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Plus/minus p-adic L-functions for GL2n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {GL}_{2n}$$\end{document}

Plus/minus p-adic L-functions for GL2n\documentclass[12pt]{minimal} \usepackage{amsmath}... We generalise Pollack’s construction of plus/minus L-functions to certain cuspidal automor- phic representations of GL using the p-adic L-functions constructed in work of Barrera 2n Salazar et al. (On p-adic l-functions for GL in finite slope shalika families, 2021). We use 2n these to prove that the complex L-functions of such representations vanish at at most finitely many twists by characters of p-power conductor. Keywords P-adic L-functions · Automorphic forms · Iwasawa theory Résumé Nous généralisons la construction des fonctions L plus/moins de Pollack à certaines représen- tations automorphes cuspidales de GL en utilisant les fonctions L p-adiques construites 2n dans les travaux de Barrera Salazar et al. [2]. Nous les utilisons pour démontrer que les fonctions L complexes de telles représentations disparaissent pour au plus un nombre fini de torsions par des caractères de conducteur une puissance de p. Mathematics Subject Classification 11G40 · 11F67 · 11R23 1 Introduction Let f = a q be a normalized cuspidal newform of weight k and level N with n=0 character ε,and let p be a prime such that p  N.Let α be a root of the Hecke polynomial 2 k−1 ¯ ∼ X − a X + p ε(p) which, after fixing an isomorphism Q C, satisfies r := v (a )< p p p p k − 1, where v is the p-adic valuation on C normalized so that v (p) = 1. From this data p p p (α) we can construct an order r locally analytic distribution L on Z whose values at special characters interpolate the critical values of the complex L-function of f and its twists. The (α) arithmetic of L is well understood in the case that f is ordinary at p i.e. when r = 0, but (α) is more mysterious in the non-ordinary case, since the unbounded growth of L means that it does not lie in the Iwasawa algebra, and hence cannot be the characteristic element of an Iwasawa module. B Rob Rockwood rob.rockwood@warwick.ac.uk Zeeman Building, University of Warwick, Coventry CV4 7HP, UK 123 R. Rockwood In [16] Pollack provides a solution to this problem in the case that a = 0 by constructing + − bounded distributions L , L each of which interpolate half the values of the complex L- p p function of f and its twists. Kobayashi [11]and Lei[13] have formulated Iwasawa main conjectures using these ‘plus/minus p-adic L-functions’, shown them to be equivalent to Kato’s main conjecture and proved one inclusion in these conjectures using Kato’s Euler system. The converse inclusion has been proved in many cases by Wan [19]. Now let Π be a cuspidal automorphic representation of GL (A ). Suppose that Π is 2n Q cohomological with respect to some pure dominant integral weight μ, and that it is the transfer of a globally generic cuspidal automorphic representation of GSpin (A ).Let p 2n+1 be a prime at which Π is unramified, and let α ,...,α be the Satake parameters at p.We 1 2n call a choice of α = α for { j ,..., j }⊂{1,..., 2n} a p-stabilisation of Π. When j 1 n i =1 i a p-stabilisation α is non-critical and under some further auxiliary technical assumptions Dimitrov, Januszewski and Raghuram [7] (ordinary case) and Barrera, Dimitrov and Williams (α) [2] construct a locally analytic distribution L on Z interpolating the L-values of Π.If we assume α satisfies a non-critical slope condition then this p-stabilisation is non-critical, although this is a stronger condition. We show that that there are at most two choices of α (α) satisfying the non-critical slope condition and thus at most two non-critical slope L can be constructed from a given Π. r × × There is an increasing filtration D (Z , C ) on the space D (Z , C ) of C -valued dis- p p p p p tributions on Z which measures the ‘growth’ of the distribution in a precise way (Definition 4). The 0th part of this filtration is the space of measures on Z . The construction of [2] shows that (α) v (α) × L ∈ D (Z , C ). p p Suppose we have two non-critical slope p-adic L-functions for a given Π and suppose the following condition, which we dub the ‘Pollack condition’, holds: Pollack condition: α + α = 0. (1) n n+1 We prove the following theorem, stated for an odd prime p: Theorem 1 Let α be a p-stabilisation satisfying the non-critical slope condition and let Crit(Π ) be the set of critical integers for Π defined in Definition 7. There exist a pair of ± v (α)−#Crit(Π )/2 × distributions L ∈ D (Z , C ) satisfying p p (α) + + − − L = log L + log L , p p p Π Π #Crit(Π )/2 × where log ∈ D (Z , C ) are distributions depending only on Crit(Π ) of order Π p n−1 #Crit(Π )/2. If the valuation of α is minimal (see Proposition 1) the distributions L i =1 p 0 × are contained in D (Z , C ). These distributions satisfy the following interpolation property for j ∈ Crit(Π ): L(Π ⊗ θ, j + 1/2) j + x θ(x )L (x ) = (∗) p + × j log (x θ) p Π for θ a Dirichlet character of conductor an even power of p, and L(Π ⊗ θ, j + 1/2) j − x θ(x )L (x ) = (∗) log (x θ) for θ a Dirichlet character of conductor an odd power of p, where the (∗) are non-zero constants. 123 Plus/minus p-adic L-functions... When p = 2 the result holds with the signs of the distributions log swapped. n−1 Remark 1 Sinceweassumethat μ is pure, the condition that α be minimal is equivalent i =1 to the statement that this p-stabilisation is P-ordinary (See [9, Sect. 6.2]) where P ⊂ GL 2n is the parabolic subgroup given by the partition 2n = (n − 1) + 2 + (n − 1). As an application we prove the following extension of the main result of [7]: Theorem 2 In the case that L are bounded distributions, the purity weight w is even, and Crit(Π ) ={w/2},wehave L(Π ⊗ θ, (w + 1)/2) = 0 for all but finitely many characters θ of p-power conductor. Remark 2 The assumption on the purity weight is to ensure that the central L-value is critical. Relation to other work: Since this paper first appeared in preprint form, Lei and Ray [14] have used the results of this paper to formulate an Iwasawa main conjecture for Π, relating the signed p-adic L-functions of Theorem 1.0.1 to signed Selmer groups. They have also generalised the construction of the signed p-adic L-functions to allow certain cases with α + α = 0, using the theory of Wach modules. n n+1 2 Preliminaries 2.1 p-adic Fourier theory We lay out the relevant theory of continuous functions on Z . The main reference for this section is [5, Sect. I.5]. Let L be a complete extension of Q , denote by C (Z , L) the Banach space of continuous p p functions on Z taking values in L. Write LA(Z , L) for the subspace of L-valued locally p p analytic functions and LA (Z , L) for the locally h-analytic functions. We give these spaces h p h −1 a valuation v in the following way: Let u = (p (1 − p)) and let v be the LA h ¯ h B(a,u ) valuation on power series which converge on B(a, u ) ={z ∈ C : v(z − a) ≥ u } given h p h by v ( f ) = inf {v (a ) + nu : f (X ) = a (X − a) }. ¯ m p m h i B(a,u ) i =0 An element f ∈ LA (Z , L) locally extends to such a power series and we define h p v ( f ) = inf v ( f ). LA a∈Z B(a,u ) h p h This gives LA(Z , L) the structure of a Fréchet space. Definition 1 Let r ∈ R .Let f : Z → L ∈ C (Z , L).Wesay that f is of order r if ≥0 p p (i ) there are functions f : Z → L such that if we define ⎛ ⎞ (i ) i ⎝ ⎠ ε ( f ) = inf v f (x + y) − f (x )y /i ! , h x ∈Z p y∈ p Z p i =0 123 R. Rockwood then ε ( f ) − rh →∞ as h →∞. We denote the set of such functions by C (Z , L). The space C (Z , L) is a Banach space with valuation given by (i ) f (x ) v ( f ) = inf inf ( ), inf ε ( f ) − r v (y) . C 0≤ j ≤ r ,x ∈Z x ,y∈Z n p p p i ! Definition 2 • Define the space of locally analytic distributions on Z to be the continuous dual of LA(Z , L), denoted D (Z , L). This space has the structure of a Fréchet space p p via the family of valuations given by restricting to LA (Z , L) and taking the dual of h p v . LA • For r ∈ R define the subspace D (Z , L) of D (Z , L) of order r distributions to ≥0 p p r 0 be the continuous dual of C (Z , L). The space D (Z , L) of bounded distributions is p p often referred to as the space of measures on Z . We equip each D (Z , L) with the p p valuation r r v (μ) = inf v (μ( f )) − v ( f ) . D f ∈C (Z ,L)\{0} p C p r r r For μ ∈ D (Z , L), f ∈ C (Z , L) we write p p μ( f ) =: f (x )μ(x ). The space D (Z , L) is given the structure of an L-algebra via convolution of distributions: f (x )(μ ∗ λ)(x ) := f (x + y)μ(x ) λ(y). Z Z Z p p p These distribution spaces will be our main object of our study. Though rather inscrutable by themselves, they become more amenable to study by identifying them with spaces of power series. For x ∈ C , a ∈ R,let B(x , a) ={y ∈ C : v (y − x)> a}.Wedefine p p p + n R = f = a ( f )X ∈ L[[X ]] : f converges on B(0, 0) n=0 and we give this space the structure of a Fréchet space via the family of valuations v . B(0,u ) Let (n) = inf{m : n < p },and for r ∈ R define ≥0 + n R ={ f = a X ∈ L[[X ]] : v (a ) + r (n) is bounded below as n →∞}. n p n n=0 We can put a valuation on these spaces v ( f ) = inf b + r (h), r h h (h) where (h) is the smallest integer satisfying p > h. However, a different valuation will be useful for our purposes. 