Extremal p-adic L-functions

In this short note we give a reinterpretation of the classical construction of cyclotomic p-adic L-functions attached to modular cuspforms. We are able to provide a genuinely new construction under the unlikely hypothesis that the Hecke polynomial has a double root. Although the fulfillment of this hypothesis contradicts Tate's conjecture in this classical setting, we focus our attention on these extremal p-adic L-functions because that scenario should not be ruled out as long as the conjecture remains open. Moreover, there are examples of Hilbert modular forms where these extremal hypothesis is satisfied and our work will provide new explicit p-adic L-functions in the Hilbert case. We study the admissibility and the interpolation properties of these extremal p-adic L-functions.


Introduction
Let f ∈ S k+2 (Γ 1 (N ), ǫ) be a modular cuspform for Γ 1 (N ) with nebentypus ǫ and weight k + 2. A very important topic in modern Number Theory is the study of the L-function L(s, π) attached to the automorphic representation π of GL 2 (A) generated by f . Understanding this complex valued analytic function is the key point for some of the most important problems in mathematics such as the Birch and Swinnerton-Dyer conjecture.
Back in the middle of the seventies, Vishik [7] and Amice-Vélu [1] defined a padic measure µ f,p of Z × p associated with f , under the hypothesis that p does not divide N . The construction of this measure was the starting point for the theory of p-adic L-functions attached to modular cuspforms. The p-adic L-function L p (f, s) is a C p -valued analytic function related with the classical L-function L(s, π) by means of the so-called interpolation properties. The function L p (f, s) is defined by means of µ f,p as where exp and log are respectively the p-adic exponential and p-adic logarithm functions.
Mazur, Tate and Teitelbaum extended in [5] the definition of µ f,p to more general situations and provided a p-adic analogue of the Birch and Swinnerton-Dyer conjecture, with the classical L-function L(π, s) replaced by L p (f, s). It has been shown that L p (f, s) is directly related with the (p-adic, or eventually l-adic) cohomology of modular curves, and this makes such p-adic Birch and Swinnerton-Dyer conjectures become more tractable. In fact, the theory of p-adic L-functions has grown tremendously during the last years. Many results, whose complex counterparts are inaccessible with current techniques, have been proven in this p-adic setting.
In this short note we give first a reinterpretation of the construction of the p-adic measures µ f,p . Our construction exploits the theory of automorphic representations and, in that sense, it is similar to the construction provided in [6] but for weights greater that 2. This opens the door to possible generalizations of p-adic measures attached to automorphic representations of GL 2 (A F ) of any weight, for any number field F .
If we go back to the classical setting treated in this note, we are able to construct µ f,p in every possible situation except when the local automorphic representation π p attached to f is supercuspidal. We hope that our work clarifies why it is not expected to find good p-adic measures in such supercuspidal case. Nevertheless, although all cases where we construct µ f,p were not considered in [5], almost all of them can be obtained as twists of the p-adic measures described there. However, we obtain a genuinely new construction that we proceed to describe: Assume that f is an eigenvector for the Hecke operator T p with eigenvalue a p , in this situation Theorem 5.1 can be expressed as: N ), ǫ) be a cuspform, and assume that Hecke polynomial P (X) := X 2 − a p X + ǫ(p)p k+1 has a double root α. Then there exists a locally analytic p-adic measure µ ext f,p of Z × p such that, for any locally polynomial character χ = χ 0 (x)x m with m ≤ k, where L (s, π, χ 0 ) is the L-function twisted by χ 0 , and τ (χ 0 ) is the Gauss sum of Definition 2.3.
We call µ ext f,p the extremal p-adic measure. Coleman and Edixhoven showed in [4] that P (X) never has double roots if the weight is 2, namely, k = 0. Moreover, they showed that assuming Tate's conjecture the polynomial P (X) can never be a square for general weights k+2. Since we believe in Tate's conjecture, we expect this situation never occur, hence surely the hypothesis of the theorem is never fulfilled and µ ext f,p can never be constructed. Since Tate's conjecture is open, we believe that it is still interesting to study this phenomena. Notice that in this unlikely situation, two p-adic measures µ f,p and µ ext f,p coexist. In fact we can define an alternative p-adic L-function called the extremal p-adic L-function, that coexists with L p (f, s), satisfying interpolation property (1.1) with completely different Euler factors e ext p (π p , χ 0 ). One can think that maybe the existence of L ext p (f, s) could provide a contradiction that would imply that the Hecke polynomial never has multiple roots, thus providing unconditional proof of this fact studied in [4].
Although the extremal situation of P (x) being a square is discarded by Tate's conjecture in this classical setting, there are examples of Hilbert modular forms satisfying this hypothesis (see [3, §3.3.1]). Since our techniques can be extended to the Hilbert setting following the work of Spiess in [6], new constructions of concrete extremal p-adic L-functions could be provided in these cases.
Acknowledgements. The author would like to thank David Loeffler and Víctor Rotger for their comments and discussions throughout the development of this paper.
The author is supported in part by DGICYT Grant MTM2015-63829-P. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 682152).
1.1. Notation. For any ring R, we denote by V (k) R := Sym k (R 2 ) the R-module of homogeneous polynomials in two variables with coefficients in R, endowed with an action of GL 2 (R): We will denote by dx the Haar measure of Q p so that vol(Z p ) = 1. Similarly, we write d × x for the Haar measure of Q × p so that vol(Z × p ) = 1. By abuse of notation, will will also denote by d × x the corresponding Haar measure of the group of ideles A × .
For any local character χ :

