aa r X i v : . [ m a t h . N T ] O c t A p -adic Stark conjecture in the rank one setting Joseph W. Ferrara
Abstract
We give a new definition of a p -adic L -function for a mixed signature character of a realquadratic field and for a nontrivial ray class character of an imaginary quadratic field. We thenstate a p -adic Stark conjecture for this p -adic L -function. We prove our conjecture in the casewhen p is split in the imaginary quadratic field by relating our construction to Katz’s p -adic L -function. We also provide numerical evidence for our conjecture in three examples. Contents p -adic L -function 8 p -adic L -function of an ordinary weight k ≥ p -adic L -functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 p -adic L -functions and p -adic Stark Conjecture 17 p -adic L -function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2 p -adic Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 F is imaginary quadratic and p splits in F p -adic L -function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.2 Definition of the period pair (Ω ∞ , Ω p ) . . . . . . . . . . . . . . . . . . . . . . . . . . 245.3 The CM Hida family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.4 Two-variable p -adic L -function of the CM family . . . . . . . . . . . . . . . . . . . . 275.5 Two-variable specialization of L p,Katz . . . . . . . . . . . . . . . . . . . . . . . . . . 285.6 Choice of periods and comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.7 Proof of the conjecture in this case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 F = Q ( √ K = Q ( p √ p = 5 . . . . . . . . . . . . . . . . . . . . . . . . . 346.2 F = Q ( √− K = Hilbert class field of F , p = 5 . . . . . . . . . . . . . . . . . . . 346.3 F = Q ( √− K = Hilbert class filed of F , p = 3 . . . . . . . . . . . . . . . . . . . 381 Introduction
Let F be a number field and let χ : G F −→ C × be a continuous one dimensional representation of the absolute Galois group of F . Let K be thefixed field of the kernel of χ . For the rest of this article, let p be an odd prime number, let Q be analgebraic closure of Q , and fix embeddings ι ∞ : Q ֒ → C and ι p : Q ֒ → C p .Via the Artin map, to χ we may associate the complex Hecke L -function, L ( χ, s ), defined bythe series L ( χ, s ) = X a ⊂ O F χ ( a )N a s for Re( s ) >
1. The function L ( χ, s ) has a meromorphic continuation to the whole complex plane.In the late 1970s, in a series of papers, Stark made precise conjectures concerning the leading termof the Taylor series expansion at s = 0 of L ( χ, s ) ([28], [29], [31]). Stark’s conjectures relate theleading term of L ( χ, s ) at s = 0 to the determinant of a matrix of linear combinations of logarithmsof units in K . His conjectures refine Dirichlet’s class number formula. Stark proved his conjectureswhen the field F is equal to Q or to an imaginary quadratic field. In general the conjectures areopen.Around the same time that Stark made his conjectures, p -adic L -functions were constructedinterpolating the critical values of complex Hecke L -functions for general number fields. This vastlygeneralized Kubota and Leopoldt’s work on the p -adic Riemann zeta function. When F is a totallyreal field and χ : G F → C × is a totally even character, Cassou-Nogues ([5]), and then Deligne andRibet ([10]) defined a p -adic meromorphic function L p ( χ, s ) : Z p −→ C p determined by the following interpolation property: for all n ∈ Z ≤ , L p ( χ, n ) = Y p | p (1 − χω n − ( p ) N p − n ) L ( χω n − , n ) (1)where ω is the Teichm¨uller character. Siegel and Klingen ([26]) showed that the values L ( χω n − , n )lie in the field obtained by adjoining the values of χω n − to Q . The equality (1) takes place in Q .Now let F be a CM field with maximal totally real subfield E . A prime p is called ordinary for F if every prime above p in E splits in F . For such primes p , Katz ([18],[19]) defined a p -adic L -function associated to any ray class character χ : G F → C × . Katz’s p -adic L -function interpolatesthe values of complex L -functions of algebraic Hecke characters with nonzero infinity type. Tospecify the interpolation property we specialize to the case that F is imaginary quadratic. Let p be a rational prime that is split in F . Let λ be a Hecke character of infinity type (1 , p -adic meromorphic function L p ( χ, t, s ) : Z p × Z p −→ C p determined by the following interpolation