123 Plus/minus p-adic L-functions... Lemma 1 A power series f ∈ L[[X ]] is in R if and only if inf (v ( f ) + rh) = h∈Z ¯ r ≥0 B(0,u ) −∞. Furthermore, the spaces R are Banach spaces when equipped with the valuation v ( f ) = inf (v ( f ) + rh). r h∈Z ≥0 B(0,u ) Moreover, v ( f ) is equivalent to v ( f ). Proof [5, Lemme II.1.1]. + + Lemma 2 If f ∈ R ,g ∈ R ,then fg ∈ R . r s r +s Proof [5, Corollaire II.1.2]. Theorem 3 Define the Amice transform: A : D (Z , L) = R μ → (1 + X ) μ(x ). The Amice transform is an isomorphism of L-algebras under which the spaces D (Z , L) and R are identified isometrically with respect to the valuations v r and v . r r Proof [5, Théorème II.2.2] and [5, Proposition II.3.1]. We now consider the multiplicative topological group Z .Let p if p odd q = 4 otherwise. We have the well-known isomorphism × × Z = (Z/q Z) × 1 + q Z , the second factor of which is topologically cyclic. Let γ be a topological generator of 1+q Z . Such a choice allows us to write any x ∈ 1 + q Z in the form x = γ for a unique s ∈ Z , p p giving us an isomorphism of topological groups 1 + pZ = Z p p γ → s. Thus Z is homeomorphic to p − 1 (resp. 2 when p = 2) copies of Z , and we can use the × × above theory of Z in this context, defining LA(Z , L), D (Z , L) in the obvious way; each p p space decomposes as a direct sum over their restrictions to each Z component and we take the infimum of the valuations on each summand. Definition 3 Define weight space to be the rigid analytic space W over Q representing × × L → Hom (Z , L ). cont Integrating characters gives a canonical identification × 0 D (Z , Q ) = H (W, O ), p W where O is the structure sheaf of W. This is isomorphism commutes with base change in the sense that for a finite extension L/Q we have × × 0 D (Z , L) = D (Z , Q )⊗ L = H (W , O ), p Q L W p p p L 123 R. Rockwood where W = W × Sp(L) and Sp(L) is the affinoid space associated to L. We identify L Q W(C ) with the set  B ,where B = B(0, 0) and the disjoint union runs over characters p ψ ψ ψ × × of Z which factor through (Z/q Z) , for details see [4, Remark B1.1]. We can thus identify × + D (Z , L) with functions on  B whichare describedbyelementsof R on each B . ψ ψ ψ Given a distribution μ ∈ D (Z , L) we write the corresponding rigid function on W as M (μ). On each B the global sections O (B ) are given (after choosing a coordinate X)by ψ W ψ precisely R . As these are quasi-Stein spaces, the topology on O (B ) is that of a Fréchet W ψ space induced by an increasing chain of affinoids Y ⊂ Y ⊂ ··· , 1 2 which we can choose to be the closed discs of radius u , whence the topology as global sections over a rigid space coincides with the topology on R given by the family of valuations v (see [17, 1C]). B(0,u ) r × × Definition 4 For r ∈ R we define a subspace D (Z , L) ⊂ D (Z , L) by ≥0 p p r × × + D (Z , L) ={μ ∈ D (Z , L) : M (μ)| ∈ R for all ψ }. p p ψ r These spaces decompose as a direct sum r × r D (Z , L) =⊕ D (Z , L) ψ p and we equip it with the valuation given by v (μ) := inf v r (μ ) D ψ D ψ where μ is the projection of μ to the ψ component. 2.2 Automorphic representations Fix n ≥ 1and set G = GL .Let Π be a cuspidal automorphic representation of G(A ). 2n Q Let T ⊂ G be the maximal diagonal torus and let 2g μ = (μ ,...,μ ) ∈ Z 1 2n be an integral weight. We say μ is dominant if μ ≥ ··· ≥ μ ,and we say μ is pure if there 1 2n is ω ∈ Z,the purity weight of μ, such that μ + μ = ω i 2n+1−i for all i = 1,..., n. Definition 5 We say that Π is cohomological with respect to a dominant integral weight μ if the (g , K )-cohomology ∞ ∞ H (g , K ,Π ⊗ V ) ∞ ∞ is non-vanishing for some q.Here g is the Lie algebra of G(R), K ⊂ G(R) is the identity component of the maximal open compact subgroup and V is the irreducible C-linear G- representation of highest weight μ. Purity of μ is a necessary condition for Π to be cohomological. 123 Plus/minus p-adic L-functions... Definition 6 We say that Π is the transfer of a globally generic cuspidal automorphic repre- sentation π of GSpin (A ) if 2n+1 Π π . for all primes  at which π is unramified. Remark 3 • For a given globally generic automorphic representation π of GSpin (A) 2n+1 the existence of such a transfer was proved by Asgari-Shahidi [1, Theorem 1.1]. • A necessary and sufficient condition for Π to be the transfer of a globally generic cuspidal automorphic representation of GSpin is that it admits a Shalika model which realises 2n+1 Π in a certain space of functions W : G(A ) → C,see [2, Sect. 2.6] for details. 2.3 p-stabilisations Let Π be a cuspidal automorphic representation of G(A ) which is cohomological with respect to a pure dominant integral weight μ and suppose that Π is the transfer of a globally generic cuspidal automorphic representation of GSpin (A ).Let B denote the upper 2n+1 triangular Borel subgroup of G. Given a prime p at which Π is unramified, define the Hodge-Tate weights of Π at p to be the integers h = μ + 2n − i , i = 1,..., 2n. (2) i i Remark 4 These weights coincide with the Hodge-Tate weights of the Galois representation associated to Π when the Hodge-Tate weight of the cyclotomic character is taken to be 1. Definition 7 Define a set Crit(Π ) ={ j ∈ Z : μ ≥ j ≥ μ }. n n+1 Remark 5 It is shownin[8, Proposition 6.1.1] that the half integers j + 1/2for j ∈ Crit(Π ) are precisely the critical points of the L-function L(s,Π) in the sense of Deligne [6, Definition 1.3]. Let p be a prime at which Π is unramified. There is an unramified character λ : T (Q ) → C p p such that Π is isomorphic to the normalised parabolic induction module 2n−1 G(Q ) Ind (|·| λ ). We define the Satake parameters at p to be the values α = λ (p), p i p,i B(Q ) where λ denotes the projection to the ith diagonal entry. After choosing an isomorphism p,i ¯ ∼ Q C,wereorder the α so that they are ordered with respect to decreasing p-adic valua- p = i tion and such that α α = λ for a fixed λ with p-adic valuation 2n − 1 + w.Thatwe i 2n+1−i can do this is a result of the transfer from GSpin ,see [1, (64)]. 2n+1 We define the Hodge polygon of Π to be the piecewise linear curve joining the following points in R : ⎧ ⎛ ⎞ ⎫ ⎨ ⎬ ⎝ ⎠ (0, 0), j , h : j = 1,..., 2n 2n+1−i ⎩ ⎭ i =1 123 R. Rockwood and define the Newton polygon on Π at p to be the piecewise linear curve joining the points ⎧ ⎛ ⎞ ⎫ ⎨ ⎬ ⎝ ⎠ (0, 0), j , v (α ) : j = 1,..., 2n . p 2n+1−i ⎩ ⎭ i =1 The following result is due in this form to Hida [9, Theorem 8.1]. Proposition 1 The Newton polygon lies on or above the Hodge polygon and the end points coincide. Definition 8 Let I = (i ,..., i ) ⊂ Z satisfy 1 ≤ i < ··· < i ≤ 2n,and set 1 n 1 n α := α ··· α . I i i 1 n We call α p-stabilisation data for Π. Let Q ⊂ GL be the parabolic subgroup given by the partition 2n = n +n. The following 2n conditions are translations of the conditions of the same name given in [2, Sect. 2.7]. Definition 9 Let I be as above. • We say that the product α is of Shalika type if I contains precisely one element of each pair {i , 2n + 1 − i } for i = 1,..., n,see [7, Definition 3.5(ii)]. • We say that α is Q-regular if it is of Shalika type and if for any other choice of J ⊂ Z satisfying the above properties a = a . This amounts to choosing a simple Hecke J I eigenvalue for the U -operator associated to Q actingonthe Q-parahoric invariants of Π ,see [7, Definition 3.5(i)] and [2, Sect. 2.7]. 2n • Set r = v (α ) − h . We say that α is non-critical slope if it satisfies I p I i I i =n+1 r < #Crit(Π ). • We say that α is minimal slope if r = #Crit(Π )/2. Remark 6 The conditions in Definition 9 are used to control a certain local twisted integral at p, attached to a choice of parahoric-invariant vector W in the Shalika model. In [7, Proposition 3.4], the authors show that this local zeta integral is an explicit multiple of W (1).In[7,Lem. 3.6], they use the Shalika-type and Q-regular conditions to exhibit an explicit vector W in the Shalika model attached to α with W (1) = 1, and hence deduce non-vanishing of the local zeta integral. We note that Π always admits p-stabilisations of Shalika type. The terminology is justified by [2, Remark 2.5], which explains that the refinements of Shalika type are exactly those that arise from refinements of GSpin . Finally, if the Satake parameter of Π is 2n+1 regular semisimple, then all stabilisations of Π are Q-regular. (α ) 1 I × × In [2] the authors construct a locally analytic distribution L ∈ D (Z , C ) with p p respect to a choice of non-critical slope Q-regular p-stabilisation data α . The distribution (α ) L is of order r andby[18, Lemma 2.10] is uniquely defined by the following interpolation p I × m property: Let θ : Z → Q be a finite-order character of conductor p , then for m ≥ 1we have c j x θ j (α ) x θ(x )L (x ) = ξ L(Π ⊗ θ, j + 1/2), j ∈ Crit(Π ), (3) ∞, j p I The authors actually construct p-adic L-functions for the wider class of non-critical p-stabilisations, but we only work with non-critical slope p-stabilisations in this paper. 