On extra-zeros of p-adic Rankin-Selberg L-functions

Abstract

We prove a version of the Extra-zero conjecture, formulated by the first named author in [8], for p-adic L-functions associated to Rankin-Selberg convolutions of modular forms of the same weight. The novelty of this result is to provide strong evidence in support of this conjecture in the non-critical case, which remained essentially unstudied. CONTENTS DENIS BENOIS AND ST ÉPHANE HORTE 6.3. The second improved p-adic L-function 47 6.4. The functional equation 48 6.5. Functional equation for zeta elements 49 7. Extra zeros of Rankin-Selberg L-functions 51 7.1. The p-adic regulator 51 7.2. The L -invariant 53 7.3. The main theorem 55 References 57 3 Here η α f denotes the canonical eigenvector associated to α( f ). See Section 3.2 below. 4 In the weight 2 case, the motivic version of this element was first constructed by Beilinson[2]. In [29], FlachProof. This theorem was first proved by Bertolini-Darmon-Rotger [15] for modular forms of weight 2. Kings, Loeffler and Zerbes extended the proof to the higher weight case [43, Theorem 7.2.6], [47, Theorem 7.1.5]. Note that the results proved in [43] and [47] are in fact more general and include also the case of modular forms of different weights.We remark that, combining this formula with the computaiton of the special value of the complex L-function in terms of the Beilinson regulator, one can write this theorem in the form (2) (see [43, Theorem 7.2.6]). Also, this result suggests that L p,α ( f , g, s + k 0 ) satisfies the conjectural interpolation properties of L p (M f ,g (k 0 ), D, s) up to "bad" Euler factors at primes dividing N f N g . 0.5. The main result. We keep previous notation and conventions. In this paper, we prove a result toward the extra-zero conjecture for the p-adic L-finction L p,α ( f , g, s) as s = k 0 . In addition to M1-2) we assume that the following conditions hold: M3) ε f , ε g and ε f ε g are primitives modulo N f , N g and lcm(N f , N g ) respectively. M4) ε f (p)ε g (p) = 1.5 More explicitly, e 1 = ε ⊗ t −1 , where ε is a compatible system of p n th roots of unity, and t = log[ε] the associated

$\mathscr{L}$-invariants of Artin motives

“…for some G p -stable linear subspace W + of W . Note that the action of ϕ on D corresponds, under (5), to the action of p…”

“…(ii) By definition W + is regular if the map r D , for D as in (5), is an isomorphism. Using (4) and the canonically isomorphism between D crys (V ) D and Hom(W + , Q p ), we are lead to study the map…”

“…It is thus a regular p-refinement by Proposition 2.9. The L -invariant L (f α ) of f α is defined as L (V, D α ) where D α is the regular submodule attached to W β (see (5)). One checks that e = 0 if α ≠ 1, in which case one has L (f α ) = 1, and that e = 1 if α = 1.…”

The derivative formula of $p$-adic $L$-functions for imaginary quadratic fields at trivial zeros

“…In[Ben11], the choice of the sign of the L -invariant is slightly different from[Ben14,BH20]. Here we follow the definition given in[Ben14,BH20].Next we consider the dual construction of the L -invariant.…”

“…In[Ben11], the choice of the sign of the L -invariant is slightly different from[Ben14,BH20]. Here we follow the definition given in[Ben14,BH20].Next we consider the dual construction of the L -invariant. Let D be a regular submodule of D cris (V ) and putD ⊥ = D ⊥ 0 = Hom E (D cris (V )/D, D cris (E(1)))andD ⊥ 1 = Hom E (D cris (V )/D −1 , D cris (E(1))).…”