property: for all k, j ∈ Z such that 1 ≤ j ≤ k − L p ( χ, k, j )Ω k − p = E p ( χ, k, j ) L ( χλ k − , j )Ω k − ∞ . (2)Here E p ( χ, k, j ) is an explicit complex number and Ω p ∈ C × p , Ω ∞ ∈ C × are p -adic and complexperiods that make both sides of (2) algebraic.In these two cases, F totally real and F imaginary quadratic with p split, p -adic Stark con-jectures have been made for L p ( χ, s ) and L p ( χ, t, s ), and some progress has been made on theseconjectures. When F is totally real and χ is totally odd Gross ([17]) stated a conjecture for the2rder of vanishing of L p ( χω, s ) at s = 0 and the leading term of the Taylor series of L p ( χω, s ) at s = 0. Progress has been made on the order of vanishing, and recently the formula for the leadingterm was proved in [8] building off of earlier work in [7]. When F is totally real and χ is totallyeven there is a conjecture for the value L p ( χ,
1) known as the Serre-Solomon-Stark conjecture ([27],[32]). This conjecture is open except in the cases when F = Q , when the formula is due to Leopoldt,and when χ is trivial, where Colmez has proven a p -adic class number formula ([6]). When F isimaginary quadratic and p is split in F , Katz stated and proved a p -adic Stark conjecture for thevalue L p ( χ, , j ) known as Katz’s p -adic Kronecker’s 2nd limit formula ([18] and see Section 5.1).One of the original motivations for Stark’s conjectures is that when the order of vanishing of L ( χ, s ) at s = 0 is exactly one, then the conjectures shed light on Hilbert’s 12th problem aboutexplicit class field theory. More precisely, when the order vanishing is exactly one then Stark’sconjectures predict the existence of a unit u ∈ O × K such that the relation L ′ ( ψ,
0) = − e X σ ∈ Gal(
K/F ) ψ ( σ ) log | σ ( u ) | (3)holds for all characters of the Galois group Gal( K/F ) and such that K ( u /e ) is an abelian extensionof F . Here e is the number of roots of unity in K and the absolute value is a particular absolutevalue on K . When F is real quadratic, ord s =0 ( L ( χ, s )) = 1 if and only if χ is mixed signature. Inthis case, we choose the absolute value on K to correspond to one of the real places of K . Then byvarying ψ and exponentiating (3) one can solve for the unit u from the L -values L ′ ( ψ, F . In Section 2, wereview the rank one abelian Stark conjecture when F is a quadratic field.The goal of this article is to define a p -adic L -function and state a p -adic Stark conjecture in thesetting when F is a quadratic field and ord s =0 ( L ( χ, s )) = 1 (the rank one setting). This is the casewhen χ is any nontrivial character if F is imaginary quadratic, and when χ is a mixed signaturecharacter when F is real quadratic. When F is imaginary quadratic and p is split in F our p -adic L -function is related to Katz’s. In the cases when F is imaginary quadratic and p is inert, as wellas when F is real quadratic and χ is mixed signature, our p -adic L -function is new. One of themain issues with defining the p -adic L -function for χ when F is quadratic and ord s =0 ( L ( χ, s )) = 1is that the complex L -function L ( χ, s ) has no critical values. Therefore the p -adic L -function of χ will not interpolate any of the special values of L ( χ, s ). In order to define the p -adic L -functionin lieu of the fact that L ( χ, s ) has no critical values we p -adically deform χ into a family of p -adicrepresentations where complex L -functions in the family do have critical values to interpolate.We now explain in more detail our definition, conjectures, and results. Let ρ = Ind G Q G F : G Q −→ GL ( C )be the induction of χ from G F to G Q . Then the q -expansion f = X a ⊂ O F χ ( a ) q N a is the q -expansion of a weight one modular form and ρ is the representation associated to f . Themodular form f has character ε = det ρ and level N = | d F | N F/ Q m where d F is the discriminant of F and m is the conductor of χ . Let x − a p ( f ) x + ε ( p ) = ( x − α )( x − β )be the Hecke polynomial of f at p . Then α and β are roots of unity, so f has two (possibly equal)ordinary p -stabilizations. Let f α ( z ) = f ( z ) − βf ( pz ) be a p -stabilization of f . Under the assumptionthat α = β , Bella¨ıche and Dmitrov ([2]) have shown that the eigencurve is smooth at the pointcorresponding to f α . We will use Bella¨ıche and Dmitrov’s result, so we assume α = β and let V bea neighborhood of f α on the eigencurve such that the weight