^{\mathfrak{a}}=\mathbb T^{\mathfrak{a}}\setminus\{1\}\), then \(U\) has a natural action of \(F\times_+\). We regard the quotient stack \(U/F\times_+\) as a generalization of \(\mathbb P^1\setminus\{0,1,\infty\}\); indeed, when \(F=\mathbb Q\), we have \(U/F\times_+\cong\mathbb G_m\setminus\{1\}=\mathbb P^1\setminus\{0,1,\infty\}\).
The polylogarithm classes of general commutative group schemes were constructed by A. Huber and G. Kings [J. Algebr. Geom. 27, No. 3, 449–495 (2018; Zbl 1464.14012)]. We consider an equivariant version of their construction in the case of \(U/F\times_+\). More precisely, we construct an element of the equivariant Deligne-Beilinson cohomology \(H^{2g-1}_{\Delta}(U/F\times_+,\mathbb L\mathrm{og})\) with coefficients in the logarithm sheaf \(\mathbb L\mathrm{og}\), which is a certain equivariant pro-unipotent variation of mixed \(\mathbb R\)-Hodge structures on \(U\). Since the general theory of equivariant variations of mixed \(\mathbb R\)-Hodge structures and equivariant Deligne-Beilinson cohomology are not yet sufficiently developed, we give a definition of equivariant Deligne-Beilinson cohomology \(H^{m}_{\Delta}(U/F\times_+,\mathbb L\mathrm{og})\) specific to our case, using the logarithmic Dolbeault complex.
We define the polylogarithm class in equivariant Deligne-Beilinson cohomology as follows.
Definition polylog. Let \(\mathrm{Cl}^+_F\) denote the narrow ideal class group of \(F\). We define the polylogarithm class \[ \operatorname{pol}\in H^{2g-1}_{\Delta}(U/F\times_+,\mathbb L\mathrm{og}) \] to be the class in the equivariant Deligne-Beilinson cohomology of \(U\) with coefficients in \(\mathbb L\mathrm{og}\) that maps to \((1,\ldots,1)\in \bigoplus_{\mathrm{Cl}^+_F}\mathbb R\) through the isomorphism \(H^{2g-1}_{\Delta}(U/F\times_+,\mathbb L\mathrm{og})\cong\bigoplus_{\mathrm{Cl}^+_F}\mathbb R\) given in Proposition (key).
When \(F=\mathbb Q\), the polylogarithm class \(\operatorname{pol}\) coincides with the Hodge realization of the polylogarithm class constructed by Beilinson and Deligne on the projective line minus three points.
Previously in [K. Bannai et al., Asian J. Math. 24, No. 1, 31–76 (2020; Zbl 1457.14020)], we constructed the Shintani generating class, which is a certain equivariant cohomology class on \(U\) generating the values of the Lerch zeta function of \(F\) at nonpositive integers. Our main theorem relates our polylogarithm class with the Shintani generating class.
Theorem 1. Under a natural inclusion \[ H^{2g-1}_{\Delta}(U/F\times_+,\mathbb L\mathrm{og})\hookrightarrow H^{2g-1}_{\mathrm{dR}}(U/F\times_+,\mathbb L\mathrm{og}\otimes\mathscr O_U), \] the polylogarithm class \(\operatorname{pol}\) maps to the de Rham Shintani class \(\mathcal S\), constructed from the Shintani generating class.
