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Automorphic Forms and Even Unimodular Lattices

Abstract

An overview of the purpose and results. The connecting thread is the p-neighbor problem for even unimodular lattices in dimensions 16 and 24.

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Notes

  1. 1.

    The notation vois comes from the French word voisin for neighbor.

  2. 2.

    The comparison of Theorem A with the formula for rLeech(p) given above leads to the “purely quadratic” relation Np(E8 ⊕E8, E16) = (9∕1456) ⋅rLeech(p) ⋅ (p 4 − 1)(p − 1)−1, which we do not know how to prove directly.

  3. 3.

    A list of these graphs can be found at http://gaetan.chenevier.perso.math.cnrs.fr/niemeier/niemeier.html.

  4. 4.

    This assertion can be proved much more directly. Indeed, if 𝜗 0 denotes the linear map \(\mathbb {C}[{\mathrm{X}}_n] \rightarrow \mathbb {C}\) that sends the class of any element of \(\mathcal {L}_n\) to 1, then we have 𝜗 0 ∘Tp =  cn(p) 𝜗 0, so that Ker 𝜗 0 is stable by Tp.

  5. 5.

    The discussion that follows does not apply verbatim to certain nonconnected group schemes that naturally occur here, such as On or PGOn. We will, when necessary, indicate any modifications needed to include them, but in this introduction we will ignore this detail.

  6. 6.

    As is customary, we denote by Π cusp(G) ⊂ Π disc(G) the subset of representations occurring in the subspace of cusp forms [92] (Sect. 4.3).

  7. 7.

    Strictly speaking, our notation includes the corresponding natural identity at the Archimedean place (Sect. 6.4.4). We also denote the summand π i[d i] simply by [d i] (resp. π i) when π i = 1 (resp. d i = 1). These conventions are used in Table 1.2.

  8. 8.

    Likewise, c∞(Δ w) has eigenvalues ± w∕2.

  9. 9.

    The definition of algebraic given in the preface, which seems more restrictive, is in fact equivalent to this one: see Remark 8.2.14. The motivic weight w(π) is also twice the greatest eigenvalue of c∞(π).

  10. 10.

    For Theorem F, we in fact prove a variant that is nearly as strong without using Arthur’s theory; see Theorem 9.3.2.

References

  1. J. Arthur, Unipotent automorphic representations: conjectures, in Orbites unipotentes et représentations II : groupes p-adiques et réels, Astérisque 171–172 (Soc. Math. France, Paris, 1989), pp. 13–71.

    Google Scholar 

  2. J. Arthur, The endoscopic classification of representations: orthogonal and symplectic groups, Colloquium Publ., vol. 61 (Amer. Math. Soc., 2013).

    Google Scholar 

  3. J. Bergström, C. Faber, G. van der Geer, Siegel modular forms of degree three and the cohomology of local systems, Selecta Math. 20 (2014), pp. 83–124.

    Article  MathSciNet  Google Scholar 

  4. S. Böcherer, Siegel modular forms and theta series, Proc. Symp. in Pure Math., vol. 49 (Bowdoin 1987, Amer. Math. Soc., 1989), pp. 3–17.

    Google Scholar 

  5. R. Borcherds, E. Freitag, R. Weissauer, A Siegel cusp form of degree 12 and weight 12, J. reine angew. Math. 494 (1998), pp. 141–153.

    MathSciNet  MATH  Google Scholar 

  6. A. Borel, Automorphic L-functions, in Automorphic forms, representations and L-functions, II (Oregon State Univ., Corvallis, Ore.), Proc. Symp. in Pure Math. XXXIII (Amer. Math. Soc., Providence, RI, 1979), pp. 27–61.

    Google Scholar 

  7. S. Breulmann, M. Kuss, On a conjecture of Duke-Imamoğlu, Proc. Amer. Math. Soc. 128 (2000), pp. 1595–1604.

    Article  MathSciNet  Google Scholar 

  8. G. Chenevier, Représentations galoisiennes automorphes et conséquences arithmétiques des conjectures de Langlands et Arthur, mémoire d’habilitation à diriger des recherches, université Paris XI (2013).