map is ´etale at all points of V except3erhaps f α . Let W be weight space. Using the constructions of [1] there exists a two-variable p -adicrigid analytic function L p ( f α , z, σ ) : V × W −→ C p such that for all classical points y ∈ V , all finite order characters ψ ∈ W ( C p ), and all integers j ,1 ≤ j ≤ k − k is the weight of y , L p ( f α , y, ψ − ( · ) h·i j − )Ω sgn ( ψ ) p,y = E p ( f α , y, ψ, j ) L ( g y , ψω j − , j )Ω sgn ( ψ ) ∞ ,y . (4)Here g y is the modular form corresponding to the point y ∈ V , L ( g y , ψω j − , j ) is the complex L -function of the modular form g y twisted by the Dirichlet character ψω j − , E p ( f α , y, ψ, j ) is anexplicit complex number, and Ω ±∞ ,y , Ω ± p,y are p -adic and complex periods respectively that makeboth sides of the equality algebraic. In Section 3, we give the background needed in order to define L p ( f α , z, σ ).Conceptually, it makes sense to define the p -adic L -function of χ as L p ( χ, α, s ) : Z p −→ C p L p ( χ, α, s ) = L p ( f α , x, h·i s − )where x ∈ V is the point corresponding to f α . The problem with this definition is that while thefunction L p ( f α , z, σ ) is determined by the above interpolation property, the triple of the function L p ( f α , z, σ ), the p -adic periods Ω ± p,y , and the complex periods Ω ±∞ ,y is not canonically defined. Thechoice of the function L p ( f α , z, σ ) may be changed by a p -adic analytic function on V for which wewould obtain a new function with new p -adic and complex periods satisfying the same interpolationformula. We would like to state a p -adic Stark conjecture for the function L p ( χ, α, s ), but becausethe function is not canonically defined it does not make sense to specify its value at any point witha precise conjecture.To define a function that does not depend on any choices, we fix two finite order Dirichletcharacters η, ψ ∈ W ( C p ) and define the p -adic L -function of χ with the auxiliary characters η and ψ as L p ( χ, α, ψω, ηω, s ) = L p ( f α , x, ψ − ω − ( · ) h·i s − ) L p ( f α , x, η − ω − ( · ) h·i s − ) . The function L p ( χ, α, ψω, ηω, s ) does not depend on the choices made to define L p ( f α , x, σ ). InSection 4, we make the following conjecture for L p ( χ, α, ψω, ηω, s ). Conjecture 1.1.
Let η, ψ ∈ W ( C p ) be of orders p m and p n respectively. Let M m and M n be thefixed fields of the kernels of the representations ρ ⊗ η and ρ ⊗ ψ respectively. Let k m and k n be thefields obtained by adjoining the values of χ , α , and ζ p m +1 and ζ p n +1 respectively to Q . Then thereexists units u ∗ χη,α ∈ k m ⊗ O × M m and u ∗ χψ,α ∈ k n ⊗ O × M n such that L p ( χ, α, ψω, ηω,
0) = (1 − βψ ( p )) (cid:16) − ψ − ( p ) αp (cid:17) τ ( ψ − ) p n +1 (1 − βη ( p )) (cid:16) − η − ( p ) αp (cid:17) τ ( η − ) p m +1 log p ( u ∗ χψ,α )log p ( u ∗ χη,α ) where τ ( ψ − ) and τ ( η − ) are the Gauss sums associated to ψ − and η − respectively. In Section 5, we prove our conjecture when F is imaginary quadratic and p is split in F bycomparing L p ( χ, α, ψω, ηω, s ) to Katz’s p -adic L -function (Theorem 5.13). We also show in Section5 that in this case, it is possible to choose the periods in (4) in such a way as to make L p ( χ, α, s )canonically defined. It is a goal of future research to explore whether or not this is possible in theother cases. 4t the outset of this project, we believed that the units u ∗ χη,α , u ∗ χψ,α would be related to theunits appearing in (3) for the characters χη and χψ (see [13] for the precise relation we expected).This is the case when F is imaginary quadratic and p is split (see Section 5.7). In the other casesit is not clear what the precise relation is or if there is a relation. In Section 6, we give evidencefor our conjecture exploring the relation between u ∗ χη,α and u ∗ χψ,α and the units that would appearin (3). Acknowledgments : I thank my adviser Samit Dasgupta for giving my the ideas that form thebasis of this material. This article is based off the work I did in my thesis ([13]). I greatly thankRalph Greenberg for sharing his work with Nike Vatsal ([15]) with me, and for reading and helpingwith an earlier version of this paper. His input was essential in finishing this project. I thank RobPollack and Rob Harron for letting me use their SAGE code for computing overconvergent modularsymbols. I also thank the anonymous referee whose comments helped improve the quality of thispaper.
