Baldi, Gregorio; Ullmo, Emmanuel Special subvarieties of non-arithmetic ball quotients and Hodge theory. (English) Zbl 1519.14025 Ann. Math. (2) 197, No. 1, 159-220 (2023); erratum ibid. 199, No. 1, 481 (2024). Let \(\Gamma \subset \text{PU} (1, n)\) be a lattice (meaning that \(\Gamma \backslash \text{PU} (1, n)\) admits a finite invariant measure). Such a lattice \(\Gamma\) is said to be arithmetic if it lies in the commensurability class of some image \(p (\mathbf{G} (\mathbb{Z}))\), where \(\mathbf{G}\) is a semisimple linear algebraic group over \(\mathbb{Q}\) and where \(p : \mathbf{G} (\mathbb{R}) \to \text{PU} (1, n)\) is a surjective homomorphism with compact kernel. This article gives a sufficient condition for \(\Gamma\) to be arithmetic in terms of the associated ball quotient \(S_{\Gamma} = \Gamma \backslash X\), where \(X\) is the hermitian symmetric space associated to \(\text{PU} (1, n)\); it does this by showing that \(\Gamma\) is arithmetic if \(S_{\Gamma}\) contains infinitely many maximal complex totally geodesic subvarieties. (The converse does not hold; in fact, there exist arithmetic \(\Gamma\) for which \(S_{\Gamma}\) does not allow any strict totally geodesic subvariety.)This main result is proved by studying unlikely intersection phenomena inside a period domain for polarised \(\mathbb{Z}\)-variations of Hodge structures. These techniques also yield an alternative proof of the characterization of arithmetic lattices by means of a commensurability criterion, a classical result that is originally due to G. A. Margulis [Discrete subgroups of semisimple Lie groups. Berlin etc.: Springer-Verlag (1991; Zbl 0732.22008)].Enabling the proof of the main result is the following idea. An argument using infinitesimal rigidity shows that the ball quotient \(S_{\Gamma}\) admits a \(\mathbb{Z}\)-variation of Hodge structures \(\widehat{\mathbb{V}}\). Complex-analytically, this gives rise to a period map \(\psi : S_{\Gamma}^{\text{an}} \to \widehat{\mathbf{G}} (\mathbb{Z}) \backslash D\). This gives rise to two bi-algebraic structures on \(S_{\Gamma}\), and hence to two notions of special (that is, bi-algebraic) subvarieties of \(S_{\Gamma}\). The first of these notions (that of \(\Gamma\)-special subvarieties of \(S_{\Gamma}\)) comes from the universal cover \(\pi : X \to S_{\Gamma}\); these are the images of the algebraic subvarieties of \(X\) along \(\pi\). The second notion (that of \(\mathbb{Z}\)-special subvarieties of \(S_{\Gamma}\)) uses the period map \(\psi\); these are the analytic irreducible components of \(\psi^{-1} (\pi_{\mathbb{Z}} (D'))\) that come from Mumford–Tate subdomains of \(D\). Arguably the main discovery of this paper, one that allows one to relate \(\mathbb{Z}\)-special subvarieties with unlikely intersection, is that the notions of \(\Gamma\)-special and \(\mathbb{Z}\)-special subvarieties of \(S_{\Gamma}\) coincide; both are, in turn, simply those subvarieties of \(S_{\Gamma}\) that are totally geodesic.The same techniques also allow the authors to apply