123 Plus/minus p-adic L-functions... where c j is a constant depending only on x θ and the infinite factor ξ is the product ∞, j x θ (α ) of a choice of period and a zeta integral at infinity. We call such a L a ‘non-critical slope p-adic L-function’. 3 Plus/Minus p-adic L-functions We construct the titular plus/minus L-functions. We first note that the condition of non-critial slope imposes strong restrictions on the number of p-adic L-functions we can construct. (α ) Theorem 4 There are at most two choices of p-stabilisation α for which L is non-critical slope. Proof Without loss of generality we may assume that μ = 0, forcing w = μ .The end 2n 1 points of the Newton and Hodge polygons coinciding implies that v (λ) = h + h , i = 1,..., n. (†) p i 2n+1−i The ‘non-critical slope’ condition for I = (i ,..., i ) is equivalent to 1 n 2n v (α ) − h < h − h . p I i n n+1 i =n+1 We observe that any I that includes a a 2-tuple of integers the form (i , 2n + 1 − i ) is not non-critical slope. Indeed, we can find an explicit I containing some (i , 2n + 1 − i ) with minimal valuation, namely (n, n + 1, n + 3,..., 2n), amongst all I containing some (i , 2n + 1 − i ). For such an I we have v (α ) − (h + h ··· + h ) p I n+1 n+2 2n ≥ h + h + h + ··· + h − (h + ··· + h ) n n+1 n+3 2n n+1 2n = h − h n n+2 > h − h , n n+1 where the first inequality is a consequence of the Newton polygon lying above the Hodge polygon and (†), and the strict inequality is due to dominance. Thus any I containing a pair of integers (i , j ) with i < j and j ≤ 2n + 1 − i cannot be non-critical slope, since any such I has greater valuation than (n, n + 1, n + 3,..., 2n). This leaves us with two choices of potential non-critical slope n-tuples: I = (n + 1, n + 2,..., 2n), n+1 and I = (n, n + 2,..., 2n). In light of Theorem 4 it is clear that the only two choices of p-stabilisation data which can give a non-critical slope distribution are α = α α ...α ,β = α α ...α . n+1 n+2 2n n n+2 2n 123 R. Rockwood Definition 10 We say that Π satisfies the ‘Pollack condition’ if α + α = 0. n n+1 Corollary 1 For a non-critical slope p-stabilisation α we have the following: • The p-stabilisation α is of Shalika type. • If we assume the Pollack condition, then α is Q-regular. Proof The first part is immediate from Theorem 4. For the second claim recall that a p-stabilisation α is Q-regular if α = α I J for all I = J . Suppose α is a non-critical slope p-stabilisation. Then by Theorem 4 α = α α ··· α or α α ··· α , I n n+2 2n n+1 n+2 n+1 and by the Pollack condition these are clearly not equal since they are non-vanishing by non- critical slope. Finally, for a critical slope p-stabilisation α we must have v (α )>v (α ) J p J p I so α = α . J I 3.1 Pollack ±-L-functions Let Π be as in the previous section. As in the previous section, set α = α α ...α ,β = α α ...α , n+1 n+2 2n n n+2 2n 2n 2n and let r = v (α) − h = v (β) − h . The Pollack condition forces p i p i i =n+1 i =n+1 r ≥ #Crit(Π )/2 since 2n r = v (α) − h ≥ v (α ) − h p i p n+1 n+1 i =n+1 h + h n n+1 = − h n+1 h − h n n+1 = #Crit(Π )/2, where the first inequality comes from Newton-above-Hodge, and the lower bound given is tight, with the bound being achieved when the end point of the segment of the Newton polygon corresponding to α ··· α touches the Hodge polygon. This justifies the use of n+2 2n the term ‘minimal slope’ in Definition 9. We assume that r < #Crit(Π ) (α) (β) so that we can construct precisely two non-critical slope p-adic L-functions L , L ∈ p p r × D (Z , C ). 123 Plus/minus p-adic L-functions... Remark 7 Unlike in the case of GL ,for n > 1 the non-critical slope condition for α, β is not necessarily implied by the Pollack condition. Indeed, suppose one has a cuspidal automorphic representation Π of GL (A ) satisfying the Pollack condition at a prime p for 2n Q which v (α ) = v (α ) for 1 ≤ i , j ≤ 2n.The value r is then the same for any choice of p- p i p j 2n stabilisation, so there are either non-critical slope p-adic L-functions or there are none. But Theorem 4 says there can be at most two choices of non-critical slope p-stabilisation. Following Pollack, we define (α) (β) L ± L p p G = , so that (α) + − L = G + G (β) + − L = G − G . (β) We note that in the case of L , the interpolation formula is given by c j x θ j (β) m x θ(x )L (x ) = (−1) ξ L(Π ⊗ θ, j + 1/2), j ∈ Crit(Π ), ∞, j from which it follows that j + m x θ(x )G (x ) = 0, if the conductor of θ is p , m odd j − m x θ(x )G (x ) = 0, if the conductor of θ is p , m even. Equivalently (noting that characters of conductor m correspond to (m − 1)th roots of unity), if ζ is any p th root of unity and p is odd, + j M (G )(γ ζ − 1) = 0for m even − j M (G )(γ ζ − 1) = 0for m odd on each of the connected components of W(C ) (which we recall we are identifying with p − 1 copies of B(0, 0)). When p = 2 the sign flips and the above vanishing is equivalent to. + j M (G )(γ ζ m − 1) = 0for m odd − j M (G )(γ ζ m − 1) = 0for m even For any j ∈ Z, Pollack defines the following power series − j 1  (γ (1 + X )) 2m log (X ) := p, j p p m=1 − j 1  (γ (1 + X )) 2m−1 log (X ) := , p, j p p m=1 in Q [[X ]],where  is the p th cyclotomic polynomial. p m We are referring to the connected components as a rigid space as opposed to those of the topology on C induced by v . 123 R. Rockwood + − + Lemma 3 The power series log (X ) (resp. log (X )) is contained in R and vanishes p, j p, j 1/2 j m at precisely the points γ ζ m − 1 for every p th root of unity ζ m with m even (resp. odd). p p Proof The statements in the lemma are proved in [16, Lemma 4.1] and [16, Lemma 4.5]. We + + − reprove that log is contained in R using the setup of Sect. 2, the result for log being p, j 1/2 p, j similar. An analysis of the Newton copolygon of the Eisenstein polynomial  givesusthat 0if h ≤ n − 1 − j v ( ((γ (1 + X ))/p) = B(0,u ) h n−h−1 p − 1 otherwise, and thus h+1 + 2m−h−1 v (log ) = p − 1 B(0,u ) p, j m=1 −(h+1) p − 1 1 h = − − , 1 − p 2 2 whence h 1 1 inf v (log ) + = − < ∞, h ¯ B(0,u ) p, j 2 p − 1 2 + + so log (X ) ∈ R by Lemma 1. p, j 1/2 We define ± ± + log (X ) = log (X ) ∈ R . Π p, j Crit(Π )/2 j ∈#Crit(Π ) ± ± By abuse of notation we will write log (X ) for the element of O (W) givenbylog (X ) Π Π on each connected component of W. Lemma 4 We have #Crit(Π ) lim sup v (log ) + h < ∞. B(0,u ) Π Proof It follows from the proof of Lemma 3 and the multiplicativity of v that B(0,u ) −(h+1) #Crit(Π ) p − 1 1 v (log ) + h = #Crit(Π ) − . B(0,u ) Π h 2 2 1 − p 2 The right side converges as h →∞ so the lim sup is finite. A similar argument works for log . It follows from the above discussion and [12, 4.7] that for odd p the rigid function log (X ) ± ± divides M (G ) in O (W),and for p = 2 we have that log (X ) divides M (G ) in O (W). Define plus/minus p-adic L-functions L (X ) to be the elements of O (W) sat- W W isfying ± ± ± M (G ) = log (X ) · L (X ) 123 Plus/minus p-adic L-functions... for p odd, and ± ∓ ± M (G ) = log (X ) · L (X ) Π p ± −1 ± for p = 2. We write L for the distribution M (L (X )). p p Proposition 2 We have ± r −#Crit(Π )/2 × L ∈ D (Z , C ). p p Proof We note that #Crit(Π ) #Crit(Π ) ± ± lim inf −v (log ) − h =− lim sup v (log ) + h > −∞. ¯ ¯ B(0,u ) Π 2 B(0,u ) Π 2 h h By the additivity of v ([5, Proposition I.4.2]) we have B(0,u ) ± #Crit(Π ) ± ± #Crit(Π ) v (L ) + (r − )h = v (G ) + rh − v (log ) − h, ¯ ¯ ¯ B(0,u ) p B(0,u ) B(0,u ) Π h 2 h h 2 ± r × ± and so since G ∈ D (Z , C ) (and thus lim inf v (G ) + rh > −∞)wehave p ¯ p B(0,u ) ± #Crit(Π ) lim inf v (L ) + (r − )h > −∞ B(0,u ) p h 2 andsobyLemma 1 we are done. #Crit(Π ) In particular, in the minimal slope case r = we get two bounded distributions. Remark 8 • One might ask if there is an analogue of the plus/minus theory for p-adic L-functions for GL . Beyond the exact methods used in the present paper, there is 2n+1 an immediate stumbling block: in general, for n ≥ 3 odd even the usual theory of p- adic L-functions is very poorly developed. For non-ordinary Π on GL , the only 2n+1 constructions of p-adic L-functions are for n = 1and Π a symmetric square lift from GL ; in this case, a study of signed Iwasawa theory has been considered in [3]. • The proofs above show that if we relax the minimal slope hypothesis we still obtain a pair of plus/minus L-functions which are unfortunately not bounded. Since the subsets of weight space on which these functions interpolate L-values are disjoint it seems that there is no hope in attempting a similar construction for these functions. • In thecasethatwehaveaGL (A ) representation admitting p-stabilisations which 2n Q are critical but not non-critical slope Theorem 4 no longer holds. As a result, for each 2n such p-stabilisation we can construct a p-adic L-function, giving us at most p-adic L-functions. It’s possible that one could generalise the methods of this paper and utilise all of these p-adic L-functions to construct bounded functions analogous to L , but this is not something we have explored. 