In this section we state the rank one abelian Stark conjecture for quadratic fields, and introducenotation that will be used in later sections. Let F be a quadratic extension of Q and let K be anontrivial finite abelian extension of F . If F is real quadratic assume that one infinite place of F stays real in K and the other becomes complex.Let S be a finite set of places of F that contains the infinite places and the places that ramify in K . Assume that | S | ≥
2. Let S K denote the places of K above those in S . Let v denote an infiniteplace of K such that v ( K ) ⊂ R if F is real quadratic. We also let v denote the infinite place of F that is v | F , so v ∈ S . Let U v,S denote the subgroup of K × defined by U v,S = ( { u ∈ K × : | u | w ′ = 1 , ∀ w ′ such that w ′ | F = v | F } if | S | ≥ { u ∈ K × : | u | w ′ = | u | w ′′ , ∀ w ′ , w ′′ | v ′ and | u | w = 1 , ∀ w S K } if S = { v, v ′ } . Let e denote the number of roots of unity in K . Let L S ( χ, s ) be the complex L -function associatedto χ with the Euler factors at the primes in S removed. Conjecture 2.1. (Stark [31] at s = 0 ) There exists u ∈ U v,S such that for all characters χ ofGal ( K/F ) , L ′ S ( χ,
0) = − e X σ ∈ Gal ( K/F ) χ ( σ ) log | σ ( u ) | v . Remark .
1. Stark conjectured the additional conclusion that K ( u /e ) is an abelian extensionof F . For our purposes we will not be considering this part of the conjecture.2. Stark proved the above conjecture when F is imaginary quadratic ([31]). The conjecture isopen when F is real quadratic.3. If | S | ≥
3, then the element u ∈ U v,S has its absolute value specified at every infinite place of K , so u if it exists is determined up to multiplication by a root of unity.4. In the real quadratic case, we can always take S to be the infinite places of F union theplaces of F that ramify in K . In this case, the conjectural u ∈ U v,S is an actual unit in O K .Similarly in the imaginary quadratic case if at least two primes of F ramify in K and we take S to be the infinite place of F union the places of F that ramify in K , then the Stark unit u ∈ U v,S is a unit in O K . Definition 2.3.
Let
K/F , S and v be as above. An element in U v,S satisfying the above conjectureis called a Stark unit for
K/F and is denoted u K . If | S | ≥
3, then u K is determined up to5ultiplication by a root of unity. When F is imaginary quadratic the units u K will be specified inSection 2.1.Now fix a character χ of Gal( K/F ). We state the rank one abelian Stark conjecture for the one L -function L S ( χ, s ).We keep the setting and notation as above for K/F , S , and v . Let χ be a character of Gal( K/F )such that ord s =0 ( L S ( χ, s )) = 1, and let k be the field obtained by adjoining the values of χ to Q .Extend log | · | v from U v,S to k ⊗ Z U v,S by k -linearity. Let( k ⊗ Z U v,S ) χ − = { u ∈ k ⊗ Z U v,S : σ ( u ) = χ − ( σ ) u, ∀ σ ∈ Gal(
K/F ) } be the χ − isotypic component of k ⊗ Z U v,S where Gal( K/F ) acts via its action on U v,S . Conjecture 2.4. (Rational Stark for χ at s = 0 ). There exists an element u χ ∈ ( k ⊗ Z U v,S ) χ − such that L ′ S ( χ,
0) = log | u χ | v . Remark .
