$p$-adic $L$-functions of Hilbert cusp forms and the trivial zero conjecture

Abstract

We prove a strong form of the trivial zero conjecture at the central point for the p-adic L-function of a non-critically refined cohomological cuspidal automorphic representation of GL2 over a totally real field, which is Iwahori spherical at places above p.In the case of a simple zero we adapt the approach of Greenberg and Stevens, based on the functional equation for the p-adic L-function of a nearly finite slope family and on improved p-adic L-functions that we construct using automorphic symbols.To compute higher order derivatives of p-adic L-functions we go beyond the Greenberg-Stevens method and study the variation of the root number in partial finite slope families to establish the vanishing of many Taylor coefficients of the p-adic L-function of the family.

$L$-functions of ${\mathrm{GL}}(2n):$ $p$-adic properties and non-vanishing of twists

“…We proceed as in the proof of [BDJ,Prop.3.5]. Pulling back the definition of the Hecke operator U • p (see §1.4) by the automorphic symbols (see §2.2.1) and the twisting operators (see §2.2.2) yields a commutative diagram (we use implicitly that p K 0 (p),K and pr β ′ ,β have the same degree as…”

“…Proof. We follow closely the proof of [BDJ,Prop.4.6]. Consider the commutative diagram: 27), the horizontal maps are induced from the morphisms of local systems written above them, the map T β is induced from the morphisms of local systems (h, v) → (hξt β p , v), and triv ′ δ is induced from the morphisms of local systems:…”

Iwasawa theory for quadratic Hilbert modular forms

“…• Taking γ = ∞ would correspond to varying k 1 while keeping k 2 fixed, and hence the liftings to GSp 4 would have weight ( 1 , 2 ) with 1 − 2 fixed. In this case, we could replace E K with a small parabolic eigenvariety for GL 2 /K (which can be seen as an instance of the "partially overconvergent cohomology" of [BDJ17]), and E(N, ψ) with a small parabolic eigenvariety for the Siegel parabolic of GSp 4 . By this method one can construct a non-trivial Euler system for ρ * π,v for any π which is ordinary at p 1 , with no assumptions on the local factor at p 2 (it could even be supercuspidal).…”

Four-variable $p$-adic triple product $L$-functions and the trivial zero conjecture

“…[Gre94b,(25), p. 166] and [Ben11, p. 1579]). Using the method of Greenberg-Stevens [GS93] and [BDJ17], we establish the trivial zero conjecture for the triple product of elliptic curves. The following result is a special case of our more general result (see Theorem 8.4).…”

“…In the case of a p-adic L-function L p (E, s) of an elliptic curve E over Q the trivial zero arises if and only if E is split multiplicative at p. An analogus formula for L ′ p (E, 1) was experimentally discovered in [MTT86] and proved in [GS93], and for Hilbert modular forms in [Mok09], [Spi14] and [BDJ17]. Our result proves the first cases of the trivial zero conjecture where multiple trivial zeros are present and the Galois representation is not of GL(2)-type.…”