3.2 An example of a GL (A ) representation satisfying the Pollack condition 4 Q We give an example of a cuspidal automorphic representation of GL (A ) satsifying the 4 Q Pollack condition and having minimal slope using the theory of twisted Yoshida lifts (see [15, Sect. 6] for an overview). Let F = Q( 5) and let σ : F → R, i = 1, 2 be the embeddings of F into R.The prime 41 splits in F and we write M for one of its prime factors. Using Magma we see that there is a weight (4, 2) cuspidal Hilbert newform f over F of level N = M and of trivial 123 R. Rockwood character and with complex multiplication by the unique extension E /F such that E /Q is not Galois and in which M ramifies. Thus there is a Hecke character ψ over E with infinity 3 2 type z → ε (z) ε (z) ε ¯ (z),where ε , ε ¯ : E → C, i = 1, 2, are the pairs of conjugate 1 2 2 i i embeddings of E and such that the Hecke eigenvalue of f at a prime ℘ of F is given by ⎪ ψ(q ) + ψ(q ) if ℘ = q q in E 1 2 1 2 a = ψ(q) if ℘ = q in E 0 otherwise. Let L be the number field generated by the Hecke eigenvalues of f . Since the weight is not parallel we have F ⊂ L.Fix arationalprime p  N and let L be the completion of L at a prime v over p. Suppose now that p splits in F and write p = ℘ · ℘ , labelled such that 1 2 σ (℘ ) is below v. i i Let ψ : G → L be the v-adic character of G = Gal(F /F ) associated to ψ by Gal,v E v F class field theory, so that V := Ind ψ is the G -representation associated to f . f ,v Gal,v F This representation is crystalline at primes not dividing N. The Hodge-Tate weights of V f ,v are given by (0, 3) at σ and (1, 2) at σ . 1 2 The restriction of V to G splits as a direct sum of characters: f ,v E V | = ψ ⊕ ψ , f ,v G Gal,v E Gal,v where c ∈ Gal(E /F ) is the non-trivial element. If a prime ℘ of F above p splits in E then a decomposition group D ⊂ G at a prime over ℘ is contained in G and we have ℘ F E D (V | ) = D (ψ | ) ⊕ D (ψ | ) whence f is ordinary at ℘ as the cris f ,v D cris Gal,v D cris D ℘ ℘ Gal,v ℘ image of the functor D is weakly admissible. Taking v to be above σ (℘) and writing the cris 1 prime decomposition of ℘ in E as ℘ = q q we have 1 2 v (ψ (q )) = 0 v 1 v (ψ (q )) = 3 v 2 up to reordering of the q . Theorem 5 Suppose π is a cuspidal automorphic representation generated by a holomorphic Hilbert modular form of weight (k , k ) over a totally real field F. Suppose that: 1 2 • For 1 = θ ∈ Gal(F /Q) we have π ≈ π, • There is a Hecke character ε over Q such that the central character ω of π satisfies ω = ε ◦ Norm . π F /Q Then there is a unique globally generic cuspidal automorphic representation (π, ω ) of k +k |k −k | 1 2 1 2 GSp (A ) (a twisted Yoshida lift) of weight ( , − 2) with central character ε 2 2 satisfying max{k , k }− 1 1 2 L(Π , s) = L π, s + . Proof See [15, Theorem 6.1.1 and Proposition 6.1.4]. Let π be the cuspidal automorphic representation of GL (A ) generated by f .Since f has 2 E non-parallel weight we see that π ≈ π for non-trivial θ ∈ Gal(F /Q). Set Π = (π, 1).Then Π a cuspidal automorphic representation of GSp (A ) of weight (3, 3) with trivial central character. This weight lies in the cohomological range and thus Π is cohomological. The Hodge-Tate weights of Π are (0, 1, 2, 3). 123 Plus/minus p-adic L-functions... Recall that we have a rational prime p such that p splits in F: pO = ℘ ℘ . F 1 2 We assume further that ℘ is inert in E and ℘ splits, and we write the factorisation of ℘ as 2 1 1 ℘ = P P . 1 1 2 Primes satisfying the above conditions are not uncommon, for example, the primes 11 and 19 admit this splitting phenomena in the tower E /F /Q. We remark that a necessary condition for such a splitting is that E is non-Galois over Q. The local L-factor of Π at such a p is given by −1 −s −s −2s L (Π , s) = (1 − ψ(P )p )(1 − ψ(P )p )(1 − ψ(℘ )p ). p 1 2 2 Choosing a prime v of L lying above p such that v lies above σ (℘ ), we deduce that Π i i satisfies the Pollack condition and has two minimal slope p-stabilisations. By Corollary 1 these p-stabilisations are Q-regular and Shalika. There is an exceptional isomorphism GSp GSpin and so we are done by applying the functorial lift from GSpin to GL . 4 5 5 4 Non-vanishing of twists We use L to show non-vanishing of the complex L-function of Π at the central value, extending work of Dimitrov, Januszewski, Raghuram [7] to a non-ordinary setting. Proposition 3 In the case that Crit(Π ) ={w/2}, we have L = 0. + − Proof We consider L , the case of L being essentially identical and we further assume p p p is odd for brevity of notation (although the same argument works in this case). Note that a × × m+1 m character θ : Z → C of conductor p corresponds to a choice of primitive p th root p p of unity ζ in a disc W determined by the restriction ψ of θ to (Z/q Z) , which gives us the θ θ identification j + + j x θ(x )log (x ) = log (ψ ,γ ζ − 1), θ θ Π Π where on the left hand side we use the description of log as a distribution and on the right + + hand side log (ψ , −) is the restriction of log to the disc in W corresponding to ψ .We θ θ Π Π adopt the analogous notation for L . It follows from Lemma 3 that log (ψ ,γ ζ − 1) = 0 p θ θ if m is odd (resp. even for p = 2). Thus for characters θ of odd p-power conductor we have the interpolation property L(Π ⊗ θ, j + 1/2) L (ψ ,γ ζ − 1) ∼ , j ∈ Crit(Π ), p θ θ log (ψ ,γ ζ − 1) θ θ where ∼ is used here to mean ‘up to non-zero constant’. By Jacquet-Shalika [10,1.3]we have L(Π ⊗ θ, s) = 0for Re(s) ≥ w/2 + 1 123 R. Rockwood for finite order characters θ, and by applying the functional equation we get non-vanishing for Re(s) ≤ w/2. Since Crit(Π ) contains an integer k not equal to w/2, the above discussion gives us L(Π , k + 1/2) = 0, and thus L = 0. Remark 9 Proposition 3 actually proves the stronger result that the power series M (L )| p ψ is non-zero for each choice of ψ. We can turn this back on itself and use L to say something about nonvanishing of L(Π ⊗ ± 0 × θ, (ω + 1)/2) in the case when L ∈ D (Z , C ). p p Theorem 6 In the case that L are bounded distributions, w is even, and Crit(Π ) ={w/2}, we have L(Π ⊗ θ, (w + 1)/2) = 0 for all but finitely many characters θ of p-power conductor. Proof Assume p odd for brevity, again noting that the argument works fine in this case. For × ± ± any character ψ of (Z/pZ) we can write M (L )| = L (ψ, T ) ∈ O [[T ]] ⊗ L for B L O p ψ p L ± ± some finite extension L/Q . We note that M (L ) = 1 L where 1 denotes the p B B p ψ p ψ indicator function on B . This power series is non-zero by Proposition 3 and Remark 9,and ± ± so Weierstrass preparation tells us that each L (ψ, T ), and thus L , has only finitely many p p zeroes. Given any character θ of p-power conductor, we have w/2 ? x θ(x )L (x ) ∼ L(Π ⊗ θ, (w + 1)/2), where + if the conductor of θ is odd p -power ? = . − otherwise Thus, for all but finitely many θ,wehave L(Π ⊗ θ, (w + 1)/2) = 0. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. 123 Plus/minus p-adic L-functions... References 1. Asgari, M., Shahidi, F.: Generic transfer for general spin groups. Duke Mathematical Journal 132(1), 137–190 (2006) 2. Barrera Salazar, D., Dimitrov, M., Williams, C.: On p-adic l-functions for GL in finite slope shalika 2n families. arXiv preprint arXiv:2103.10907 (2021). https://arxiv.org/abs/2103.10907 3. Büyükboduk, K., Lei, A., Venkat, G.: Iwasawa theory for symmetric square of non- p-ordinary eigenforms. arXiv preprint arXiv:1807.11517 (2018) 4. Coleman, R.F.: p-adic banach spaces and families of modular forms. Inventiones mathematicae 127(3), 417–479 (1997) 5. Colmez, P.: Fonctions d’une variable p-adique. Astérisque 330, 13–59 (2010) 6. Deligne, P.: Valeurs de fonctions L et périodes d’intégrales. In: Proc. Symp. Pure Math, vol. 33, pp. 313–346 (1979). https://doi.org/10.1090/pspum/033.2/546622 7. Dimitrov, M., Januszewski, F., Raghuram, A.: L-functions of GL : p-adic properties and non- 2n vanishing of twists. Compositio Mathematica 156(12), 2437–2468 (2020). https://doi.org/10.1112/ s0010437x20007551 8. Grobner, H., Raghuram, A.: On the arithmetic of Shalika models and the critical values of L-functions for GL . American Journal of Mathematics 136(3), 675–728 (2014). URL https://doi.org/10.1353/ajm. 2n 2014.0021 9. Hida, H.: Automorphic induction and Leopoldt type conjectures for GL(n). Asian Journal of Mathematics 2(4), 667–710 (1998). URL https://doi.org/10.4310/ajm.1998.v2.n4.a5 10. Jacquet, H., Shalika, J.A.: A non-vanishing theorem for zeta functions of GL . Inventiones mathematicae 38(1), 1–16 (1976) 11. Kobayashi, S.i.: Iwasawa theory for elliptic curves at supersingular primes. Inventiones mathematicae 152(1), 1–36 (2003). URL https://doi.org/10.1007/s00222-002-0265-4 12. Lazard, M.: Les zéros d’une fonction analytique d’une variable sur un corps valué complet. Publications Mathématiques de l’IHÉS 14, 47–75 (1962) 13. Lei, A.: Iwasawa theory for modular forms at supersingular primes. Compositio Mathematica 147(3), 803–838 (2011). URL https://doi.org/10.1112/s0010437x10005130 14. Lei, A., Ray, J.: Iwasawa theory of automorphic representations of GL at non-ordinary primes. arXiv 2n preprint arXiv:2010.00715 (2020) 15. Loeffler, D., Zerbes, S.L.: Iwasawa theory for quadratic hilbert modular forms. arXiv preprint arXiv:2006.14491 (2020) 16. Pollack, R.: On the p-adic L-function of a modular form at a supersingular prime. Duke Mathematical Journal 118(3), 523–558 (2003). URL https://doi.org/10.1215/s0012-7094-03-11835-9 17. Pottharst, J.: Analytic families of finite-slope selmer groups. Algebra & Number Theory 7(7), 1571–1612 (2013) 18. Visik, M.: Non-archimedean measures connected with Dirichlet series. Sbornik: Mathematics 28, 216– 228 (1976). URL https://doi.org/10.1070/sm1976v028n02abeh001648 19. Wan, X.: Iwasawa main conjecture for non-ordinary modular forms. arXiv preprint arXiv:1607.07729 (2016) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annales mathématiques du Québec Springer Journals

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Rockwood, Rob
Annales mathématiques du Québec , Volume 47 (1) – Apr 1, 2023
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Abstract

We generalise Pollack’s construction of plus/minus L-functions to certain cuspidal automor- phic representations of GL using the p-adic L-functions constructed in work of Barrera 2n Salazar et al. (On p-adic l-functions for GL in finite slope shalika families, 2021). We use 2n these to prove that the complex L-functions of such representations vanish at at most finitely many twists by characters of p-power conductor. Keywords P-adic L-functions · Automorphic forms · Iwasawa theory Résumé Nous généralisons la construction des fonctions L plus/moins de Pollack à certaines représen- tations automorphes cuspidales de GL en utilisant les fonctions L p-adiques construites 2n dans les travaux de Barrera Salazar et al. [2]. Nous les utilisons pour démontrer que les fonctions L complexes de telles représentations disparaissent pour au plus un nombre fini de torsions par des caractères de conducteur une puissance de p. Mathematics Subject Classification 11G40 · 11F67 · 11R23 1 Introduction Let f = a q be a normalized cuspidal newform of weight k and level N with n=0 character ε,and let p be a prime such that p  N.Let α be a root of the Hecke polynomial 2 k−1 ¯ ∼ X − a X + p ε(p) which, after fixing an isomorphism Q C, satisfies r := v (a )< p p p p k − 1, where v is the p-adic valuation on C normalized so that v (p) = 1. From this data p p p (α) we can construct an order r locally analytic distribution L on Z whose values at special characters interpolate the critical values of the complex L-function of f and its twists. The (α) arithmetic of L is well understood in the case that f is ordinary at p i.e. when r = 0, but (α) is more mysterious in the non-ordinary case, since the unbounded growth of L means that it does not lie in the Iwasawa algebra, and hence cannot be the characteristic element of an Iwasawa module. B Rob Rockwood rob.rockwood@warwick.ac.uk Zeeman Building, University of Warwick, Coventry CV4 7HP, UK 123 R. Rockwood In [16] Pollack provides a solution to this problem in the case that a = 0 by constructing + − bounded distributions L , L each of which interpolate half the values of the complex L- p p function of f and its twists. Kobayashi [11]and Lei[13] have formulated Iwasawa main conjectures using these ‘plus/minus p-adic L-functions’, shown them to be equivalent to Kato’s main conjecture and proved one inclusion in these conjectures using Kato’s Euler system. The converse inclusion has been proved in many cases by Wan [19]. Now let Π be a cuspidal automorphic representation of GL (A ). Suppose that Π is 2n Q cohomological with respect to some pure dominant integral weight μ, and that it is the transfer of a globally generic cuspidal automorphic representation of GSpin (A ).Let p 2n+1 be a prime at which Π is unramified, and let α ,...,α be the Satake parameters at p.We 1 2n call a choice of α = α for { j ,..., j }⊂{1,..., 2n} a p-stabilisation of Π. When j 1 n i =1 i a p-stabilisation α is non-critical and under some further auxiliary technical assumptions Dimitrov, Januszewski and Raghuram [7] (ordinary case) and Barrera, Dimitrov and Williams (α) [2] construct a locally analytic distribution L on Z interpolating the L-values of Π.If we assume α satisfies a non-critical slope condition then this p-stabilisation is non-critical, although this is a stronger condition. We show that that there are at most two choices of α (α) satisfying the non-critical slope condition and thus at most two non-critical slope L can be constructed from a given Π. r × × There is an increasing filtration D (Z , C ) on the space D (Z , C ) of C -valued dis- p p p p p tributions on Z which measures the ‘growth’ of the distribution in a precise way (Definition 4). The 0th part of this filtration is the space of measures on Z . The construction of [2] shows that (α) v (α) × L ∈ D (Z , C ). p p Suppose we have two non-critical slope p-adic L-functions for a given Π and suppose the following condition, which we dub the ‘Pollack condition’, holds: Pollack condition: α + α = 0. (1) n n+1 We prove the following theorem, stated for an odd prime p: Theorem 1 Let α be a p-stabilisation satisfying the non-critical slope condition and let Crit(Π ) be the set of critical integers for Π defined in Definition 7. There exist a pair of ± v (α)−#Crit(Π )/2 × distributions L ∈ D (Z , C ) satisfying p p (α) + + − − L = log L + log L , p p p Π Π #Crit(Π )/2 × where log ∈ D (Z , C ) are distributions depending only on Crit(Π ) of order Π p n−1 #Crit(Π )/2. If the valuation of α is minimal (see Proposition 1) the distributions L i =1 p 0 × are contained in D (Z , C ). These distributions satisfy the following interpolation property for j ∈ Crit(Π ): L(Π ⊗ θ, j + 1/2) j + x θ(x )L (x ) = (∗) p + × j log (x θ) p Π for θ a Dirichlet character of conductor an even power of p, and L(Π ⊗ θ, j + 1/2) j − x θ(x )L (x ) = (∗) log (x θ) for θ a Dirichlet character of conductor an odd power of p, where the (∗) are non-zero constants. 123 Plus/minus p-adic L-functions... When p = 2 the result holds with the signs of the distributions log swapped. n−1 Remark 1 Sinceweassumethat μ is pure, the condition that α be minimal is equivalent i =1 to the statement that this p-stabilisation is P-ordinary (See [9, Sect. 6.2]) where P ⊂ GL 2n is the parabolic subgroup given by the partition 2n = (n − 1) + 2 + (n − 1). As an application we prove the following extension of the main result of [7]: Theorem 2 In the case that L are bounded distributions, the purity weight w is even, and Crit(Π ) ={w/2},wehave L(Π ⊗ θ, (w + 1)/2) = 0 for all but finitely many characters θ of p-power conductor. Remark 2 The assumption on the purity weight is to ensure that the central L-value is critical. Relation to other work: Since this paper first appeared in preprint form, Lei and Ray [14] have used the results of this paper to formulate an Iwasawa main conjecture for Π, relating the signed p-adic L-functions of Theorem 1.0.1 to signed Selmer groups. They have also generalised the construction of the signed p-adic L-functions to allow certain cases with α + α = 0, using the theory of Wach modules. n n+1 2 Preliminaries 2.1 p-adic Fourier theory We lay out the relevant theory of continuous functions on Z . The main reference for this section is [5, Sect. I.5]. Let L be a complete extension of Q , denote by C (Z , L) the Banach space of continuous p p functions on Z taking values in L. Write LA(Z , L) for the subspace of L-valued locally p p analytic functions and LA (Z , L) for the locally h-analytic functions. We give these spaces h p h −1 a valuation v in the following way: Let u = (p (1 − p)) and let v be the LA h ¯ h B(a,u ) valuation on power series which converge on B(a, u ) ={z ∈ C : v(z − a) ≥ u } given h p h by v ( f ) = inf {v (a ) + nu : f (X ) = a (X − a) }. ¯ m p m h i B(a,u ) i =0 An element f ∈ LA (Z , L) locally extends to such a power series and we define h p v ( f ) = inf v ( f ). LA a∈Z B(a,u ) h p h This gives LA(Z , L) the structure of a Fréchet space. Definition 1 Let r ∈ R .Let f : Z → L ∈ C (Z , L).Wesay that f is of order r if ≥0 p p (i ) there are functions f : Z → L such that if we define ⎛ ⎞ (i ) i ⎝ ⎠ ε ( f ) = inf v f (x + y) − f (x )y /i ! , h x ∈Z p y∈ p Z p i =0 123 R. Rockwood then ε ( f ) − rh →∞ as h →∞. We denote the set of such functions by C (Z , L). The space C (Z , L) is a Banach space with