Derivative of the standard $p$-adic $L$-function associated with a Siegel form
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Abstract:
In this paper we first construct a two-variable $p$-adic $L$-function for the standard representation associated with a Hida family of parallel weight genus $g$ Siegel forms, using a method developed by Böcherer–Schmidt in one variable. When a form $f$ has weight $g+1$ a non-crystalline trivial zero could appear. In this case, using the two-variable $p$-adic $L$-function we have constructed, we can apply the method of Greenberg–Stevens to calculate the first derivative of the $p$-adic $L$-function for $f$ and show that it has the form predicted by a conjecture of Greenberg on trivial zeros.References
- Denis Benois, A generalization of Greenberg’s $\scr L$-invariant, Amer. J. Math. 133 (2011), no. 6, 1573–1632. MR 2863371, DOI 10.1353/ajm.2011.0043
- S. Böcherer and C.-G. Schmidt, $p$-adic measures attached to Siegel modular forms, Ann. Inst. Fourier (Grenoble) 50 (2000), no. 5, 1375–1443 (English, with English and French summaries). MR 1800123, DOI 10.5802/aif.1796
- John Coates, Motivic $p$-adic $L$-functions, $L$-functions and arithmetic (Durham, 1989) London Math. Soc. Lecture Note Ser., vol. 153, Cambridge Univ. Press, Cambridge, 1991, pp. 141–172. MR 1110392, DOI 10.1017/CBO9780511526053.006
- Ellen E. Eischen, $p$-adic differential operators on automorphic forms on unitary groups, Ann. Inst. Fourier (Grenoble) 62 (2012), no. 1, 177–243 (English, with English and French summaries). MR 2986270, DOI 10.5802/aif.2704
- Stephen Gelbart, Ilya Piatetski-Shapiro, and Stephen Rallis, Explicit constructions of automorphic $L$-functions, Lecture Notes in Mathematics, vol. 1254, Springer-Verlag, Berlin, 1987. MR 892097, DOI 10.1007/BFb0078125
- Ralph Greenberg, Trivial zeros of $p$-adic $L$-functions, $p$-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991) Contemp. Math., vol. 165, Amer. Math. Soc., Providence, RI, 1994, pp. 149–174. MR 1279608, DOI 10.1090/conm/165/01606
- Michael Harris, Arithmetic vector bundles and automorphic forms on Shimura varieties. I, Invent. Math. 82 (1985), no. 1, 151–189. MR 808114, DOI 10.1007/BF01394784
- Michael Harris, Arithmetic vector bundles and automorphic forms on Shimura varieties. II, Compositio Math. 60 (1986), no. 3, 323–378. MR 869106
- Haruzo Hida, A $p$-adic measure attached to the zeta functions associated with two elliptic modular forms. II, Ann. Inst. Fourier (Grenoble) 38 (1988), no. 3, 1–83. MR 976685, DOI 10.5802/aif.1141
- Haruzo Hida, Control theorems of coherent sheaves on Shimura varieties of PEL type, J. Inst. Math. Jussieu 1 (2002), no. 1, 1–76. MR 1954939, DOI 10.1017/S1474748002000014
- Zheng Liu, Nearly overconvergent Siegel modular forms, available at http://www.math.columbia.edu/~zliu/Files/NHF, 2016.
- Zheng Liu, $p$-adic $L$-functions for ordinary families of symplectic groups, available at http://math.columbia.edu/~zliu/Files/SLF, 2016.
- B. Mazur, J. Tate, and J. Teitelbaum, On $p$-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. Math. 84 (1986), no. 1, 1–48. MR 830037, DOI 10.1007/BF01388731
- Bernadette Perrin-Riou, Fonctions $L$ $p$-adiques des représentations $p$-adiques, Astérisque 229 (1995), 198 (French, with English and French summaries). MR 1327803
- Vincent Pilloni, Prolongement analytique sur les variétés de Siegel, Duke Math. J. 157 (2011), no. 1, 167–222 (French, with English and French summaries). MR 2783930, DOI 10.1215/00127094-2011-004
- Vincent Pilloni, Sur la théorie de Hida pour le groupe $\textrm {GSp}_{2g}$, Bull. Soc. Math. France 140 (2012), no. 3, 335–400 (French, with English and French summaries). MR 3059119, DOI 10.24033/bsmf.2630
- Brooks Roberts and Ralf Schmidt, Local newforms for GSp(4), Lecture Notes in Mathematics, vol. 1918, Springer, Berlin, 2007. MR 2344630, DOI 10.1007/978-3-540-73324-9
- Giovanni Rosso, Derivative of symmetric square $p$-adic $L$-functions via pull-back formula, Arithmetic and geometry, London Math. Soc. Lecture Note Ser., vol. 420, Cambridge Univ. Press, Cambridge, 2015, pp. 373–400. MR 3467131
- Giovanni Rosso, $\mathcal L$-invariant for Siegel-Hilbert forms, Doc. Math. 20 (2015), 1227–1253. MR 3424479
- Peter Scholze, On torsion in the cohomology of locally symmetric varieties, Ann. of Math. (2) 182 (2015), no. 3, 945–1066. MR 3418533, DOI 10.4007/annals.2015.182.3.3
- Goro Shimura, On Eisenstein series, Duke Math. J. 50 (1983), no. 2, 417–476. MR 705034
- Goro Shimura, On a class of nearly holomorphic automorphic forms, Ann. of Math. (2) 123 (1986), no. 2, 347–406. MR 835767, DOI 10.2307/1971276
- Sug Woo Shin, Automorphic Plancherel density theorem, Israel J. Math. 192 (2012), no. 1, 83–120. MR 3004076, DOI 10.1007/s11856-012-0018-z
Additional Information
- Giovanni Rosso
- Affiliation: DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, United Kingdom
- Address at time of publication: Department of Mathematics and Statistics, Concordia University, Montreal H3G 1M8, Canada
- MR Author ID: 1025013
- ORCID: 0000-0002-4707-0386
- Email: gr385@cam.ac.uk
- Received by editor(s): September 2, 2016
- Received by editor(s) in revised form: November 11, 2016, November 23, 2016, and November 28, 2016
- Published electronically: April 4, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 6469-6491
- MSC (2010): Primary 11F33, 11F67
- DOI: https://doi.org/10.1090/tran/7138
- MathSciNet review: 3814337