work by Bakker-Tsimerman on the Ax-Schanuel conjecture [B. Bakker and J. Tsimerman, Invent. Math. 217, No. 1, 77–94 (2019; Zbl 1420.14021)] in the context of the current paper. This yields a proof of the following non-arithmetic version of the Ax-Schanuel conjecture : Let \(W \subset X \times S_{\Gamma}\) be an algebraic subvariety, and let \(\Pi : X \times S_{\Gamma}\) be the graph of the universal cover \(\pi : X \to S_{\Gamma}\). Then if \(U\) is an irreducible component of \(W \cap \Pi\) such that \(\dim (U) > \dim (W) - \dim (S_{\Gamma})\), we have that the projection of \(U\) to \(S_{\Gamma}\) is either zero-dimensional or contained in a strict totally geodesic subvariety of \(S_{\Gamma}\). Another consequence of the results in this article is that if \(H \subset \text{PU} (1, n)\) is of hermitian type, then the intersection \(\Gamma \cap H\) is a lattice in \(H\) provided that it is Zariski dense in \(H\). Reviewer: Jeroen Sijsling (Ulm) Cited in 1 ReviewCited in 4 Documents MSC: 14G35 Modular and Shimura varieties 22E40 Discrete subgroups of Lie groups 03C64 Model theory of ordered structures; o-minimality 14P10 Semialgebraic sets and related spaces 32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables Keywords:non-arithmetic lattices; rigidity; Hodge theory and Mumford-Tate domains; functional transcendence; unlikely intersections Citations:Zbl 0732.22008; Zbl 1420.14021 × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Andr\'{e}, Yves, Mumford-{T}ate groups of mixed {H}odge structures and the theorem of the fixed part, Compositio Math.. Compositio Mathematica, 82, 1-24 (1992) · Zbl 0770.14003 [2] Ash, Avner; Mumford, David; Rapoport, Michael; Tai, Yung-Sheng, Smooth {C}ompactifications of {L}ocally {S}ymmetric {V}arieties, Cambridge Mathematical Library, x+230 pp. (2010) · Zbl 1209.14001 · doi:10.1017/CBO9780511674693 [3] Bader, Uri; Fisher, David; Miller, Nicholas; Stover, Matthew, Arithmeticity, superrigidity and totally geodesic submanifolds of complex hyperbolic manifolds (2020) · Zbl 1534.53056 [4] Bader, Uri; Fisher, David; Miller, Nicholas; Stover, Matthew, Arithmeticity, {S}uperrigidity, and {T}otally {G}eodesic {S}ubmanifolds, Ann. of Math. (2). Annals of Mathematics. Second Series, 193, 837-861 (2021) · Zbl 07353243 · doi:10.4007/annals.2021.193.3.4 [5] Baily, Jr., W. L.; Borel, A., Compactification of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2). Annals of Mathematics. Second Series, 84, 442-528 (1966) · Zbl 0154.08602 · doi:10.2307/1970457 [6] Bakker, B.; Klingler, B.; Tsimerman, J., Tame topology of arithmetic quotients and algebraicity of {H}odge loci, J. Amer. Math. Soc.. Journal of the Amer. Math. Soc., 33, 917-939 (2020) · Zbl 1460.14027 · doi:10.1090/jams/952 [7] Bakker, Benjamin; Tsimerman, Jacob, The {A}x-{S}chanuel conjecture for variations of {H}odge structures, Invent. Math.. Inventiones Mathematicae, 217, 77-94 (2019) · Zbl 1420.14021 · doi:10.1007/s00222-019-00863-8 [8] Bass, H., Groups of integral representation type, Pacific J. Math., 