valuation given by (i ) f (x ) v ( f ) = inf inf ( ), inf ε ( f ) − r v (y) . C 0≤ j ≤ r ,x ∈Z x ,y∈Z n p p p i ! Definition 2 • Define the space of locally analytic distributions on Z to be the continuous dual of LA(Z , L), denoted D (Z , L). This space has the structure of a Fréchet space p p via the family of valuations given by restricting to LA (Z , L) and taking the dual of h p v . LA • For r ∈ R define the subspace D (Z , L) of D (Z , L) of order r distributions to ≥0 p p r 0 be the continuous dual of C (Z , L). The space D (Z , L) of bounded distributions is p p often referred to as the space of measures on Z . We equip each D (Z , L) with the p p valuation r r v (μ) = inf v (μ( f )) − v ( f ) . D f ∈C (Z ,L)\{0} p C p r r r For μ ∈ D (Z , L), f ∈ C (Z , L) we write p p μ( f ) =: f (x )μ(x ). The space D (Z , L) is given the structure of an L-algebra via convolution of distributions: f (x )(μ ∗ λ)(x ) := f (x + y)μ(x ) λ(y). Z Z Z p p p These distribution spaces will be our main object of our study. Though rather inscrutable by themselves, they become more amenable to study by identifying them with spaces of power series. For x ∈ C , a ∈ R,let B(x , a) ={y ∈ C : v (y − x)> a}.Wedefine p p p + n R = f = a ( f )X ∈ L[[X ]] : f converges on B(0, 0) n=0 and we give this space the structure of a Fréchet space via the family of valuations v . B(0,u ) Let (n) = inf{m : n < p },and for r ∈ R define ≥0 + n R ={ f = a X ∈ L[[X ]] : v (a ) + r (n) is bounded below as n →∞}. n p n n=0 We can put a valuation on these spaces v ( f ) = inf b + r (h), r h h (h) where (h) is the smallest integer satisfying p > h. However, a different valuation will be useful for our purposes. 123 Plus/minus p-adic L-functions... Lemma 1 A power series f ∈ L[[X ]] is in R if and only if inf (v ( f ) + rh) = h∈Z ¯ r ≥0 B(0,u ) −∞. Furthermore, the spaces R are Banach spaces when equipped with the valuation v ( f ) = inf (v ( f ) + rh). r h∈Z ≥0 B(0,u ) Moreover, v ( f ) is equivalent to v ( f ). Proof [5, Lemme II.1.1]. + + Lemma 2 If f ∈ R ,g ∈ R ,then fg ∈ R . r s r +s Proof [5, Corollaire II.1.2]. Theorem 3 Define the Amice transform: A : D (Z , L) = R μ → (1 + X ) μ(x ). The Amice transform is an isomorphism of L-algebras under which the spaces D (Z , L) and R are identified isometrically with respect to the valuations v r and v . r r Proof [5, Théorème II.2.2] and [5, Proposition II.3.1]. We now consider the multiplicative topological group Z .Let p if p odd q = 4 otherwise. We have the well-known isomorphism × × Z = (Z/q Z) × 1 + q Z , the second factor of which is topologically cyclic. Let γ be a topological generator of 1+q Z . Such a choice allows us to write any x ∈ 1 + q Z in the form x = γ for a unique s ∈ Z , p p giving us an isomorphism of topological groups 1 + pZ = Z p p γ → s. Thus Z is homeomorphic to p − 1 (resp. 2 when p = 2) copies of Z , and we can use the × × above theory of Z in this context, defining LA(Z , L), D (Z , L) in the obvious way; each p p space decomposes as a direct sum over their restrictions to each Z component and we take the infimum of the valuations on each summand. Definition 3 Define weight space to be the rigid analytic space W over Q representing × × L → Hom (Z , L ). cont Integrating characters gives a canonical identification × 0 D (Z , Q ) = H (W, O ), p W where O is the structure sheaf of W. This is isomorphism commutes with base change in the sense that for a finite extension L/Q we have × × 0 D (Z , L) = D (Z , Q )⊗ L = H (W , O ), p Q L W p p p L 123 R. Rockwood where W = W × Sp(L) and Sp(L) is the affinoid space associated to L. We identify L Q W(C ) with the set  B ,where B = B(0, 0) and the disjoint union runs over characters p ψ ψ ψ × × of Z which factor through (Z/q Z) , for details see [4, Remark B1.1]. We can thus identify × + D (Z , L) with functions on  B whichare describedbyelementsof R on each B . ψ ψ ψ Given a distribution μ ∈ D (Z , L) we write the corresponding rigid function on W as M (μ). On each B the global sections O (B ) are given (after choosing a coordinate X)by ψ W ψ precisely R . As these are quasi-Stein spaces, the topology on O (B ) is that of a Fréchet W ψ space induced by an increasing chain of affinoids Y ⊂ Y ⊂ ··· , 1 2 which we can choose to be the closed discs of radius u , whence the topology as global sections over a rigid space coincides with the topology on R given by the family of valuations v (see [17, 1C]). B(0,u ) r × × Definition 4 For r ∈ R we define a subspace D (Z , L) ⊂ D (Z , L) by ≥0 p p r × × + D (Z , L) ={μ ∈ D (Z , L) : M (μ)| ∈ R for all ψ }. p p ψ r These spaces decompose as a direct sum r × r D (Z , L) =⊕ D (Z , L) ψ p and we equip it with the valuation given by v (μ) := inf v r (μ ) D ψ D ψ where μ is the projection of μ to the ψ component. 2.2 Automorphic representations Fix n ≥ 1and set G = GL .Let Π be a cuspidal automorphic representation of G(A ). 2n Q Let T ⊂ G be the maximal diagonal torus and let 2g μ = (μ ,...,μ ) ∈ Z 1 2n be an integral weight. We say μ is dominant if μ ≥ ··· ≥ μ ,and we say μ is pure if there 1 2n is ω ∈ Z,the purity weight of μ, such that μ + μ = ω i 2n+1−i for all i = 1,..., n. Definition 5 We say that Π is cohomological with respect to a dominant integral weight μ if the (g , K )-cohomology ∞ ∞ H (g , K ,Π ⊗ V ) ∞ ∞ is non-vanishing for some q.Here g is the Lie algebra of G(R), K ⊂ G(R) is the identity component of the maximal open compact subgroup and V is the irreducible C-linear G- representation of highest weight μ. Purity of μ is a necessary condition for Π to be cohomological. 123 Plus/minus p-adic L-functions... Definition 6 We say that Π is the transfer of a globally generic cuspidal automorphic repre- sentation π of GSpin (A ) if 2n+1 Π π . for all primes  at which π is unramified. Remark 3 • For a given globally generic automorphic representation π of GSpin (A) 2n+1 the existence of such a transfer was proved by Asgari-Shahidi [1, Theorem 1.1]. • A necessary and sufficient condition for Π to be the transfer of a globally generic cuspidal automorphic representation of GSpin is that it admits a Shalika model which realises 2n+1 Π in a certain space of functions W : G(A ) → C,see [2, Sect. 2.6] for details. 2.3 p-stabilisations Let Π be a cuspidal automorphic representation of G(A ) which is cohomological with respect to a pure dominant integral weight μ and suppose that Π is the transfer of a globally generic cuspidal automorphic representation of GSpin (A ).Let B denote the upper 2n+1 triangular Borel subgroup of G. Given a prime p at which Π is unramified, define the Hodge-Tate weights of Π at p to be the integers h = μ + 2n − i , i = 1,..., 2n. (2) i i Remark 4 These weights coincide with the Hodge-Tate weights of the Galois representation associated to Π when the Hodge-Tate weight of the cyclotomic character is taken to be 1. Definition 7 Define a set Crit(Π ) ={ j ∈ Z : μ ≥ j ≥ μ }. n n+1 Remark 5 It is shownin[8, Proposition 6.1.1] that the half integers j + 1/2for j ∈ Crit(Π ) are precisely the critical points of the L-function L(s,Π) in the sense of Deligne [6, Definition 1.3]. Let p be a prime at which Π is unramified. There is an unramified character λ : T (Q ) → C p p such that Π is isomorphic to the normalised parabolic induction module 2n−1 G(Q ) Ind (|·| λ ). We define the Satake parameters at p to be the values α = λ (p), p i p,i B(Q ) where λ denotes the projection to the ith diagonal entry. After choosing an isomorphism p,i ¯ ∼ Q C,wereorder the α so that they are ordered with respect to decreasing p-adic valua- p = i tion and such that α α = λ for a fixed λ with p-adic valuation 2n − 1 + w.Thatwe i 2n+1−i can do this is a result of the transfer from GSpin ,see [1, (64)]. 2n+1 We define the Hodge polygon of Π to be the piecewise linear curve joining the following points in R : ⎧ ⎛ ⎞ ⎫ ⎨ ⎬ ⎝ ⎠ (0, 0), j , h : j = 1,..., 2n 2n+1−i ⎩ ⎭ i =1 123 R. Rockwood and define the Newton polygon on Π at p to be the piecewise linear curve joining the points ⎧ ⎛ ⎞ ⎫ ⎨ ⎬ ⎝ ⎠ (0, 0), j , v (α ) : j = 1,..., 2n . p 2n+1−i ⎩ ⎭ i =1 The following result is due in this form to Hida [9, Theorem 8.1]. Proposition 1 The Newton polygon lies on or above the Hodge polygon and the end points coincide. Definition 8 Let I = (i ,..., i ) ⊂ Z satisfy 1 ≤ i < ··· < i ≤ 2n,and set 1 n 1 n α := α ··· α . I i i 1 n We call α p-stabilisation data for Π. Let Q ⊂ GL be the parabolic subgroup given by the partition 2n = n +n. The following 2n conditions are translations of the conditions of the same name given in [2, Sect. 2.7]. Definition 9 Let I be as above. • We say that the product α is of Shalika type if I contains precisely one element of each pair {i , 2n + 1 − i } for i = 1,..., n,see [7, Definition 3.5(ii)]. • We say that α is Q-regular if it is of Shalika type and if for any other choice of J ⊂ Z satisfying the above properties a = a . This amounts to choosing a simple Hecke J I eigenvalue for the U -operator associated to Q actingonthe Q-parahoric invariants of Π ,see [7, Definition 3.5(i)] and [2, Sect. 2.7]. 