Theorem 1 indicates that our polylogarithm class is related to the critical values of Hecke \(L\)-functions, since these values are written in terms of the Lerch zeta values at nonpositive integers. We further expect that our polylogarithm class is related to the Lerch zeta values at positive integers, and hence to the noncritical Hecke \(L\)-values. Let \(\xi\) be a nontrivial torsion point in \(U\), and let \(\xi\Delta\) be the \(\Delta\) orbit of \(\xi\) with respect to the action of \(\Delta=\mathcal{O}_{F+}^\times\). Assuming further the existence of a conjectural equivariant plectic Hodge theory, fitting in with the formalism of the category of mixed plectic \(\mathbb R\)-Hodge structures \(\mathrm{MHS}^{\boxtimes I}_{\mathbb R}\) defined by J. Nekovář and A. J. Scholl [Contemp. Math. 664, 321–337 (2016; Zbl 1402.11092)], we may define the specialization \[ i_{\xi\Delta}^*\colon H^{2g-1}_{\Delta}(U/F\times_+,\mathbb L\mathrm{og}) \rightarrow H^{2g-1}_{\Delta^I}(\xi\Delta/\Delta,i_{\xi\Delta}^*\mathbb L\mathrm{og}) \cong\prod_{n=1}^\infty(2\pi i)^{(n-1)g}\mathbb R\tag{1} \] with respect to the equivariant morphism \(i_{\xi\Delta}\colon\xi\Delta\rightarrow U\). We propose the following:
Conjecture 1. Let \(\xi\) be a torsion point in \(U\), and we let \(\xi\Delta\) be the orbit of \(\xi\) in \(U\). Then \(i_{\xi\Delta}^*\operatorname{pol}\in H^{2g-1}_{\Delta^I}(\xi\Delta/\Delta,i_{\xi\Delta}^*\mathbb L\mathrm{og})=\prod_{n=1}^\infty(2\pi i)^{(n-1)g}\mathbb R\) satisfies \[ i_{\xi\Delta}^*\operatorname{pol}=(d_F^{1/2}\mathcal{L}^\infty(\xi\Delta,n))_{n=1}^\infty\in\prod_{n=1}^\infty(2\pi i)^{(n-1)g}\mathbb R. \] Here \(d_F\) denotes the discriminant of \(F\) and \[ \mathcal{L}^\infty(\xi\Delta,n)=\begin{cases} \mathrm{Re}\mathcal{L}(\xi\Delta,n) & \text{ if }(n-1)g\text{ is even}, \\ i\,\mathrm{Im}\mathcal{L}(\xi\Delta,n) & \text{ if }(n-1)g\text{ is odd}, \end{cases} \] where \(\mathcal L(\xi\Delta,s)\) is the Lerch zeta function corresponding to the point \(\xi\) (see Definition 1).
In a subsequent research, we will also consider a syntomic version of this conjecture.
If \(F=\mathbb Q\), then the Lerch zeta function for a nontrivial torsion point \(\xi\in\mathbb G_m(\mathbb C)\) is given as \(\mathcal L(\xi,s)=\sum_{n=1}^\infty \xi^n n^{-s}\), hence we have \(\mathcal L(\xi,k)=\mathrm{Li}_k(\xi)\), where \(\mathrm{Li}_k(\xi)\) is the value at \(\xi\) of the \(k\)-th polylogarithm function given by the series \(\mathrm{Li}_k(t)=\sum_{n=1}^\infty t^n n^{-k}\). Therefore Conjecture 1 is a direct generalization of the result of Beilinson and Deligne on the calculation of the specialization of \(\operatorname{pol}\) at nontrivial roots of unity. Recall that this calculation and the motivicity of the polylogarithm class lead to a conceptual proof of the Beilinson conjecture for the Dirichlet \(L\)-functions.
In this article, we prospect a similar strategy for Hecke \(L\)-functions of totally real fields, using an equivariant version of the polylogarithm for certain algebraic tori associated to totally real fields.
Let \(F\) be a totally real field and \(g=[F\colon\mathbb Q]\). We denote by \(\mathfrak{I}\) the set of nonzero fractional ideals of \(F\), and by the multiplicative group of totally positive elements of \(F\). For any \(\mathfrak{a}\in\mathfrak{I}\), we let \(\mathbb T^{\mathfrak{a}}=\mathrm{Hom}(\mathfrak{a},\mathbb G_m),\) which is a \(g\)-dimensional affine group scheme over \(\mathbb Q\). If we let \begin{align*} U&=\mathrm{coprod}_{\mathfrak{a}\in\mathfrak{I}}U^{\mathfrak{a}} \end{align*} for \(U^{\mathfrak{a}}=\mathbb T^{\mathfrak{a}}\setminus\{1\}\), then \(U\) has a natural action of \(F\times_+\). We regard the quotient stack \(U/F\times_+\) as a generalization of \(\mathbb P^1\setminus\{0,1,\infty\}\); indeed, when \(F=\mathbb Q\), we have \(U/F\times_+\cong\mathbb G_m\setminus\{1\}=\mathbb P^1\setminus\{0,1,\infty\}\).