86, 15-51 (1980) · Zbl 0444.20006 · doi:10.2140/pjm.1980.86.15 [9] Borel, Armand, Introduction aux groupes arithm\'{e}tiques, Publications de l’Institut de Math\'{e}matique de l’Universit\'{e} de Strasbourg, XV. Actualit\'{e}s Scientifiques et Industrielles, No. 1341, 125 pp. (1969) · Zbl 0186.33202 [10] Calabi, Eugenio; Vesentini, Edoardo, On compact, locally symmetric {K}\"{a}hler manifolds, Ann. of Math. (2). Annals of Mathematics. Second Series, 71, 472-507 (1960) · Zbl 0100.36002 · doi:10.2307/1969939 [11] Cattani, Eduardo; Deligne, Pierre; Kaplan, Aroldo, On the locus of {H}odge classes, J. Amer. Math. Soc.. Journal of the Amer. Math. Soc., 8, 483-506 (1995) · Zbl 0851.14004 · doi:10.2307/2152824 [12] Chan, Shan Tai; Mok, Ngaiming, Asymptotic total geodesy of local holomorphic curves exiting a bounded symmetric domain and applications to a uniformization problem for algebraic subsets, J. Differential Geom.. Journal of Differential Geometry, 120, 1-49 (2022) · Zbl 1497.32007 · doi:10.4310/jdg/1641413830 [13] Clozel, Laurent; Ullmo, Emmanuel, \'{E}quidistribution de sous-vari\'{e}t\'{e}s sp\'{e}ciales, Ann. of Math. (2). Annals of Mathematics. Second Series, 161, 1571-1588 (2005) · Zbl 1099.11031 · doi:10.4007/annals.2005.161.1571 [14] Cohen, Paula; Wolfart, J\"{u}rgen, Modular embeddings for some nonarithmetic {F}uchsian groups, Acta Arith.. Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica, 56, 93-110 (1990) · Zbl 0717.14014 · doi:10.4064/aa-56-2-93-110 [15] Cohen, Paula; Wolfart, J\"{u}rgen, Monodromie des fonctions d’{A}ppell, vari\'{e}t\'{e}s ab\'{e}liennes et plongement modulaire. Journ\'{e}es Arithm\'{e}tiques, 1989, Ast\'{e}risque. Ast\'{e}risque, 97-105 (1991) · Zbl 0769.14017 [16] Corlette, Kevin, Archimedean superrigidity and hyperbolic geometry, Ann. of Math. (2). Annals of Mathematics. Second Series, 135, 165-182 (1992) · Zbl 0768.53025 · doi:10.2307/2946567 [17] Corlette, Kevin; Simpson, Carlos, On the classification of rank-two representations of quasiprojective fundamental groups, Compos. Math.. Compositio Mathematica, 144, 1271-1331 (2008) · Zbl 1155.58006 · doi:10.1112/S0010437X08003618 [18] Daw, Christopher; Ren, Jinbo, Applications of the hyperbolic {A}x-{S}chanuel conjecture, Compos. Math.. Compositio Mathematica, 154, 1843-1888 (2018) · Zbl 1427.11057 · doi:10.1112/s0010437x1800725x [19] Deligne, Pierre; Mostow, G. D., Monodromy of hypergeometric functions and nonlattice integral monodromy, Inst. Hautes \'{E}tudes Sci. Publ. Math.. Institut des Hautes \'{E}tudes Scientifiques. Publications Math\'{e}matiques, 5-89 (1986) · Zbl 0615.22008 [20] Deligne, Pierre, Th\'{e}orie de {H}odge. {II}, Inst. Hautes \'{E}tudes Sci. Publ. Math.. Institut des Hautes \'{E}tudes Scientifiques. Publications Math\'{e}matiques, 5-57 (1971) · Zbl 0219.14007 [21] Author’s review, Travaux de {S}himura. S\'{e}minaire {B}ourbaki, 23\`eme ann\'{e}e (1970/71), {E}xp. {N}o. 389, Lecture Notes in Math., 244, 123-165 (1971) · Zbl 0224.00002 [22] Deligne, Pierre, Vari\'{e}t\'{e}s de {S}himura: interpr\'{e}tation modulaire, et techniques de construction de mod\`eles canoniques. Automorphic {F}orms, {R}epresentations and {\(L\)}-functions ({P}roc. {S}ympos. {P}ure {M}ath., {O}regon {S}tate {U}niv., {C}orvallis, {O}re., 1977), {P}art 2, Proc. Sympos. Pure Math., XXXIII, 247-289 (1979) [23] Delp, K.; Hoffoss, D.; Fox Manning, J., Problems in groups, geometry, and three-manifolds (2015) [24] Deng, Y., Big {P}icard theorem and algebraic hyperbolicity for varieties admitting a variation of {H}odge structures (2020) [25] Deraux, Martin, A new nonarithmetic lattice in {\({\rm PU}(3,1)\)}, Algebr. Geom. Topol.. Algebraic & Geometric Topology, 20, 925-963 (2020) · Zbl 1457.22003 · doi:10.2140/agt.2020.20.925 [26] Deraux, Martin; Parker, John R.; Paupert, Julien, New non-arithmetic complex hyperbolic lattices, Invent. Math.. Inventiones Mathematicae, 203, 681-771 (2016) · Zbl 1337.22007 · doi:10.1007/s00222-015-0600-1 [27] Deraux, Martin; Parker, John R.; Paupert, Julien, New nonarithmetic complex hyperbolic lattices {II}, Michigan Math. J.. Michigan Mathematical Journal, 70, 135-205 (2021) · Zbl 1542.22019 · doi:10.1307/mmj/1592532044 [28] Dimitrov, Mladen; Ramakrishnan, Dinakar, Arithmetic quotients of the complex ball and a conjecture of {L}ang, Doc. Math.. Documenta Mathematica, 20, 1185-1205 (2015) · Zbl 1351.11039 [29] van den Dries, Lou, Tame Topology and o-Minimal Structures, London Math. Soc. Lecture Note Ser., 248, x+180 pp. (1998) · Zbl 0953.03045 · doi:10.1017/CBO9780511525919 [30] Edixhoven, Bas; Yafaev, Andrei, Subvarieties of {S}himura varieties, Ann. of Math. (2). Annals of Mathematics. Second Series, 157, 621-645 (2003) · Zbl 1053.14023 · doi:10.4007/annals.2003.157.621 [31] Eisenman, Donald A., Intrinsic Measures on Complex Manifolds and Holomorphic Mappings, Mem. Amer. Math. Soc., 96, i+80 pp. (1970) · Zbl 0197.05901 [32] Esnault, H\'{e}l\`ene; Groechenig, Michael, Cohomologically rigid local systems and integrality, Selecta Math. (N.S.). Selecta Mathematica. New Series, 24, 4279-4292 (2018) · Zbl 1408.14037 · doi:10.1007/s00029-018-0409-z [33] Faltings, G., Arithmetic varieties and rigidity. Seminar on {N}umber {T}heory, {P}aris 1982-83 ({P}aris, 1982/1983), Progr. Math., 51, 63-77 (1984) · Zbl 0549.14010 [34] Garland, H.; Raghunathan, M. S., Fundamental domains for lattices in ({R}-)rank {\(1\)} semisimple {L}ie groups, Ann. of Math. (2). Annals of Mathematics. Second Series, 92, 279-326 (1970) · Zbl 0206.03603 · doi:10.2307/1970838 [35] Goldman, William M., Complex {H}yperbolic {G}eometry, Oxford Math. Monogrl, xx+316 pp. (1999) · Zbl 0939.32024 [36] Gonz\'{a}lez-Diez, Gabino, Variations on {B}elyi’s theorem, Q. J. Math.. The Quarterly Journal of Mathematics, 57, 339-354 (2006) · Zbl 1123.14016 · doi:10.1093/qmath/hai021 [37] Green, Mark; Griffiths, Phillip; Kerr, Matt, Mumford-{T}ate Groups and Domains. Their Geometry and Arithmetic, Ann. of Math. Stud., 183, viii+289 pp. (2012) · Zbl 1248.14001 [38] Griffiths, Phillip A., Complex-analytic properties of certain {Z}ariski