1. As it happens with Conjecture 2.1, Conjecture 2.4 is open when F is realquadratic and χ is mixed signature.2. Since we are assuming ord s =0 ( L S ( χ, s )) = 1, the k -dimension of ( k ⊗ Z U v,S ) χ − is one.3. Conjecture 2.1 implies Conjecture 2.4 by taking u χ = − e X σ ∈ Gal(
K/F ) χ ( σ ) ⊗ σ ( u ) ∈ ( k ⊗ Z U v,S ) χ − where u ∈ U v,S is the unit satisfying Conjecture 2.1. In this section we define the Stark units that exist in the imaginary quadratic case of the rank oneabelian Stark conjecture. These units will be used in later sections.Let L = Z ω + Z ω ⊂ C be a lattice in C with ordered basis so that τ = ω /ω is in the upperhalf plane. Define the sigma and delta functions of a complex number z and lattice L to be σ ( z, L ) = z Y ω ∈ Lω =0 (cid:16) − zω (cid:17) e zω + ( zω ) ∆( L ) = (cid:18) πiω (cid:19) e πiτ ∞ Y n =1 (1 − e πinτ ) . Let A ( L ) = ω ω − ω ω πi so A ( L ) the area of C /L divided by π . Further let η ( L ) = ω X n ∈ Z X m ∈ Z ,m =0 mω + nω ) and η ( L ) = ω X m ∈ Z X n ∈ Z ,n =0 mω + nω ) and define η ( z, L ) = ω η − ω η πiA ( L ) z + ω η − ω η πiA ( L ) z. θ ( z, L ) = ∆( L ) exp( − η ( z, L ) z ) σ ( z, L ) . We now define Robert’s units associated to an integral ideal of an imaginary quadratic field([23]). For the rest of this section we fix the following notation. Let F be any imaginary quadraticfield, f a non-trivial integral ideal of F , F ( f ) the ray class field of F of conductor f , G f = Gal( F ( f ) /F ), f the least positive integer in f ∩ Z , and w f the number of roots of unity in F congruent to 1 mod f .For a fractional ideal a coprime to f , let σ a ∈ G f be the image of a under the Artin map. Let S bethe set consisting of the infinite place of F and the places dividing f , and let v be the infinite placeof F ( f ) induced by ι ∞ . Definition 2.6.
Define for σ ∈ G f , the Robert unit associated to σ by E ( σ ) = θ (1 , fc − ) f where σ c = σ . Proposition 2.7. ([14]) For all σ ∈ G f ,(i) E ( σ ) ∈ F ( f ) .(ii) For all σ ′ ∈ G f , σ ′ ( E ( σ )) = E ( σ ′ σ ) .(iii) If f is divisible by two distinct primes then E ( σ ) is a unit in F ( f ) . If f = q n for a prime q of F , then E ( σ ) is a q -unit. Theorem 2.8. (Kronecker’s second limit formula) For all characters χ of G f , L ′ S ( χ,
0) = − f w f X σ ∈ G f log | E ( σ ) | v . When Stark stated his conjectures, he recast this theorem using the following lemma.
Lemma 2.9. (Lemma 9 on page 225 of [31]) Let K ⊂ F ( f ) be a subfield of F ( f ) that is a nontrivialextension of F . Let J ⊂ G f be the subgroup such that G f /J = Gal ( K/F ) , and define for σJ ∈ G f /JE ( σJ ) = Y σ ′ ∈ σJ E ( σ ′ ) = N F ( f ) /K ( E ( σ )) . Let e be the number of roots of unity in K . Then E ( σJ ) e is a f w f power in K . Definition 2.10.
