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S ‐adic L ‐Functions Attached to the Symmetric Square of a Newform
Author(s) -
Dabrowski A,
Delbourgo D
Publication year - 1997
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611597000191
Subject(s) - mathematics , iwasawa theory , multiplicative function , conjecture , pure mathematics , square (algebra) , bounded function , function (biology) , modular form , mathematical analysis , geometry , evolutionary biology , biology
In order to apply the ideas of Iwasawa theory to the symmetric square of a newform, we need to be able to define non‐archimedean analogues of its complex L ‐series. The interpolated p ‐adic L ‐function is closely connected via a “Main Conjecture” with certain Selmer groups over the cyclotomic Z p ‐extension of Q . In the p ‐ordinary case these functions are well understood. In this article we extend the interpolation to an arbitrary set S of good primes (not necessarily satisfying ordinarity conditions). The corresponding S ‐adic functions can be characterised in terms of certain admissibility criteria. We also allow interpolation at particular primes dividing the level of the newform. One interesting application is to the symmetric square of a modular elliptic curve E defined over Q . Our constructions yield p ‐adic L ‐functions at all primes of stable or semi‐stable reduction. If p is ordinary or multiplicative the corresponding analytic function is bounded; if p is supersingular our function behaves like log 2 (1 + T). 1991 Mathematics Subject Classification : 11F67, 11F66, 11F33, 11F30