The polylogarithm classes of general commutative group schemes were constructed by A. Huber and G. Kings [J. Algebr. Geom. 27, No. 3, 449–495 (2018; Zbl 1464.14012)]. We consider an equivariant version of their construction in the case of \(U/F\times_+\). More precisely, we construct an element of the equivariant Deligne-Beilinson cohomology \(H^{2g-1}_{\Delta}(U/F\times_+,\mathbb L\mathrm{og})\) with coefficients in the logarithm sheaf \(\mathbb L\mathrm{og}\), which is a certain equivariant pro-unipotent variation of mixed \(\mathbb R\)-Hodge structures on \(U\). Since the general theory of equivariant variations of mixed \(\mathbb R\)-Hodge structures and equivariant Deligne-Beilinson cohomology are not yet sufficiently developed, we give a definition of equivariant Deligne-Beilinson cohomology \(H^{m}_{\Delta}(U/F\times_+,\mathbb L\mathrm{og})\) specific to our case, using the logarithmic Dolbeault complex.
We define the polylogarithm class in equivariant Deligne-Beilinson cohomology as follows.
Definition polylog. Let \(\mathrm{Cl}^+_F\) denote the narrow ideal class group of \(F\). We define the polylogarithm class \[ \operatorname{pol}\in H^{2g-1}_{\Delta}(U/F\times_+,\mathbb L\mathrm{og}) \] to be the class in the equivariant Deligne-Beilinson cohomology of \(U\) with coefficients in \(\mathbb L\mathrm{og}\) that maps to \((1,\ldots,1)\in \bigoplus_{\mathrm{Cl}^+_F}\mathbb R\) through the isomorphism \(H^{2g-1}_{\Delta}(U/F\times_+,\mathbb L\mathrm{og})\cong\bigoplus_{\mathrm{Cl}^+_F}\mathbb R\) given in Proposition (key).
When \(F=\mathbb Q\), the polylogarithm class \(\operatorname{pol}\) coincides with the Hodge realization of the polylogarithm class constructed by Beilinson and Deligne on the projective line minus three points.
Previously in [K. Bannai et al., Asian J. Math. 24, No. 1, 31–76 (2020; Zbl 1457.14020)], we constructed the Shintani generating class, which is a certain equivariant cohomology class on \(U\) generating the values of the Lerch zeta function of \(F\) at nonpositive integers. Our main theorem relates our polylogarithm class with the Shintani generating class.
Theorem 1. Under a natural inclusion \[ H^{2g-1}_{\Delta}(U/F\times_+,\mathbb L\mathrm{og})\hookrightarrow H^{2g-1}_{\mathrm{dR}}(U/F\times_+,\mathbb L\mathrm{og}\otimes\mathscr O_U), \] the polylogarithm class \(\operatorname{pol}\) maps to the de Rham Shintani class \(\mathcal S\), constructed from the Shintani generating class.