Let K ⊂ F ( f ) be a nontrivial extension of F such that Gal( K/F ) = G f /J . Let e be the number of roots of unity in K . Define the Stark unit of the extension
K/F , denoted u K to be an element of K such that u fw f K = E ( J ) e where E ( J ) = Q σ ∈ J E ( σ ). Such an element u K exists by the previous lemma and is unique up tomultiplication by a root of unity in K . Theorem 2.11. ([31] Stark’s Conjecture when F is imaginary quadratic) Keeping the notation asin the previous definition, for all characters χ of Gal ( K/F ) , L ′ S ( χ,
0) = − e X σ ∈ Gal ( K/F ) χ ( σ ) log | σ ( u K ) | v and K ( u /eK ) is an abelian extension of F . Background for definition of the p -adic L -function In this section, we set some notation and conventions that will be fixed throughout for modularforms and modular symbols. We also state some relevant definitions for later reference.Fix a positive integer N such that p ∤ N and let Γ be either Γ ( N ) or Γ ( N ) ∩ Γ ( p ). Our Heckeactions are defined via the double coset algebra of Γ in GL ( Q ). Let T ℓ denote the Hecke operatorat ℓ for ℓ ∤ N p . If Γ = Γ ( N ), let T p denote the Hecke operator at p , while if Γ = Γ ( N ) ∩ Γ ( p ), let U p denote the Hecke operator at p . Let ι denote the operator for the double coset correspondingto (cid:18) − (cid:19) . For a ∈ ( Z /N Z ) × , let [ a ] denote the diamond operators. Define the Hecke algebra tobe the algebra H = ( Z [ T ℓ , ℓ ∤ N p, U p , [ a ] , a ∈ ( Z /N Z ) × ] if Γ = Γ ( N ) ∩ Γ ( p ) Z [ T ℓ , ℓ ∤ N, [ a ] , a ∈ ( Z /N Z ) × ] if Γ = Γ ( N ) . If Σ is a subsemigroup of GL ( Q ) containing the matrices needed to define H , then we also consider H as a subalgebra of the double coset algebra of Γ in Σ.For k ≥
1, let S k (Γ , Q ) denote the space of holomorphic weight k and level Γ cusp forms withalgebraic q -expansions, and let S k ( N, ε, Q ) ⊂ S k (Γ ( N ) , Q ) be the space of holomorphic cuspformsof level N and nebentypus ε with algebraic q -expansions. Let S k (Γ , C p ) = S k (Γ , Q ) ⊗ Q C p and S k (Γ , C ) = S k (Γ , Q ) ⊗ Q C and similarly let S k ( N, ε, C p ) = S k ( N, ε, Q ) ⊗ Q C p and S k ( N, ε, C ) = S k ( N, ε, Q ) ⊗ Q C . Let F be the set of holomorphic functions f on the upper half plane such that for all c ∈ P ( Q ),lim z → c | f ( z ) | = 0, where to make sense of the limit, we view P ( Q ) and the upper half planeas subsets of P ( C ). For k ≥
1, we define the following weight- k action of GL +2 ( Q ) on F : for γ = (cid:18) a bc d (cid:19) ∈ GL +2 ( Q ), f ∈ F , f | γ,k ( z ) = ( cz + d ) k f (cid:18) az + bcz + d (cid:19) . The space S k (Γ , C ), of holomorphic cusp forms of weight k and level Γ is the set of invariants of Γwith respect to this weight- k action. Let Σ + = GL +2 ( Q ) ∩ M ( Z ) ⊂ GL ( Q ). The action of Σ + on F induces an action of H on S k (Γ , C ) which leaves the space S k (Γ , Q ) invariant, defining an actionof H on S k (Γ , Q ). We extend this action to S k (Γ , C p ) by linearity.For the rest of this article, we adopt the notation that Γ = Γ ( N ) and Γ = Γ ( N ) ∩ Γ ( p ). Definition 3.1. A Hecke eigenform (or just eigenform ) of level N and character ε is an element f ∈ S k ( N, ε, C p ) which is an eigenvector for all the elements of H . A normalized eigenform is aHecke eigenform f ∈ S k ( N, ε, C p ) such that the leading term of the q -expansion of f is 1. If f is anormalized eigenform, then f ∈ S k ( N, ε, Q ) and so we may also view f as an element of S k ( N, ε, C ).If f ∈ S k ( N, ε, Q ) is a normalized eigenform that is new at level N , we call f a newform . Definition 3.2.