    Google Scholar 

  9. G. Chenevier, J. Lannes, Kneser neighbours and orthogonal Galois representations in dimensions 16 and 24, Algebraic Number Theory, Oberwolfach Rep. 31 (2011).

    Google Scholar 

  10. G. Chenevier, D. Renard, Level one algebraic cusp form of classical groups of small rank, Mem. Amer. Math. Soc., vol. 1121 (Amer. Math. Soc., Providence, RI, 2015).

    Google Scholar 

  11. J. H. Conway, N. J. A. Sloane, Twenty-three constructions for the Leech lattice, Proc. Roy. Soc. London 381 (1982), pp. 275–283.

    Article  MathSciNet  Google Scholar 

  12. J. H. Conway, N. J. A. Sloane, Sphere packings, lattices and groups, Grundlehren math. Wiss., vol. 290 (Springer-Verlag, New York, 1999).

    Google Scholar 

  13. W. Duke, Ö. Imamoğlu, Siegel modular forms of small weight, Math. Ann. 310 (1998), pp. 73–82.

    Article  MathSciNet  Google Scholar 

  14. V. Erokhin, Theta series of even unimodular 24-dimensional lattices (in Russian), Zap. Naučn. Sem. Leningrad Otdel. Mat. Inst. Steklov. (LOMI) 86 (1979), pp. 82–93.; English translation in J. Soviet Math. 17 (1981), pp. 1999–2008.

    Google Scholar 

  15. C. Faber, Modular forms and the cohomology of moduli spaces, in Automorphic forms and automorphic L-functions, Surikaisekikenkyusho Kokyuroku 1826 (2013), pp. 129–137.

    Google Scholar 

  16. C. Faber, G. van der Geer, Sur la cohomologie des systèmes locaux sur les espaces de modules des courbes de genre 2 et des surfaces abéliennes, I et II, C. R. Acad. Sci. Paris, Sér. I 338 (2004), pp. 381–384.

    Article  MathSciNet  Google Scholar 

  17. S. Fermigier, Annulation de la cohomologie cuspidale de sous-groupes de congruence de \(\mathrm {GL}_n(\mathbb {Z})\), Math. Ann. 306 (1996), pp. 247–256.

    Google Scholar 

  18. G. van der Geer, Siegel modular forms and their applications, in The 1–2–3 of modular forms, ed. by J. H. Bruinier, G. van der Geer, G. Harder, D. Zagier, Universitext (Springer Verlag, Berlin, 2008), pp. 181–245.

    MATH  Google Scholar 

  19. S. Gelbart, H. Jacquet, A relation between automorphic representations of GL(2) and GL(3), Ann. Sci. Éc. Norm. Sup. 11 (1978), pp. 1–73.

    Article  MathSciNet  Google Scholar 

  20. I. M. Gel’fand, M. I. Graev, I. I. Pyatetskii-Shapiro, Representation theory and automorphic functions (Academic Press, 1990).

    Google Scholar 

  21. B. Gross, On the Satake isomorphism, in Galois representations in arithmetic algebraic geometry, ed. by A. Scholl, R. Taylor (Cambridge Univ. Press, 1998).

    Google Scholar 

  22. G. Harder, A congruence between a Siegel and an elliptic modular form, in The 1–2–3 of modular forms, ed. by J. H. Bruinier, G. van der Geer, G. Harder, D. Zagier (Universitext, Springer Verlag, Berlin, 2008), pp. 247–260.

    MATH  Google Scholar 

  23. T. Ikeda, On the lifting of elliptic cusp forms to Siegel cusp forms of degree 2n, Ann. of Math. 154 (2001), pp. 641–681.

    Article  MathSciNet  Google Scholar 

  24. T. Ikeda, Pullback of the lifting of elliptic cusp forms and Miyawaki’s conjecture, Duke Math. J. 131 (2006), pp. 469–497.

    Article  MathSciNet  Google Scholar 

  25. T. Ikeda, On the lifting of automorphic representations from \(\mbox{PGL}_2(\mathbb {A})\) to \(\mbox{Sp}_{2n}(\mathbb {A})\) or \(\widetilde {\mbox{Sp}_{2n+1}(\mathbb {A})}\) over a totally real field, available at https://www.math.kyoto-u.ac.jp/~ikeda/.

