On the \(p\)-adic \(L\)-function of Hilbert modular forms at supersingular primes. (English) Zbl 1219.11135
The \(p\)-adic \(L\)-function \(L_p(f,\cdot)\) attached to an elliptic modular form \(f=\sum_n a_nq^n\) at good prime \(p\) by Amice-Vélu, Vishik, and Mazur-Tate-Teitelbaum, is in general unbounded (but \(h\)-admissible in the sense of Amice-Vélu and Vishik). R. Pollack [Duke Math. J. 118, No. 3, 523–558 (2003; Zbl 1074.11061)] proved that the function \(L_p(f,\cdot)\) at the most supersingular prime \(p\) (i.e. \(a_p= 0\)) is controlled by two Iwasawa functions (plus/minus \(p\)-adic \(L\)-functions) and by two half-logarithms.
The author uses the \(p\)-adic \(L\)-function constructed by A. Dąbrowski [Ann. Inst. Fourier 44, No. 4, 1025–1041 (1994; Zbl 0808.11035)] to obtain plus/minus \(p\)-adic \(L\)-functions for Hilbert modular forms. The construction is a generalization of Pollack’s method. She uses the constructed plus/minus \(p\)-adic \(L\)-functions to formulate the Iwasawa main conjecture for the supersingular elliptic curves defined over totally real number fields.
Remark 1. J. Park and S. Shahabi [‘Plus/minus \(p\)-adic \(L\)-functions for Hilbert modular forms’, preprint (2009)] also used the \(p\)-adic \(L\)-function constructed by the reviewer to give plus/minus \(p\)-adic \(L\)-functions for Hilbert modular forms.
Remark 2. A. Dąbrowski [C. R., Math., Acad. Sci. Paris 349, No. 7–8, 365–368 (2011; Zbl 1219.11075)] has formulated a (conjectural) generalization of Pollack’s result to \(p\)-adic \(L\)-functions attached to motives.
The author uses the \(p\)-adic \(L\)-function constructed by A. Dąbrowski [Ann. Inst. Fourier 44, No. 4, 1025–1041 (1994; Zbl 0808.11035)] to obtain plus/minus \(p\)-adic \(L\)-functions for Hilbert modular forms. The construction is a generalization of Pollack’s method. She uses the constructed plus/minus \(p\)-adic \(L\)-functions to formulate the Iwasawa main conjecture for the supersingular elliptic curves defined over totally real number fields.
Remark 1. J. Park and S. Shahabi [‘Plus/minus \(p\)-adic \(L\)-functions for Hilbert modular forms’, preprint (2009)] also used the \(p\)-adic \(L\)-function constructed by the reviewer to give plus/minus \(p\)-adic \(L\)-functions for Hilbert modular forms.
Remark 2. A. Dąbrowski [C. R., Math., Acad. Sci. Paris 349, No. 7–8, 365–368 (2011; Zbl 1219.11075)] has formulated a (conjectural) generalization of Pollack’s result to \(p\)-adic \(L\)-functions attached to motives.
Reviewer: Andrzej Dąbrowski (Szczecin)
MSC:
| 11M38 | Zeta and \(L\)-functions in characteristic \(p\) |
| 11F41 | Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces |
| 11R23 | Iwasawa theory |
Keywords:
\(p\)-adic \(L\)-function; Hilbert modular form; \(h\)-admissible measure; supersingular prime; elliptic curves plus/minus \(p\)-adic \(L\)-function; Iwasawa main conjecture; half-logarithmReferences:
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