Let f = ∞ X n =1 a n q n ∈ S k ( N, ε, Q ) be a newform. Then the Hecke polynomial of f at p is the polynomial x − a p x + ε ( p ) p k − . Let α and β be the roots of this polynomial. Definethe p -stabilizations of f to be f α ( z ) := f ( z ) − βf ( pz ) and f β ( z ) := f ( z ) − αf ( pz ).8he p -stabilizations f α , f β are elements of S k (Γ , Q ), and are eigenvectors for the action of H .The T ℓ eigenvalues of f α (respectively f β ) are the same as for f when ℓ = p , and the U p -eigenvalueof f α (respectively f β ) is α (respectively β ). Definition 3.3.
Let S ordk ( N, ε, C p ) (respectively S ordk (Γ , C p )) denote the maximal invariant sub-space of S k ( N, ε, C p ) (respectively S k (Γ , C p )) with respect to the action of T p (respectively U p )such that the characteristic polynomial of T p (respectively U p ) restricted to this subspace has rootswhich are p -adic units. We call the subspace S ordk ( N, ε, C p ) (respectively S ordk (Γ , C p )) the ordi-nary subspace of S k ( N, ε, C p ) (respectively S k (Γ , C p )). A cuspform f ∈ S k ( N, ε, C p ) (respec-tively S k (Γ , C p )) is called p -ordinary if f is an element of the subspace S ordk ( N, ε, C p ) (respectively S ordk (Γ , C p )).We remark that if f ∈ S ordk ( N, ε, C p ) is a newform and k ≥
2, then there is a unique p -ordinary p -stabilization of f , while if f ∈ S ( N, ε, C p ) is a weight one newform, then there are two (possiblyequal) p -ordinary p -stabilizations of f .We now introduce modular symbols. Let △ = Div ( P ( Q )) be the set of degree zero divisors on P ( Q ) and view △ as a GL ( Q )-module via the action of linear fractional transformations. Let V be a right Γ module. We define a right action of Γ on Hom( △ , V ) via the rule for ϕ ∈ Hom( △ , V ), γ ∈ Γ, and D ∈ △ : ( ϕ | γ )( D ) = ϕ ( γD ) | γ. Definition 3.4.
The set of V -valued modular symbols on Γ, denoted Symb Γ ( V ), is the set ofall ϕ ∈ Hom( △ , V ) that are invariant under the action of Γ.In the cases we consider, V has an action of a submonoid of GL ( Q ) which defines an actionof H on Symb Γ ( V ) through a double coset algebra. When 2 acts invertibly on V and ι acts onSymb Γ ( V ), we get a decomposition of Symb Γ ( V ) into the direct sum of the 1 and − ι , denoted Symb +Γ ( V ) , Symb − Γ ( V ) ⊂ Symb Γ ( V ). If ϕ ∈ Symb Γ ( V ), then we write ϕ ± for theprojection of ϕ onto Symb ± Γ ( V ). In this section we introduce overconvergent modular symbols following the notation and conventionsof [1] and [21].Let W = Hom cont ( Z × p , G m ) denote weight space as a Q p -rigid analytic space, and let R denotethe ring of Q p -rigid analytic functions on W . Let ω ∈ W be the Teichm¨uller character. For m with0 ≤ m ≤ p −
2, let W m ⊂ W denote the subset of W consisting of characters whose restriction to µ p − ⊂ Z × p is equal to ω m .We give an explicit description of certain admissible open subsets of the Q p -points of W m . Forany κ ∈ W m ( Q p ) and any r ≥
1, let W ( κ, /p r ) denote the closed disk of radius 1 /p r in W m around κ . Then W ( κ, /p r )( C p ) = { κ ′ ∈ W m ( C p ) : | κ ′ ( γ ) − κ ( γ ) | ≤ /p r } , and W ( κ, /p r ) is an an admissible open subset of W m . The ring of Q p -rigid analytic functions on W ( κ, /p r ) is the Q p -algebra R = ( ∞ X n =0 a n ( w − ( κ ( γ ) − n ∈ Q p [[ w − ( κ ( γ ) − | a n p rn | → n → ∞ ) and W ( κ, /p r ) = Sp R ⊂ W . We remark that R is isomorphic to the Tate algebra Q p h T i = ( ∞ X n =0 a n T n ∈ Q p [[ T ]] : | a n | → n → ∞ )