Local integrals
2.1. Gauss sums. In this section ψ : Q p → C × will be a non-trivial additive character such that ker(ψ) = Z p .
In particular, To deduce the second part, notice that hence the result follows.
Hence the integral I : and the result follows.
We now define the Gauss sum: For any character χ of conductor n ≥ 0, 3. Classical cyclotomic p-adic L-function 3.1. Classical Modular symbols. Let f ∈ S k+2 (N, ǫ) be a modular cuspidal newform of weight (k + 2) level Γ 1 (N ) and nebentypus ǫ. By definition, we have for any P ∈ V (k). Hence, if we denote by ∆ 0 the group of degree zero divisors of P 1 (Q) with the natural action of GL 2 (Q), we obtain the Modular Symbol : Notice that Γ 1 (N )-equivariance follows from the fact that the above equality implies The following result is well known and classical:

3.2.
Classical p-adic distributions. Given f ∈ S k+2 (N, ǫ), we will assume that f is an eigenvector for the Hecke operator T p with eigenvalue a p . Let α be a non zero root of the Hecke polynomial X 2 − a p X + ǫ(p)p k+1 We will construct a distribution µ f,α of locally polynomial functions of Z × p of degree less that k attached to f (and α in case p ∤ N ). Since the open sets U (a, n) = a + p n Z p (a ∈ Z × p and n ∈ N) form a basis of Z × p , it is enough to define the image of P 1, x−a p n 1 U(a,n) (x), for any P ∈ V (k) with integer coefficients: It defines a distribution because µ ± f,α satisfies additivity, namely, since The following result shows that, under certain hypothesis, we can extend µ ± f,α to a locally analytic measure.
If we assume that there exists such a root α with ord p α < k + 1, then we define the (cyclotomic) p-adic L-function:

p-adic L-functions
In this section we provide a reinterpretation of the distributions µ ± f,αp . Let f ∈ S k+2 (Γ 1 (N ), ǫ) be a cuspidal newform as above and let p be any prime. Fix the embedding where L is certain finite extension of the coefficient field Q({a n } n ), and V is certain model over L of the local automorphic representation π p generated by f . Assume also that, for big enough n, where m is fixed, V i ∈ V do not depend neither s nor n, and c i (s, n) ∈ O L .
4.1. p-adic distributions. Let us consider the subgroup Again by strong approximation we have that GL 2 (A f ) = GL 2 (Q) +K 1 (N ). Thus, for any GL 2 (A f ) ∋ g = h g k g , where h g ∈ GL 2 (Q) + , k g ∈K 1 (N ) are well defined up to multiplication by Γ 1 (N ) = GL 2 (Q) + ∩K 1 (N ). Write K :=K 1 (N ) ∩ GL 2 (Z p ). By strong multiplicity one π K p is one dimensional. Therefore V K = Lw 0 and V = L[GL 2 (Q p )]w 0 . Notice that we have a natural morphism This implies that, for any h ∈ GL 2 (Q) For any z ∈ Q × p such that z = p n u where u ∈ Z × p , we can choose d ∈ Z such that d ≡ u −1 mod N Z p and d ≡ p n mod N Z ℓ , for ℓ = p. Let us choose A = ( a b c d ) ∈ Γ 0 (N ), and we have This implies that, if ε p is the central character of π p , Again let C k (Z × p , C p ) be the space of locally polynomial functions of Z × p of degree less that k. Recall the Z × p -equivariant isomorphism Fixing L ֒→ C p , we define the distributions µ ± f,p attached to f and δ:

Admissible Distributions.
We have just constructed a distribution µ ± f,p : C k (Z × p , C p ) −→ C p . This section is devoted to extend this distribution to a locally analytic measure µ ± f,p ∈ Hom C loc−an (Z × p , C p ), C p .
for some fixed A ∈ C p , and any g ∈ C k (Z × p , O Cp ) which is polynomical in a small enough U (a, n) ⊆ Z × p . We will denote previous relation by Proposition 4.5. If h < k + 1, a h-admissible the distribution µ can be extended to a locally analytic measure such that The limit converge because {a n } n is Cauchy, indeed by additivity we have that a n+1 − a n n → 0. It is clear by the definition that µ(P a,N m ) ∈ A · p −N h O Cp for all m, a and N . Moreover, it extends to a locally analytic measure by continuity which is determined by the image of locally polynomial functions of degree at most h.
Notice that, for all m ≤ k, Using property (4.6) and Remarks 4.2 and Remark 4.3, we compute that Notice where the subindex ǫ indicates that the action of Γ 1 (N p r )/Γ 0 (N p r ) is given by the character ǫ. By Manin's trick we have that and some fixed A ∈ L. We deduce the following result.
Theorem 4.6. Fix an embedding L ֒→ C p . We have that µ ± f,p is v p (γ)-admissible.

Interpolation properties.
Given the modular form f ∈ S k+2 (Γ 1 (N )), let us consider the automorphic form φ : GL 2 (Q)\GL 2 (A) → C, characterized by its restriction to GL 2 (R) + × GL 2 (A f ): Given ϕ f,p = ϕ + f,p and g ∈ GL 2 (Q p ), we compute This implies that, if we consider the automorphic representation π generated by φ, and the GL 2 (Q p )-equivariant morphism we have that Let H be the maximum subgroup of Z × p such that h | sH is constant, for all by means of (4.5). Hence we have ϕ f,p (δ(h))(0 − ∞)(P ) = be a locally polynomial character. This implies that χ(x) = χ 0 (x)x m , for some natural m ≤ k and some locally constant character χ 0 . This implies that χ = ı(χ 0 ⊗ Y m X k−m ). We deduce that whereχ 0 (y) := χ 0 (y p |y|y −1 ∞ ). Let ψ : A/Q → C × be a global additive character and we define the Whittaker model element

This element admits a expression W H
. Moreover by [2,Theorem 3.5.5], it provides the Fourier expansion We compute By definition of δ, when v = p the element W H v correspond to the new-vector, thus by [2, Proposition 3.5.3] We conclude using the results explained in [2, §3.5] where the Euler factor 4.4. The morphisms δ. In this section we will construct morphisms δ satisfying Assumption 4.1. The only case that will be left is the case when π p is supercuspidal, in this situation we will not be able to construct admissible p-adic distributions. Let π p be the local representation. Let W : π p → C be the Whittaker functional, and let us consider the Kirillov model K given by the embedding Recall that the Kirillov model lies in the space of locally constant functions φ : Q × p → C endowed with the action (4.10) 1 x 1 φ(y) = ψ(xy)φ(y), a 1 φ(y) = φ(ay).