2n • Set r = v (α ) − h . We say that α is non-critical slope if it satisfies I p I i I i =n+1 r < #Crit(Π ). • We say that α is minimal slope if r = #Crit(Π )/2. Remark 6 The conditions in Definition 9 are used to control a certain local twisted integral at p, attached to a choice of parahoric-invariant vector W in the Shalika model. In [7, Proposition 3.4], the authors show that this local zeta integral is an explicit multiple of W (1).In[7,Lem. 3.6], they use the Shalika-type and Q-regular conditions to exhibit an explicit vector W in the Shalika model attached to α with W (1) = 1, and hence deduce non-vanishing of the local zeta integral. We note that Π always admits p-stabilisations of Shalika type. The terminology is justified by [2, Remark 2.5], which explains that the refinements of Shalika type are exactly those that arise from refinements of GSpin . Finally, if the Satake parameter of Π is 2n+1 regular semisimple, then all stabilisations of Π are Q-regular. (α ) 1 I × × In [2] the authors construct a locally analytic distribution L ∈ D (Z , C ) with p p respect to a choice of non-critical slope Q-regular p-stabilisation data α . The distribution (α ) L is of order r andby[18, Lemma 2.10] is uniquely defined by the following interpolation p I × m property: Let θ : Z → Q be a finite-order character of conductor p , then for m ≥ 1we have c j x θ j (α ) x θ(x )L (x ) = ξ L(Π ⊗ θ, j + 1/2), j ∈ Crit(Π ), (3) ∞, j p I The authors actually construct p-adic L-functions for the wider class of non-critical p-stabilisations, but we only work with non-critical slope p-stabilisations in this paper. 123 Plus/minus p-adic L-functions... where c j is a constant depending only on x θ and the infinite factor ξ is the product ∞, j x θ (α ) of a choice of period and a zeta integral at infinity. We call such a L a ‘non-critical slope p-adic L-function’. 3 Plus/Minus p-adic L-functions We construct the titular plus/minus L-functions. We first note that the condition of non-critial slope imposes strong restrictions on the number of p-adic L-functions we can construct. (α ) Theorem 4 There are at most two choices of p-stabilisation α for which L is non-critical slope. Proof Without loss of generality we may assume that μ = 0, forcing w = μ .The end 2n 1 points of the Newton and Hodge polygons coinciding implies that v (λ) = h + h , i = 1,..., n. (†) p i 2n+1−i The ‘non-critical slope’ condition for I = (i ,..., i ) is equivalent to 1 n 2n v (α ) − h < h − h . p I i n n+1 i =n+1 We observe that any I that includes a a 2-tuple of integers the form (i , 2n + 1 − i ) is not non-critical slope. Indeed, we can find an explicit I containing some (i , 2n + 1 − i ) with minimal valuation, namely (n, n + 1, n + 3,..., 2n), amongst all I containing some (i , 2n + 1 − i ). For such an I we have v (α ) − (h + h ··· + h ) p I n+1 n+2 2n ≥ h + h + h + ··· + h − (h + ··· + h ) n n+1 n+3 2n n+1 2n = h − h n n+2 > h − h , n n+1 where the first inequality is a consequence of the Newton polygon lying above the Hodge polygon and (†), and the strict inequality is due to dominance. Thus any I containing a pair of integers (i , j ) with i < j and j ≤ 2n + 1 − i cannot be non-critical slope, since any such I has greater valuation than (n, n + 1, n + 3,..., 2n). This leaves us with two choices of potential non-critical slope n-tuples: I = (n + 1, n + 2,..., 2n), n+1 and I = (n, n + 2,..., 2n). In light of Theorem 4 it is clear that the only two choices of p-stabilisation data which can give a non-critical slope distribution are α = α α ...α ,β = α α ...α . n+1 n+2 2n n n+2 2n 123 R. Rockwood Definition 10 We say that Π satisfies the ‘Pollack condition’ if α + α = 0. n n+1 Corollary 1 For a non-critical slope p-stabilisation α we have the following: • The p-stabilisation α is of Shalika type. • If we assume the Pollack condition, then α is Q-regular. Proof The first part is immediate from Theorem 4. For the second claim recall that a p-stabilisation α is Q-regular if α = α I J for all I = J . Suppose α is a non-critical slope p-stabilisation. Then by Theorem 4 α = α α ··· α or α α ··· α , I n n+2 2n n+1 n+2 n+1 and by the Pollack condition these are clearly not equal since they are non-vanishing by non- critical slope. Finally, for a critical slope p-stabilisation α we must have v (α )>v (α ) J p J p I so α = α . J I 3.1 Pollack ±-L-functions Let Π be as in the previous section. As in the previous section, set α = α α ...α ,β = α α ...α , n+1 n+2 2n n n+2 2n 2n 2n and let r = v (α) − h = v (β) − h . The Pollack condition forces p i p i i =n+1 i =n+1 r ≥ #Crit(Π )/2 since 2n r = v (α) − h ≥ v (α ) − h p i p n+1 n+1 i =n+1 h + h n n+1 = − h n+1 h − h n n+1 = #Crit(Π )/2, where the first inequality comes from Newton-above-Hodge, and the lower bound given is tight, with the bound being achieved when the end point of the segment of the Newton polygon corresponding to α ··· α touches the Hodge polygon. This justifies the use of n+2 2n the term ‘minimal slope’ in Definition 9. We assume that r < #Crit(Π ) (α) (β) so that we can construct precisely two non-critical slope p-adic L-functions L , L ∈ p p r × D (Z , C ). 123 Plus/minus p-adic L-functions... Remark 7 Unlike in the case of GL ,for n > 1 the non-critical slope condition for α, β is not necessarily implied by the Pollack condition. Indeed, suppose one has a cuspidal automorphic representation Π of GL (A ) satisfying the Pollack condition at a prime p for 2n Q which v (α ) = v (α ) for 1 ≤ i , j ≤ 2n.The value r is then the same for any choice of p- p i p j 2n stabilisation, so there are either non-critical slope p-adic L-functions or there are none. But Theorem 4 says there can be at most two choices of non-critical slope p-stabilisation. Following Pollack, we define (α) (β) L ± L p p G = , so that (α) + − L = G + G (β) + − L = G − G . (β) We note that in the case of L , the interpolation formula is given by c j x θ j (β) m x θ(x )L (x ) = (−1) ξ L(Π ⊗ θ, j + 1/2), j ∈ Crit(Π ), ∞, j from which it follows that j + m x θ(x )G (x ) = 0, if the conductor of θ is p , m odd j − m x θ(x )G (x ) = 0, if the conductor of θ is p , m even. Equivalently (noting that characters of conductor m correspond to (m − 1)th roots of unity), if ζ is any p th root of unity and p is odd, + j M (G )(γ ζ − 1) = 0for m even − j M (G )(γ ζ − 1) = 0for m odd on each of the connected components of W(C ) (which we recall we are identifying with p − 1 copies of B(0, 0)). When p = 2 the sign flips and the above vanishing is equivalent to. + j M (G )(γ ζ m − 1) = 0for m odd − j M (G )(γ ζ m − 1) = 0for m even For any j ∈ Z, Pollack defines the following power series − j 1  (γ (1 + X )) 2m log (X ) := p, j p p m=1 − j 1  (γ (1 + X )) 2m−1 log (X ) := , p, j p p m=1 in Q [[X ]],where  is the p th cyclotomic polynomial. p m We are referring to the connected components as a rigid space as opposed to those of the topology on C induced by v . 123 R. Rockwood + − + Lemma 3 The power series log (X ) (resp. log (X )) is contained in R and vanishes p, j p, j 1/2 j m at precisely the points γ ζ m − 1 for every p th root of unity ζ m with m even (resp. odd). p p Proof The statements in the lemma are proved in [16, Lemma 4.1] and [16, Lemma 4.5]. We + + − reprove that log is contained in R using the setup of Sect. 2, the result for log being p, j 1/2 p, j similar. An analysis of the Newton copolygon of the Eisenstein polynomial  givesusthat 0if h ≤ n − 1 − j v ( ((γ (1 + X ))/p) = B(0,u ) h n−h−1 p − 1 otherwise, and thus h+1 + 2m−h−1 v (log ) = p − 1 B(0,u ) p, j m=1 −(h+1) p − 1 1 h = − − , 1 − p 2 2 whence h 1 1 inf v (log ) + = − < ∞, h ¯ B(0,u ) p, j 2 p − 1 2 + + so log (X ) ∈ R by Lemma 1. p, j 1/2 We define ± ± + log (X ) = log (X ) ∈ R . Π p, j Crit(Π )/2 j ∈#Crit(Π ) ± ± By abuse of notation we will write log (X ) for the element of O (W) givenbylog (X ) Π Π on each connected component of W. Lemma 4 We have #Crit(Π ) lim sup v (log ) + h < ∞. B(0,u ) Π Proof It follows from the proof of Lemma 3 and the multiplicativity of v that B(0,u ) −(h+1) #Crit(Π ) p − 1 1 v (log ) + h = #Crit(Π ) − . B(0,u ) Π h 2 2 1 − p 2 The right side converges as h →∞ so the lim sup is finite. A similar argument works for log . It follows from the above discussion and [12, 4.7] that for odd p the rigid function log (X ) ± ± divides M (G ) in O (W),and for p = 2 we have that log (X ) divides M (G ) in O (W). Define plus/minus p-adic L-functions L (X ) to be the elements of O (W) sat- W W isfying ± ± ± M (G ) = log (X ) · L (X ) 123 Plus/minus p-adic L-functions... for p odd, and ± ∓ ± M (G ) = log (X ) · L (X ) Π p ± −1 ± for p = 2. We write L for the distribution M (L (X )). p p Proposition 2 We have ± r −#Crit(Π )/2 × L ∈ D (Z , C ). p p Proof We note that #Crit(Π ) #Crit(Π ) ± ± lim inf −v (log ) − h =− lim sup v (log ) + h > −∞. ¯ ¯ B(0,u ) Π 2 B(0,u ) Π 2 h h By