S -adic L -Functions Attached to the Symmetric Square of a Newform

Abstract

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 Zp‐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 log2(1 + T). 1991 Mathematics Subject Classification: 11F67, 11F66, 11F33, 11F30

The exceptional zero conjecture for Hilbert modular forms

“…Exceptional zero for the symmetric square of an elliptic curve. DabrowskiDelbourgo [DD97] constructed a p-adic L-function attached to the symmetric square of an elliptic modular form, using Rankin's method (by convoluting with a half-integral weight theta series). Their method should generalize to give a two-variable p-adic L-function of the symmetric square, Hilbert modular forms.…”

Bounded p-adic L-functions of motives at supersingular primes

“…(ii) The condition in part (iii) of Conjecture 1 is called the condition of Dąbrowski-Panchishkin (see also [16]). Here is an example where P N,p (d + , M) = P H (d + , M), but P N,p (u, M) ≡ P H (u, M): M = M( f ) ⊗ M(g), where f , g are elliptic cusp forms of weights w( f ) > w(g) and where p is ordinary for f but supersingular for g. (iii) Conjecture 1 has been proved for Tate motive, and in the following cases: M = Sym m M( f ), m = 1, 2, 3 (see [1,18,11,6,2]), M = M( f ) ⊗ M(g), w( f ) > w(g) (see [12]), and M = M( f 1 [2]). …”

Derivative of symmetric square p-adic L-functions via pull-back formula

“…We point out that the analytic L-invariant for CM forms has already been studied in literature [DD97,Har12,HL13]. Note also that our choice of periods is not optimal [Ros13b, §6].…”