  26. O. King, A mass formula for unimodular lattices with no roots, Math. Comp. 72, no. 242 (2003), pp. 839–863.

    Article  MathSciNet  Google Scholar 

  27. B. Kostant, The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math. 81 (1959), pp. 973–1032.

    Article  MathSciNet  Google Scholar 

  28. R. Langlands, Problems in the theory of automorphic forms, Lecture Notes in Math., vol. 170 (Springer Verlag, 1970).

    Google Scholar 

  29. J.-F. Mestre, Formules explicites et minorations de conducteurs des variétés algébriques, Compos. Math. 58 (1986), pp. 209–232.

    MATH  Google Scholar 

  30. S. Miller, The highest-lowest zero and other applications of positivity, Duke Math. J. 112 (2002), pp. 83–116.

    Article  MathSciNet  Google Scholar 

  31. S.-I. Mizumoto, Poles and residues of standard L-functions attached to Siegel modular forms, Math. Ann. 289 (1991), pp. 589–612.

    Article  MathSciNet  Google Scholar 

  32. C. Moeglin, J.-L. Waldspurger, Stabilisation de la formule des traces tordue VI & X, preprint available at http://webusers.imj-prg.fr/~jean-loup.waldspurger/ (2014).

  33. G. Nebe, B. Venkov, On Siegel modular forms of weight 12, J. reine angew. Math. 351 (2001), pp. 49–60.

    MathSciNet  MATH  Google Scholar 

  34. H.-V. Niemeier, Definite quadratische Formen der Dimension 24 und Diskriminante 1, J. Number Theory 5 (1973), pp. 142–178.

    Article  MathSciNet  Google Scholar 

  35. C. Poor, D. S. Yuen, Dimensions of spaces of Siegel modular forms of low weight in degree four, Bull. Austral. Math. Soc. 54, no. 2 (1996), pp. 309–315.

    Article  MathSciNet  Google Scholar 

  36. C. Poor, D. S. Yuen, Computations of spaces of Siegel modular cusp forms, J. Math. Soc. Japan 59 (2007), pp. 185–222.

    Article  MathSciNet  Google Scholar 

  37. J.-P. Serre, Cours d’arithmétique (Publ. Univ. France, Paris, 1970).

    MATH  Google Scholar 

  38. O. Taïbi, Dimensions of spaces of level one automorphic forms for split classical groups using the trace formula, preprint available at http://arxiv.org/abs/1406.4247 (2014).

  39. R. Tsushima, An explicit dimension formula for the spaces of generalized automorphic forms with respect to \(\mbox{Sp}(2,\mathbb {Z})\), Proc. Japan Acad., vol. 59 (1983).

    Google Scholar 

  40. B. Venkov, On the classification of integral even unimodular 24-dimensional quadratic forms, Chapter 18 in J. H. Conway, N. J. A. Sloane, Sphere packings, lattices and groups, Grundlehren math. Wiss., vol. 290 (Springer-Verlag, New York, 1999).

    Google Scholar 

  41. J.-L. Waldspurger, Engendrement par des séries thêta de certains espaces de formes modulaires, Invent. Math. 50 (1979), pp. 135–168.

    Article  MathSciNet  Google Scholar 

  42. J.-L. Waldspurger, Stabilisation de la formule des traces tordues I, II, III, IV, V, VII, VIII, IX, preprints available at http://webusers.imj-prg.fr/~jean-loup.waldspurger/ (2014).

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Authors and Affiliations

  1. CNRS, Institut de Mathématique d’Orsay, Université Paris-Sud, Orsay, France

    Gaëtan Chenevier

  2. Institut de Mathématiques de Jussieu, Université Paris Diderot, Paris, France

    Jean Lannes

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  1. Gaëtan Chenevier
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Chenevier, G., Lannes, J. (2019). Introduction. In: Automorphic Forms and Even Unimodular Lattices. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol 69. Springer, Cham. https://doi.org/10.1007/978-3-319-95891-0_1

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