open sets on algebraic varieties, Ann. of Math. (2). Annals of Mathematics. Second Series, 94, 21-51 (1971) · Zbl 0221.14008 · doi:10.2307/1970733 [39] Gromov, M.; Piatetski-Shapiro, I., Non-arithmetic groups in {L}obachevsky spaces, Inst. Hautes \'{E}tudes Sci. Publ. Math.. Institut des Hautes \'{E}tudes Scientifiques. Publications Math\'{e}matiques, 93-103 (1988) · Zbl 0649.22007 [40] Gromov, Mikhail; Schoen, Richard, Harmonic maps into singular spaces and {\(p\)}-adic superrigidity for lattices in groups of rank one, Inst. Hautes \'{E}tudes Sci. Publ. Math.. Institut des Hautes \'{E}tudes Scientifiques. Publications Math\'{e}matiques, 165-246 (1992) · Zbl 0896.58024 [41] Habegger, Philipp; Pila, Jonathan, O-minimality and certain atypical intersections, Ann. Sci. \'{E}c. Norm. Sup\'{e}r. (4). Annales Scientifiques de l’\'{E}cole Normale Sup\'{e}rieure. Quatri\`eme S\'{e}rie, 49, 813-858 (2016) · Zbl 1364.11110 · doi:10.24033/asens.2296 [42] Hummel, Christoph; Schroeder, Viktor, Cusp closing in rank one symmetric spaces, Invent. Math.. Inventiones Mathematicae, 123, 283-307 (1996) · Zbl 0860.53025 · doi:10.1007/s002220050027 [43] Katz, Nicholas M., Rigid Local Systems, Ann. of Math. Stud., 139, viii+223 pp. (1996) · Zbl 0864.14013 · doi:10.1515/9781400882595 [44] Kazhdan, David, On arithmetic varieties. {II}, Israel J. Math.. Israel Journal of Mathematics, 44, 139-159 (1983) · Zbl 0543.14030 · doi:10.1007/BF02760617 [45] Klingler, B.; Ullmo, E.; Yafaev, A., The hyperbolic {A}x-{L}indemann-{W}eierstrass conjecture, Publ. Math. Inst. Hautes \'{E}tudes Sci.. Publications Math\'{e}matiques. Institut de Hautes \'{E}tudes Scientifiques, 123, 333-360 (2016) · Zbl 1372.14016 · doi:10.1007/s10240-015-0078-9 [46] Klingler, B.; Ullmo, E.; Yafaev, A., Bi-algebraic geometry and the {A}ndr\'{e}-{O}ort conjecture. Algebraic Geometry: {S}alt {L}ake {C}ity 2015, Proc. Sympos. Pure Math., 97, 319-359 (2018) · Zbl 1372.14016 · doi:10.1007/s10240-015-0078-9 [47] Klingler, B., Hodge loci and atypical intersections: conjectures (2017) [48] Klingler, B.; Otwinowska, A.; Urbanik, D., On the fields of definition of {H}odge loci (2020) [49] Koziarz, Vincent; Maubon, Julien, Finiteness of totally geodesic exceptional divisors in {H}ermitian locally symmetric spaces, Bull. Soc. Math. France. Bulletin de la Soci\'{e}t\'{e} Math\'{e}matique de France, 146, 613-631 (2018) · Zbl 1420.32017 · doi:10.24033/bsmf.2767 [50] Langer, Adrian; Simpson, Carlos, Rank 3 rigid representations of projective fundamental groups, Compos. Math.. Compositio Mathematica, 154, 1534-1570 (2018) · Zbl 1409.14039 · doi:10.1112/s0010437x18007182 [51] Margulis, G. A., Discrete Subgroups of Semisimple {L}ie Groups, Ergeb. Math. Grenzgeb., 17, x+388 pp. (1991) · Zbl 0732.22008 · doi:10.1007/978-3-642-51445-6 [52] Mohammadi, Amir; Margulis, Gregorii, Arithmeticity of hyperbolic 3-manifolds containing infinitely many totally geodesic surfaces, Ergodic Theory Dynam. Systems. Ergodic Theory and Dynamical Systems, 42, 1188-1219 (2022) · Zbl 1483.57017 · doi:10.1017/etds.2021.21 [53] McReynolds, D. B.; Reid, A. W., The genus spectrum of a hyperbolic 3-manifold, Math. Res. Lett.. Mathematical Research Letters, 21, 169-185 (2014) · Zbl 1301.53039 · doi:10.4310/MRL.2014.v21.n1.a14 [54] Mochizuki, Takuro, Asymptotic behaviour of tame harmonic bundles and an application to pure twistor {\(D\)}-modules. {I}, Mem. Amer. Math. Soc.. Memoirs of the Amer. Math. Soc., 185, xii+324 pp. (2007) · Zbl 1259.32005 · doi:10.1090/memo/0869 [55] Mok, Ngaiming, Projective algebraicity of minimal compactifications of complex-hyperbolic space forms of finite volume. Perspectives in Analysis, Geometry, and Topology, Progr. Math., 296, 331-354 (2012) · Zbl 1253.32006 · doi:10.1007/978-0-8176-8277-4_14 [56] Mok, Ngaiming, Zariski closures of images of algebraic subsets under the uniformization map on finite-volume quotients of the complex unit ball, Compos. Math.. Compositio Mathematica, 155, 2129-2149 (2019) · Zbl 1439.32055 · doi:10.1112/s0010437x19007577 [57] Mok, Ngaiming; Pila, Jonathan; Tsimerman, Jacob, Ax-{S}chanuel for {S}himura varieties, Ann. of Math. (2). Annals of Mathematics. Second Series, 189, 945-978 (2019) · Zbl 1481.14048 · doi:10.4007/annals.2019.189.3.7 [58] Mok, Ngaiming; Yeung, Sai-Kee, Geometric realizations of uniformization of conjugates of {H}ermitian locally symmetric manifolds. Complex {A}nalysis and {G}eometry, Univ. Ser. Math., 253-270 (1993) · Zbl 0793.32011 [59] Morris, Dave Witte, Introduction to Arithmetic Groups, xii+475 pp. (2015) · Zbl 1319.22007 [60] Mostow, G. D., On a remarkable class of polyhedra in complex hyperbolic space, Pacific J. Math.. Pacific Journal of Mathematics, 86, 171-276 (1980) · Zbl 0456.22012 · doi:10.2140/pjm.1980.86.171 [61] Mumford, D., Hirzebruch’s proportionality theorem in the noncompact case, Invent. Math.. Inventiones Mathematicae, 42, 239-272 (1977) · Zbl 0365.14012 · doi:10.1007/BF01389790 [62] Nori, Madhav V., A nonarithmetic monodromy group, C. R. Acad. Sci. Paris S\'{e}r. I Math.. Comptes Rendus des S\'{e}ances de l’Acad\'{e}mie des Sciences. S\'{e}rie I. Math\'{e}matique, 302, 71-72 (1986) · Zbl 0602.14025 [63] Peters, Chris, On rigidity of locally symmetric spaces, M\"{u}nster J. Math.. M\"{u}nster Journal of Mathematics, 10, 277-286 (2017) · Zbl 1453.14128 · doi:10.17879/80299606895 [64] Peterzil, Ya’acov; Starchenko, Sergei, Complex analytic geometry and analytic-geometric categories, J. Reine Angew. Math.. Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle’s Journal], 626, 39-74 (2009) · Zbl 1168.03029 · doi:10.1515/CRELLE.2009.002 [65] Raghunathan, M. S., Discrete Subgroups of {L}ie Groups, Ergeb. Math. Grenzgeb., 68, ix+227 pp. (1972) · Zbl 0254.22005 [66] Sarnak, Peter, Notes on thin matrix groups. Thin {G}roups and {S}uperstrong {A}pproximation, Math. Sci. Res. Inst. Publ., 61, 343-362 (2014) · Zbl 1365.11039 [67] Shimura, Goro, Algebraic varieties without deformation and the {C}how variety, J. Math. Soc. Japan. Journal of the Mathematical Society of Japan, 20, 336-341 (1968) · Zbl 