Gehrmann, Lennart

Functoriality of Automorphic \(\mathrm{L}\)-Invariants and Applications

Doc. Math. 24, 1225-1243 (2019)
DOI: 10.25537/dm.2019v24.1225-1243
Communicated by Don Blasius


We study the behaviour of automorphic \(\mathrm{L}\)-invariants associated to cuspidal representations of \(\mathrm{GL}(2)\) of cohomological weight 0 under abelian base change and Jacquet-Langlands lifts to totally definite quaternion algebras. Under a standard non-vanishing hypothesis on automorphic \(\mathrm{L}\)-functions and some technical restrictions on the automorphic representation and the base field we get a simple proof of the equality of automorphic and arithmetic \(\mathrm{L}\)-invariants. This together with Spieß' results on \(p\)-adic \(\mathrm{L}\)-functions yields a new proof of the exceptional zero conjecture for modular elliptic curves -- at least, up to sign.

Mathematics Subject Classification

11F41, 11F67, 11F75, 11F85, 11G05


\(p\)-adic periods, modular forms, automorphic representations


  • 1. A. Akbary. Simultaneous non-vanishing of twists. Proceedings of the American Mathematical Society, 134(11):3143-3151, 2006. DOI 10.1090/S0002-9939-06-08369-9; zbl 1173.11327; MR2231896.
  • 2. J. Arthur and L. Clozel. Simple algebras, base change, and the advanced theory of the trace formula, volume 120 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1989. DOI 10.1515/9781400882403; zbl 0682.10022; MR1007299.
  • 3. K. Barré-Sirieix, G. Diaz, F. Gramain, and G. Philibert. Une preuve de la conjecture de Mahler-Manin. Inventiones mathematicae, 124(1):1-9, 1996. DOI 10.1007/s002220050044; zbl 0853.11059; MR1369409.
  • 4. D. Barrera, M. Dimitrov, and A. Jorza. \(p\)-adic \(L\)-functions of Hilbert cusp forms and the trivial zero conjecture. Sept. 2017. arxiv 1709.08105.
  • 5. D. Barrera and C. Williams. \( \mathcalL \)-invariants and exceptional zeros of Bianchi modular forms. Trans. Amer. Math. Soc., 372(1):1-34, 2019. DOI 10.1090/tran/7436; zbl 07076381; MR3968760; arxiv 1707.04049.
  • 6. F. Bergunde. On leading terms of quaternionic Stickelberger elements over number fields. PhD thesis, University Bielefeld, 2017.
  • 7. F. Bergunde and L. Gehrmann. Leading terms of anticyclotomic Stickelberger elements and \(p\)-adic periods. Trans. Amer. Math. Soc., 370(9):6297-6329, 2018. DOI 10.1090/tran/7120; zbl 06891709; MR3814331; arxiv 1607.01197.
  • 8. M. Bertolini, H. Darmon, and A. Iovita. Families of automorphic forms on definite quaternion algebras and Teitelbaum's conjecture. Astérisque, 331:29-64, 2010. zbl 1251.11033; MR2667886.
  • 9. C. Breuil. Invariant \(L\) et série spéciale p-adique. Annales scientifiques de l'École Normale Supérieure, 37(4):559-610, 2004. DOI 10.1016/j.ansens.2004.02.001; zbl 1166.11331; MR2097893.
  • 10. C. Breuil. Série spéciale p-adique et cohomologie étale complétée. Astérisque, 331:65-115, 2010. zbl 1246.11106; MR2667887.
  • 11. L. Clozel. Motifs et formes automorphes: applications du principe de fonctorialité. In Automorphic forms, Shimura varieties, and L-functions, Vol. I (Ann Arbor, MI, 1988), volume 10 of Perspect. Math., pages 77-159. Academic Press, Boston, MA, 1990. MR1044819.
  • 12. H. Darmon. Integration on \(H_ptimes H\) and arithmetic applications. Ann. of Math. (2), 154(3):589-639, 2001. DOI 10.2307/3062142; zbl 1035.11027; MR1884617.
  • 13. S. Dasgupta and M. Greenberg. L-invariants and Shimura curves. Algebra and Number Theory, 6(3):455-485, 2012. DOI 10.2140/ant.2012.6.455; zbl 1285.11072; MR2966706.
  • 14. H. Deppe. p-adic L-functions of automorphic forms and exceptional zeros. Documenta Mathematica, 21:689-734, 2016.; zbl 06638963; MR3522253.
  • 15. D. Disegni. On the p-adic Birch and Swinnerton-Dyer conjecture for elliptic curves over number fields. Kyoto Journal of Mathematics, to appear. arxiv 1609.02528.
  • 16. R. Greenberg and G. Stevens. p-adic L-functions and p-adic periods of modular forms. Inventiones mathematicae, 111(1):407-447, 1993. DOI 10.1007/BF01231294; zbl 0778.11034; MR1198816.
  • 17. X. Guitart, M. Masdeu, and M. H. Şengün. Darmon points on elliptic curves over number fields of arbitrary signature. Proceedings of the London Mathematical Society, 111(2):484-518, 2015. DOI 10.1112/plms/pdv033; zbl 1391.11081; MR3384519; arxiv 1404.6650.
  • 18. H. Hida. L-invariants of Tate curves. Pure and Applied Math. Quarterly, 5:1343-1384, 2009. DOI 10.4310/PAMQ.2009.v5.n4.a6; zbl 1244.11056; MR2560319.
  • 19. L. Orton. An elementary proof of a weak exceptional zero conjecture. Canadian Journal of Mathematics, 56:373-405, 2004. DOI 10.4153/CJM-2004-018-4; zbl 1060.11030; MR2040921.
  • 20. D. E. Rohrlich. Nonvanishing of L-functions and structure of Mordell-Weil groups. Journal für die reine und angewandte Mathematik, 417:1-26, 1991. DOI 10.1515/crll.1991.417.1; zbl 0731.11030; MR1103904.
  • 21. M. Spieß. On special zeros of p-adic L-functions of Hilbert modular forms. Inventiones mathematicae, 196(1):69-138, 2014. DOI 10.1007/s00222-013-0465-0; zbl 1392.11027; MR3179573; arxiv 1207.2289.
  • 22. J. T. Teitelbaum. Values of p-adic L-functions and a p-adic Poisson kernel. Inventiones mathematicae, 101(2):395-410, 1990. DOI 10.1007/BF01231508; zbl 0731.11065; MR1062968.


Gehrmann, Lennart
Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann-Straße 9, 45127 Essen, Germany