Theorem 1 indicates that our polylogarithm class is related to the critical values of Hecke \(L\)-functions, since these values are written in terms of the Lerch zeta values at nonpositive integers. We further expect that our polylogarithm class is related to the Lerch zeta values at positive integers, and hence to the noncritical Hecke \(L\)-values. Let \(\xi\) be a nontrivial torsion point in \(U\), and let \(\xi\Delta\) be the \(\Delta\) orbit of \(\xi\) with respect to the action of \(\Delta=\mathcal{O}_{F+}^\times\). Assuming further the existence of a conjectural equivariant plectic Hodge theory, fitting in with the formalism of the category of mixed plectic \(\mathbb R\)-Hodge structures \(\mathrm{MHS}^{\boxtimes I}_{\mathbb R}\) defined by J. Nekovář and A. J. Scholl [Contemp. Math. 664, 321–337 (2016; Zbl 1402.11092)], we may define the specialization \[ i_{\xi\Delta}^*\colon H^{2g-1}_{\Delta}(U/F\times_+,\mathbb L\mathrm{og}) \rightarrow H^{2g-1}_{\Delta^I}(\xi\Delta/\Delta,i_{\xi\Delta}^*\mathbb L\mathrm{og}) \cong\prod_{n=1}^\infty(2\pi i)^{(n-1)g}\mathbb R\tag{1} \] with respect to the equivariant morphism \(i_{\xi\Delta}\colon\xi\Delta\rightarrow U\). We propose the following:
Conjecture 1. Let \(\xi\) be a torsion point in \(U\), and we let \(\xi\Delta\) be the orbit of \(\xi\) in \(U\). Then \(i_{\xi\Delta}^*\operatorname{pol}\in H^{2g-1}_{\Delta^I}(\xi\Delta/\Delta,i_{\xi\Delta}^*\mathbb L\mathrm{og})=\prod_{n=1}^\infty(2\pi i)^{(n-1)g}\mathbb R\) satisfies \[ i_{\xi\Delta}^*\operatorname{pol}=(d_F^{1/2}\mathcal{L}^\infty(\xi\Delta,n))_{n=1}^\infty\in\prod_{n=1}^\infty(2\pi i)^{(n-1)g}\mathbb R. \] Here \(d_F\) denotes the discriminant of \(F\) and \[ \mathcal{L}^\infty(\xi\Delta,n)=\begin{cases} \mathrm{Re}\mathcal{L}(\xi\Delta,n) & \text{ if }(n-1)g\text{ is even}, \\ i\,\mathrm{Im}\mathcal{L}(\xi\Delta,n) & \text{ if }(n-1)g\text{ is odd}, \end{cases} \] where \(\mathcal L(\xi\Delta,s)\) is the Lerch zeta function corresponding to the point \(\xi\) (see Definition 1).
In a subsequent research, we will also consider a syntomic version of this conjecture.
If \(F=\mathbb Q\), then the Lerch zeta function for a nontrivial torsion point \(\xi\in\mathbb G_m(\mathbb C)\) is given as \(\mathcal L(\xi,s)=\sum_{n=1}^\infty \xi^n n^{-s}\), hence we have \(\mathcal L(\xi,k)=\mathrm{Li}_k(\xi)\), where \(\mathrm{Li}_k(\xi)\) is the value at \(\xi\) of the \(k\)-th polylogarithm function given by the series \(\mathrm{Li}_k(t)=\sum_{n=1}^\infty t^n n^{-k}\). Therefore Conjecture 1 is a direct generalization of the result of Beilinson and Deligne on the calculation of the specialization of \(\operatorname{pol}\) at nontrivial roots of unity. Recall that this calculation and the motivicity of the polylogarithm class lead to a conceptual proof of the Beilinson conjecture for the Dirichlet \(L\)-functions.
