Derivative of the standard 𝑝-adic 𝐿-function associated with a Siegel form

Rosso, Giovanni

doi:10.1090/tran/7138

Derivative of the standard $p$-adic $L$-function associated with a Siegel form
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Trans. Amer. Math. Soc. 370 (2018), 6469-6491 Request permission

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