an:00223670
Zbl 0811.11040
Hida, Haruzo
Modular \(p\)-adic \(L\)-functions and \(p\)-adic Hecke algebras
EN
Transl., Ser. 2, Am. Math. Soc. 160, 125-154 (1994); translation from S??gaku 44, No. 4, 289-305 (1992).
00324299
1994
j
11F85 11S37 20G25
survey; \(p\)-adic \(L\)-function of algebraic groups; \(p\)-adic Hecke algebras; cuspidal irreducible representations
This is a survey article for \(p\)-adic \(L\)-functions of algebraic groups and \(p\)-adic Hecke algebras.
The author discusses how the number of variables for \(L\)-functions is determined by algebraic groups. Starting with Riemann's zeta function and the Kubota-Leopoldt \(p\)-adic \(L\)-function, he interprets them as functions on characters for some suitable groups. Then he goes on to consider \(L\)-functions for cuspidal irreducible representations appearing in the space of cusp forms \(L_ 2^ 0 (\xi)\) from the right regular representation of \(\text{GL}_ n (\mathbb{A})\) where \(\mathbb{A}\) is the ring of adeles of \(\mathbb{Q}\). He confirms that one variable in enough for the above \(L\)-functions. After explaining the Langlands conjecture, he introduces a \(p\)-analogue of \(L_ 2^ 0 (\xi)\) and the \(p\)-adic Hecke algebras for \(\mathbb{Q}\) and then for arbitrary number fields. There he finds \(p\)-adic \(L\)-functions whose number of variables needs to be more than one. Several results are stated with short proofs. Along the line he also discusses under certain conditions how to construct the maximal GL(2) extension unramified outside \(p\) over a number field.
Nine open questions and one conjecture are given.
K.I.Ohta (Tokyo)