Reviewer: Mohammad Reza Rahmati (León)
MSC:
| 11G55 | Polylogarithms and relations with \(K\)-theory |
| 14C30 | Transcendental methods, Hodge theory (algebro-geometric aspects) |
| 11M35 | Hurwitz and Lerch zeta functions |
| 11R42 | Zeta functions and \(L\)-functions of number fields |
| 11R80 | Totally real fields |
Keywords:
polylogarithm; totally real fields; Hecke \(L\)-function; Lerch zeta function; equivariance; algebraic torusCitations:
Zbl 1169.11050; Zbl 0382.57027; Zbl 0588.14013; Zbl 0651.12009; Zbl 0799.19004; Zbl 0742.14022; Zbl 0906.19004; Zbl 0888.14002; Zbl 0721.14004; Zbl 0721.14005; Zbl 0817.14014; Zbl 1464.14012; Zbl 1457.14020; Zbl 1402.11092References:
| [1] | Bannai, K.; Kobayashi, S.; Tsuji, T., On the de Rhamand p-adic realizations of the elliptic polylogarithm for CM elliptic curves, Ann. Sci. Éc. Norm. Supér. (4), 43, 2, 185-234, 2010, English, with English and French summaries, MR2662664 · Zbl 1197.11073 |
| [2] | Bannai, K.; Hagihara, K.; Kobayashi, S.; Yamada, K.; Yamamoto, S.; Yasuda, S., Category of mixed plectic Hodge structures, Asian J. Math., 24, 1, 31-76, 2020, MR4143982 · Zbl 1457.14020 |
| [3] | Bannai, K.; Hagihara, K.; Yamada, K.; Yamamoto, S., The Hodge realization of the polylogarithm on the product of multiplicative groups, Math. Z., 296, 3-4, 1787-1817, 2020, MR4159851 · Zbl 1469.11223 |
| [4] | Bannai, K.; Hagihara, K.; Yamada, K.; Yamamoto, S., p-adic polylogarithms and p-adic Hecke L-functions for totally real fields, J. Reine Angew. Math., 791, 53-87, 2022, MR4489625 · Zbl 1507.11108 |
| [5] | Bannai, K.; Hagihara, K.; Yamada, K.; Yamamoto, S., Canonical equivariant cohomology classes generating zeta values of totally real fields, Trans. Amer. Math. Soc. Ser. B, 10, 613-635, 2023, MR4583122 · Zbl 1531.11082 |
| [6] | Beĭlinson, A., Higher Regulators and Values of L-Functions, Current Problems in Mathematics, vol. 24, 181-238, 1984, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform: Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform Moscow, Russian, MR760999 |
| [7] | Beĭlinson, A., Notes on absolute Hodge cohomology, (Applications of Algebraic K-Theory to Algebraic Geometry and Number Theory, Part I, II. Applications of Algebraic K-Theory to Algebraic Geometry and Number Theory, Part I, II, Boulder, Colo., 1983. Applications of Algebraic K-Theory to Algebraic Geometry and Number Theory, Part I, II. Applications of Algebraic K-Theory to Algebraic Geometry and Number Theory, Part I, II, Boulder, Colo., 1983, Contemp. Math., vol. 55, 1986, Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 35-68, MR862628 · Zbl 0621.14011 |
| [8] | Beĭlinson, A.; Deligne, P., Interprétation motivique de la conjecture de Zagier reliant polylogarithmes et régulateurs, (Proc. Sympos. Pure Math., Motives. Proc. Sympos. Pure Math., Motives, Seattle, WA, 1991, vol. 55, 1994, Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 97-121, French, MR1265552 · Zbl 0799.19004 |
| [9] | Beĭlinson, A.; Kings, G.; Levin, A., Topological polylogarithms and p-adic interpolation of L-values of totally real fields, Math. Ann., 371, 3-4, 1449-1495, 2018, MR3831278 · Zbl 1448.11122 |
| [10] | Beĭlinson, A.; Levin, A., The elliptic polylogarithm, motives, (Proc. Sympos. Pure Math.. Proc. Sympos. Pure Math., Seattle, WA, 1991, vol. 55, 1994, Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 123-190, MR1265553 · Zbl 0817.14014 |
| [11] | Blottière, D., Réalisation de Hodge du polylogarithme d’un schéma abélien, J. Inst. Math. Jussieu, 8, 1, 1-38, 2009, French, with English and French summaries, MR2461901 · Zbl 1160.14033 |