0197.17202 · doi:10.2969/jmsj/02010336 [68] Simpson, Carlos T., Higgs bundles and local systems, Inst. Hautes \'{E}tudes Sci. Publ. Math.. Institut des Hautes \'{E}tudes Scientifiques. Publications Math\'{e}matiques, 5-95 (1992) · Zbl 0814.32003 [69] Siu, Yum Tong; Yau, Shing Tung, Compactification of negatively curved complete {K}\"{a}hler manifolds of finite volume. Seminar on {D}ifferential {G}eometry, Ann. of Math. Stud., 102, 363-380 (1982) · Zbl 0471.00020 · doi:10.1515/9781400881918 [70] To, Wing-Keung, Total geodesy of proper holomorphic immersions between complex hyperbolic space forms of finite volume, Math. Ann.. Mathematische Annalen, 297, 59-84 (1993) · Zbl 0788.32018 · doi:10.1007/BF01459488 [71] Tsimerman, Jacob, The {A}ndr\'{e}-{O}ort conjecture for {\( \mathcal{A}_g\)}, Ann. of Math. (2). Annals of Mathematics. Second Series, 187, 379-390 (2018) · Zbl 1415.11086 · doi:10.4007/annals.2018.187.2.2 [72] Tsuji, Hajime, A characterization of ball quotients with smooth boundary, Duke Math. J.. Duke Mathematical Journal, 57, 537-553 (1988) · Zbl 0671.14002 · doi:10.1215/S0012-7094-88-05724-9 [73] Ullmo, Emmanuel, Applications du th\'{e}or\`eme d’{A}x-{L}indemann hyperbolique, Compos. Math.. Compositio Mathematica, 150, 175-190 (2014) · Zbl 1326.14062 · doi:10.1112/S0010437X13007446 [74] Ullmo, Emmanuel, Structures sp\'{e}ciales et probl\`eme de {Z}ilber-{P}ink. Around the {Z}ilber-{P}ink Conjecture/{A}utour de la Conjecture de {Z}ilber-{P}ink, Panor. Synth\`eses, 52, 1-30 (2017) · Zbl 1375.11003 [75] Ullmo, Emmanuel; Yafaev, Andrei, A characterization of special subvarieties, Mathematika. Mathematika. A Journal of Pure and Applied Mathematics, 57, 263-273 (2011) · Zbl 1236.14029 · doi:10.1112/S0025579311001628 [76] Vinberg, {\`E}. B., Rings of definition of dense subgroups of semisimple linear groups, Izv. Akad. Nauk SSSR Ser. Mat.. Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, 35, 45-55 (1971) · Zbl 0252.20043 · doi:10.070/IM1971v005n01ABEH001006 [77] Weil, A., Algebras with involutions and the classical groups, J. Indian Math. Soc. (N.S.). The Journal of the Indian Mathematical Society. New Series, 24, 589-623 (1960) · Zbl 0109.02101 · doi:10.18311/jims/1960/16935 [78] Weil, A., On discrete subgroups of {L}ie groups, Ann. of Math. (2). Annals of Mathematics. Second Series, 72, 369-384 (1960) · Zbl 0131.26602 · doi:10.2307/1970140 [79] Weil, A., On discrete subgroups of {L}ie groups. {II}, Ann. of Math. (2). Annals of Mathematics. Second Series, 75, 578-602 (1962) · Zbl 0131.26602 · doi:10.2307/1970212 [80] Weil, A., Remarks on the cohomology of groups, Ann. of Math. (2). Annals of Mathematics. Second Series, 80, 149-157 (1964) · Zbl 0192.12802 · doi:10.2307/1970495 [81] Weil, A., Adeles and Algebraic Groups, Progr. Math., 23, iii+126 pp. (1982) · Zbl 0493.14028 [82] Wolfart, J\"{u}rgen, Werte hypergeometrischer {F}unktionen, Invent. Math.. Inventiones Mathematicae, 92, 187-216 (1988) · Zbl 0649.10022 · doi:10.1007/BF01393999 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.