On congruence prime criterion for cusp forms on $GL_2$ over number fields (Automorphic forms, trace formulas and zeta functions)

Content Provider	Semantic Scholar
Author	健一, 並川
Copyright Year	2011
Abstract	Around 1980, Hida proved an analogue of the class number formula for certain degree 3 L-functions, that is, he found a meaning of the special value of the adjoint Lfunctions of cusp forms. Let us recall briefly his discovery. We denote by $L(s$ , Ad$(f))$ the adjoint L-function associated with a normalized eigen cusp form $f$ on $GL_{2}$ over the rational number field. For a fixed odd prime $p$ , there are p-optimal periods $\Omega_{f,\pm}$ of $f$ (see Section 3) which are well-defined up to p-adic units. Then Hida proved that $C_{f,p}$ $:=\Gamma$ (1, Ad$(f)$ ) $L(1$ , Ad$(f))/\Omega_{f,+}\Omega_{f}$,-is a p-adic integer. Moreover, we can consider the p-adic valuation of $C_{f,p}$ since the periods $\Omega_{f,\pm}$ are well-defined up to p-adic units. Hence, if we consider an analogue of the class number formula, we expect that the quantity $C_{f,p}$ have some algebraic nature. Hida proved the following theorem.
Starting Page	44
Ending Page	53
Page Count	10
File Format	PDF HTM / HTML
Volume Number	1767
Alternate Webpage(s)	http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1767-05.pdf
Language	English
Access Restriction	Open
Content Type	Text
Resource Type	Article