| [12] | Borel, A., Cohomologie de \(\operatorname{S} L_n\) et valeurs de fonctions zeta aux points entiers, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), 4, 4, 613-636, 1977, French, MR506168 · Zbl 0382.57027 |
| [13] | Burgos, J. I., A \(\mathcal{C}^\infty\) logarithmic Dolbeault complex, Compos. Math., 92, 1, 61-86, 1994, MR1275721 · Zbl 0826.32007 |
| [14] | Burgos, J. I.; Wildeshaus, J., Hodge modules on Shimura varieties and their higher direct images in the Baily-Borel compactification, Ann. Sci. Éc. Norm. Supér. (4), 37, 3, 363-413, 2004, English, with English and French summaries, MR2060481 · Zbl 1073.14036 |
| [15] | Brylinski, J.-L.; Zucker, S., An Overview of Recent Advances in Hodge Theory, Several Complex Variables, VI, Encyclopaedia Math. Sci., vol. 69, 39-142, 1990, Springer: Springer Berlin, MR1095090 · Zbl 0711.00012 |
| [16] | Cirici, J.; Guillén, F., Homotopy theory of mixed Hodge complexes, Tohoku Math. J. (2), 68, 3, 349-375, 2016, MR3550924 · Zbl 1359.55013 |
| [17] | Deligne, P., Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, vol. 163, 1970, Springer-Verlag: Springer-Verlag Berlin-New York, French, MR0417174 · Zbl 0244.14004 |
| [18] | Deligne, P., Théorie de Hodge. II, Publ. Math. Inst. Hautes Études Sci., 40, 5-57, 1971, French, MR0498551 · Zbl 0219.14007 |
| [19] | Deligne, P., Théorie de Hodge. III, Publ. Math. Inst. Hautes Études Sci., 44, 5-77, 1974, French, MR0498552 · Zbl 0237.14003 |
| [20] | Deligne, P., Le groupe fondamental de la droite projective moins trois points, Galois groups over \(\mathbb{Q} \), (Math. Sci. Res. Inst. Publ.. Math. Sci. Res. Inst. Publ., Berkeley, CA, 1987, vol. 16, 1989, Springer: Springer New York), 79-297, French, MR1012168 · Zbl 0742.14022 |
| [21] | Deninger, C., Higher regulators and Hecke L-series of imaginary quadratic fields. I, Invent. Math., 96, 1, 1-69, 1989, MR981737 · Zbl 0721.14004 |
| [22] | Deninger, C., Higher regulators and Hecke L-series of imaginary quadratic fields. II, Ann. Math. (2), 132, 1, 131-158, 1990, MR1059937 · Zbl 0721.14005 |
| [23] | Deninger, C., Extensions of motives associated to symmetric powers of elliptic curves and to Hecke characters of imaginary quadratic fields, (Arithmetic Geometry. Arithmetic Geometry, Cortona, 1994. Arithmetic Geometry. Arithmetic Geometry, Cortona, 1994, Sympos. Math., vol. XXXVII, 1997, Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 99-137, MR1472494 · Zbl 0888.14002 |
| [24] | Griffiths, P.; Harris, J., Principles of Algebraic Geometry, Pure and Applied Mathematics, 1978, Wiley-Interscience [John Wiley & Sons]: Wiley-Interscience [John Wiley & Sons] New York, MR507725 · Zbl 0408.14001 |
| [25] | Gross, B. H., On the values of Artin L-functions, preprint, available at · Zbl 1169.11050 |
| [26] | Grothendieck, A., Sur quelques points d’algèbre homologique, Tohoku Math. J. (2), 9, 119-221, 1957, French, MR102537 · Zbl 0118.26104 |
| [27] | Hain, R. M.; Zucker, S., Unipotent variations of mixed Hodge structure, Invent. Math., 88, 1, 83-124, 1987, MR877008 · Zbl 0622.14007 |
| [28] | Huber, A., Calculation of derived functors via Ind-categories, J. Pure Appl. Algebra, 90, 1, 39-48, 1993, MR1246272 · Zbl 0793.18007 |
| [29] | Huber, A., Mixed Motives and Their Realization in Derived Categories, Lecture Notes in Mathematics, vol. 1604, 1995, Springer- Verlag: Springer- Verlag Berlin, MR1439046 · Zbl 0938.14008 |
| [30] | Huber, A.; Kings, G., Polylogarithm for families of commutative group schemes, J. Algebraic Geom., 27, 449-495, 2018 · Zbl 1464.14012 |