the additivity of v ([5, Proposition I.4.2]) we have B(0,u ) ± #Crit(Π ) ± ± #Crit(Π ) v (L ) + (r − )h = v (G ) + rh − v (log ) − h, ¯ ¯ ¯ B(0,u ) p B(0,u ) B(0,u ) Π h 2 h h 2 ± r × ± and so since G ∈ D (Z , C ) (and thus lim inf v (G ) + rh > −∞)wehave p ¯ p B(0,u ) ± #Crit(Π ) lim inf v (L ) + (r − )h > −∞ B(0,u ) p h 2 andsobyLemma 1 we are done. #Crit(Π ) In particular, in the minimal slope case r = we get two bounded distributions. Remark 8 • One might ask if there is an analogue of the plus/minus theory for p-adic L-functions for GL . Beyond the exact methods used in the present paper, there is 2n+1 an immediate stumbling block: in general, for n ≥ 3 odd even the usual theory of p- adic L-functions is very poorly developed. For non-ordinary Π on GL , the only 2n+1 constructions of p-adic L-functions are for n = 1and Π a symmetric square lift from GL ; in this case, a study of signed Iwasawa theory has been considered in [3]. • The proofs above show that if we relax the minimal slope hypothesis we still obtain a pair of plus/minus L-functions which are unfortunately not bounded. Since the subsets of weight space on which these functions interpolate L-values are disjoint it seems that there is no hope in attempting a similar construction for these functions. • In thecasethatwehaveaGL (A ) representation admitting p-stabilisations which 2n Q are critical but not non-critical slope Theorem 4 no longer holds. As a result, for each 2n such p-stabilisation we can construct a p-adic L-function, giving us at most p-adic L-functions. It’s possible that one could generalise the methods of this paper and utilise all of these p-adic L-functions to construct bounded functions analogous to L , but this is not something we have explored. 3.2 An example of a GL (A ) representation satisfying the Pollack condition 4 Q We give an example of a cuspidal automorphic representation of GL (A ) satsifying the 4 Q Pollack condition and having minimal slope using the theory of twisted Yoshida lifts (see [15, Sect. 6] for an overview). Let F = Q( 5) and let σ : F → R, i = 1, 2 be the embeddings of F into R.The prime 41 splits in F and we write M for one of its prime factors. Using Magma we see that there is a weight (4, 2) cuspidal Hilbert newform f over F of level N = M and of trivial 123 R. Rockwood character and with complex multiplication by the unique extension E /F such that E /Q is not Galois and in which M ramifies. Thus there is a Hecke character ψ over E with infinity 3 2 type z → ε (z) ε (z) ε ¯ (z),where ε , ε ¯ : E → C, i = 1, 2, are the pairs of conjugate 1 2 2 i i embeddings of E and such that the Hecke eigenvalue of f at a prime ℘ of F is given by ⎪ ψ(q ) + ψ(q ) if ℘ = q q in E 1 2 1 2 a = ψ(q) if ℘ = q in E 0 otherwise. Let L be the number field generated by the Hecke eigenvalues of f . Since the weight is not parallel we have F ⊂ L.Fix arationalprime p  N and let L be the completion of L at a prime v over p. Suppose now that p splits in F and write p = ℘ · ℘ , labelled such that 1 2 σ (℘ ) is below v. i i Let ψ : G → L be the v-adic character of G = Gal(F /F ) associated to ψ by Gal,v E v F class field theory, so that V := Ind ψ is the G -representation associated to f . f ,v Gal,v F This representation is crystalline at primes not dividing N. The Hodge-Tate weights of V f ,v are given by (0, 3) at σ and (1, 2) at σ . 1 2 The restriction of V to G splits as a direct sum of characters: f ,v E V | = ψ ⊕ ψ , f ,v G Gal,v E Gal,v where c ∈ Gal(E /F ) is the non-trivial element. If a prime ℘ of F above p splits in E then a decomposition group D ⊂ G at a prime over ℘ is contained in G and we have ℘ F E D (V | ) = D (ψ | ) ⊕ D (ψ | ) whence f is ordinary at ℘ as the cris f ,v D cris Gal,v D cris D ℘ ℘ Gal,v ℘ image of the functor D is weakly admissible. Taking v to be above σ (℘) and writing the cris 1 prime decomposition of ℘ in E as ℘ = q q we have 1 2 v (ψ (q )) = 0 v 1 v (ψ (q )) = 3 v 2 up to reordering of the q . Theorem 5 Suppose π is a cuspidal automorphic representation generated by a holomorphic Hilbert modular form of weight (k , k ) over a totally real field F. Suppose that: 1 2 • For 1 = θ ∈ Gal(F /Q) we have π ≈ π, • There is a Hecke character ε over Q such that the central character ω of π satisfies ω = ε ◦ Norm . π F /Q Then there is a unique globally generic cuspidal automorphic representation (π, ω ) of k +k |k −k | 1 2 1 2 GSp (A ) (a twisted Yoshida lift) of weight ( , − 2) with central character ε 2 2 satisfying max{k , k }− 1 1 2 L(Π , s) = L π, s + . Proof See [15, Theorem 6.1.1 and Proposition 6.1.4]. Let π be the cuspidal automorphic representation of GL (A ) generated by f .Since f has 2 E non-parallel weight we see that π ≈ π for non-trivial θ ∈ Gal(F /Q). Set Π = (π, 1).Then Π a cuspidal automorphic representation of GSp (A ) of weight (3, 3) with trivial central character. This weight lies in the cohomological range and thus Π is cohomological. The Hodge-Tate weights of Π are (0, 1, 2, 3). 123 Plus/minus p-adic L-functions... Recall that we have a rational prime p such that p splits in F: pO = ℘ ℘ . F 1 2 We assume further that ℘ is inert in E and ℘ splits, and we write the factorisation of ℘ as 2 1 1 ℘ = P P . 1 1 2 Primes satisfying the above conditions are not uncommon, for example, the primes 11 and 19 admit this splitting phenomena in the tower E /F /Q. We remark that a necessary condition for such a splitting is that E is non-Galois over Q. The local L-factor of Π at such a p is given by −1 −s −s −2s L (Π , s) = (1 − ψ(P )p )(1 − ψ(P )p )(1 − ψ(℘ )p ). p 1 2 2 Choosing a prime v of L lying above p such that v lies above σ (℘ ), we deduce that Π i i satisfies the Pollack condition and has two minimal slope p-stabilisations. By Corollary 1 these p-stabilisations are Q-regular and Shalika. There is an exceptional isomorphism GSp GSpin and so we are done by applying the functorial lift from GSpin to GL . 4 5 5 4 Non-vanishing of twists We use L to show non-vanishing of the complex L-function of Π at the central value, extending work of Dimitrov, Januszewski, Raghuram [7] to a non-ordinary setting. Proposition 3 In the case that Crit(Π ) ={w/2}, we have L = 0. + − Proof We consider L , the case of L being essentially identical and we further assume p p p is odd for brevity of notation (although the same argument works in this case). Note that a × × m+1 m character θ : Z → C of conductor p corresponds to a choice of primitive p th root p p of unity ζ in a disc W determined by the restriction ψ of θ to (Z/q Z) , which gives us the θ θ identification j + + j x θ(x )log (x ) = log (ψ ,γ ζ − 1), θ θ Π Π where on the left hand side we use the description of log as a distribution and on the right + + hand side log (ψ , −) is the restriction of log to the disc in W corresponding to ψ .We θ θ Π Π adopt the analogous notation for L . It follows from Lemma 3 that log (ψ ,γ ζ − 1) = 0 p θ θ if m is odd (resp. even for p = 2). Thus for characters θ of odd p-power conductor we have the interpolation property L(Π ⊗ θ, j + 1/2) L (ψ ,γ ζ − 1) ∼ , j ∈ Crit(Π ), p θ θ log (ψ ,γ ζ − 1) θ θ where ∼ is used here to mean ‘up to non-zero constant’. By Jacquet-Shalika [10,1.3]we have L(Π ⊗ θ, s) = 0for Re(s) ≥ w/2 + 1 123 R. Rockwood for finite order characters θ, and by applying the functional equation we get non-vanishing for Re(s) ≤ w/2. Since Crit(Π ) contains an integer k not equal to w/2, the above discussion gives us L(Π , k + 1/2) = 0, and thus L = 0. Remark 9 Proposition 3 actually proves the stronger result that the power series M (L )| p ψ is non-zero for each choice of ψ. We can turn this back on itself and use L to say something about nonvanishing of L(Π ⊗ ± 0 × θ, (ω + 1)/2) in the case when L ∈ D (Z , C ). p p Theorem 6 In the case that L are bounded distributions, w is even, and Crit(Π ) ={w/2}, we have L(Π ⊗ θ, (w + 1)/2) = 0 for all but finitely many characters θ of p-power conductor. Proof Assume p odd for brevity, again noting that the argument works fine in this case. For × ± ± any character ψ of (Z/pZ) we can write M (L )| = L (ψ, T ) ∈ O [[T ]] ⊗ L for B L O p ψ p L ± ± some finite extension L/Q . We note that M (L ) = 1 L where 1 denotes the p B B p ψ p ψ indicator function on B . This power series is non-zero by Proposition 3 and Remark 9,and ± ± so Weierstrass preparation tells us that each L (ψ, T ), and thus L , has only finitely many p p zeroes. Given any character θ of p-power conductor, we have w/2 ? x θ(x )L (x ) ∼ L(Π ⊗ θ, (w + 1)/2), where + if the conductor of θ is odd p -power ? = . − otherwise Thus, for all but finitely many θ,wehave L(Π ⊗ θ, (w + 1)/2) = 0. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. 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Journal

Annales mathématiques du Québec – Springer Journals

Published: Apr 1, 2023

Keywords: P-adic L-functions; Automorphic forms; Iwasawa theory; 11G40; 11F67; 11R23

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