| [31] | Huber, A.; Wildeshaus, J., Classical motivic polylogarithm according to Beilinson and Deligne, Doc. Math., 3, 27-133, 1998, MR1643974 · Zbl 0906.19004 |
| [32] | Kashiwara, M., A study of variation of mixed Hodge structure, Publ. Res. Inst. Math. Sci., 22, 5, 991-1024, 1986, MR866665 · Zbl 0621.14007 |
| [33] | Kashiwara, M.; Schapira, P., Categories and Sheaves, Grundlehren der mathematischen Wissenschaften, vol. 332, 2006, Springer-Verlag: Springer-Verlag Berlin, MR2182076 · Zbl 1118.18001 |
| [34] | Levin, A., Elliptic polylogarithms: an analytic theory, Compos. Math., 106, 3, 267-282, 1997, MR1457106 · Zbl 0905.11028 |
| [35] | Levin, A., Polylogarithmic currents on abelian varieties, (Regulators in Analysis, Geometry and Number Theory. Regulators in Analysis, Geometry and Number Theory, Progr. Math., vol. 171, 2000, Birkhäuser Boston: Birkhäuser Boston Boston, MA), 207-229, MR1724893 · Zbl 1135.11325 |
| [36] | Liu, Q., Algebraic Geometry and Arithmetic Curves, Oxford Graduate Texts in Mathematics, vol. 6, 2002, Oxford University Press: Oxford University Press Oxford, translated from the French by Reinie Erné, Oxford Science Publications, MR1917232 · Zbl 0996.14005 |
| [37] | Milne, J. S., Étale Cohomology, Princeton Mathematical Series, vol. 33, 1980, Princeton University Press: Princeton University Press Princeton, N.J., MR559531 · Zbl 0433.14012 |
| [38] | Nekovář, J.; Scholl, A. J., Introduction to Plectic Cohomology, Advances in the Theory of Automorphic Forms and Their L-Functions, Contemp. Math., vol. 664, 321-337, 2016, Amer. Math. Soc.: Amer. Math. Soc. Providence, RI, MR3502988 · Zbl 1402.11092 |
| [39] | Neukirch, J., The Beilinson Conjecture for Algebraic Number Fields, Beilinson’s Conjectures on Special Values of L-Functions, Perspect. Math., vol. 4, 193-247, 1988, Academic Press: Academic Press Boston, MA, MR944995 · Zbl 0651.12009 |
| [40] | Neukirch, J., Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften, vol. 322, 1999, Springer-Verlag: Springer-Verlag Berlin, translated from the 1992 German original and with a note by Norbert Schappacher; with a foreword by G. Harder, MR1697859 · Zbl 0956.11021 |
| [41] | Peters, C. A.M.; Steenbrink, J. H.M., Mixed Hodge Structures, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 52, 2008, Springer-Verlag: Springer-Verlag Berlin, MR2393625 · Zbl 1138.14002 |
| [42] | (Rapoport, M.; Schappacher, N.; Schneider, P., Beilinson’s Conjectures on Special Values of L-Functions, Perspectives in Mathematics, vol. 4, 1988, Academic Press, Inc.: Academic Press, Inc. Boston, MA), MR944987 · Zbl 0635.00005 |
| [43] | Saito, M., Mixed Hodge modules, Publ. Res. Inst. Math. Sci., 26, 2, 221-333, 1990, MR1047415 · Zbl 0727.14004 |
| [44] | Scheider, R., The de Rham realization of the elliptic polylogarithm in families, 2014, available at |
| [45] | Schneider, P., Introduction to the Beilinson Conjectures, Beilinson’s Conjectures on Special Values of L-Functions, Perspect. Math., vol. 4, 1-35, 1988, Academic Press: Academic Press Boston, MA, MR944989 |
| [46] | Sprang, J., The algebraic de Rham realization of the elliptic polylogarithm via the Poincaré bundle, Algebra Number Theory, 14, 3, 545-585, 2020, MR4113775 · Zbl 1469.11225 |
| [47] | Zagier, D., Polylogarithms, Dedekind zeta functions and the algebraic K-theory of fields, (Arithmetic Algebraic Geometry. Arithmetic Algebraic Geometry, Texel, 1989. Arithmetic Algebraic Geometry. Arithmetic Algebraic Geometry, Texel, 1989, Progr. Math., vol. 89, 1991, Birkhäuser Boston: Birkhäuser Boston Boston, MA), 391-430, MR1085270 · Zbl 0728.11062 |
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