9y setting T = ( x − ( κ ( γ ) − /p r . The sets W ( κ, /p r ) form a basis of admissible open neighborhoodsof κ in W m .For each r ∈ | C × p | = p Q , let B [ Z p , r ] = { z ∈ C p : ∃ a ∈ Z p , | z − a | ≤ r } .B [ Z p , r ] is the set of C p -points of the Q p -rigid analytic space which is the union of the closed unitballs of radius r around each point in Z p . Let A [ r ] be the Q p -algebra of rigid analytic functions on B [ Z p , r ]. The sup norm on A [ r ] makes A [ r ] a Q p -Banach space. Explicitly the norm is given for f ∈ A [ r ] by k f k r = sup z ∈ B [ Z p ,r ] | f ( z ) | . Let D [ r ] = Hom Q p ( A [ r ] , Q p ) be the continuous Q p -dual of A [ r ]. The space D [ r ] is a Q p -Banachspace with norm given by k µ k r = sup f ∈ A [ r ] ,f =0 | µ ( f ) |k f k r for µ ∈ D [ r ]. For r > r , restriction of functions gives a map A [ r ] → A [ r ]. This map is injective,has dense image, and is compact. The dual map D [ r ] → D [ r ] is injective and compact. For anyreal number r ≥ A † [ r ] = lim −→ s>r A [ s ] and D † [ r ] = lim ←− s>r D [ s ] . We give A † [ r ] the inductive limit topology and D † [ r ] the projective limit topology. For the remainderof this article, we write A = A † [0] and D = D † [0]. We remark that D is the continuous Q p -lineardual to A , and that A may be identified with the set of locally analytic functions on Z p and D theset of locally analytic distributions.Given µ ∈ D , via integration µ determines a Q p -rigid analytic function on W , which we call the p -adic Mellin transform of µ . We denote the map corresponding to the p -adic Mellin transform L : D −→ R . For µ ∈ D and χ ∈ W ( C p ), we use the integration symbol for µ evaluated at χ : L ( µ )( χ ) = Z Z × p χ ( z ) dµ ( z ) . We now define overconvergent modular symbols. LetΣ ( p ) = (cid:26)(cid:18) a bc d (cid:19) ∈ M ( Z p ) : p ∤ a, p | c and ad − bc = 0 (cid:27) . For any integer k ∈ Z , we define a weight k action of Σ ( p ) on A [ r ] for r < p as follows. For γ = (cid:18) a bc d (cid:19) ∈ Σ ( p ), f ∈ A [ r ], let( γ · k f )( z ) = ( a + cz ) k f (cid:18) b + dza + cz (cid:19) . This induces an action of Σ ( p ) on D [ r ] on the right via( µ | k γ )( f ) = µ ( γ · k f )for µ ∈ D [ r ]. These actions induce actions of Σ ( p ) on A and D . When we consider A and D withtheir weight k actions, we write k in the subscript, A k , D k . The spaces of modular symbols ofinterest are Symb Γ ( D k ). These space are Hecke modules via the action of Σ ( p ) on D k . Definition 3.5.
Let k ∈ Z . The space of overconvergent modular symbols of weight k isdefined to be Symb Γ ( D k ). 10 efinition 3.6. Let ϕ ∈ Symb Γ ( D k ) be an overconvergent modular symbol of weight k . We definethe p -adic L -function of ϕ by composing the following two maps: first evaluation at { } − {∞} , andthen the map L from before. The composition is called the Mellin transform of ϕ and denotedby Λ k : Λ k : Symb Γ ( D k ) → R . For ϕ ∈ Symb Γ ( D k ) and χ ∈ W ( C p ),Λ k ( ϕ )( χ ) = Z Z × p χ ( z ) d ( ϕ ( { } − {∞} ))( z ) . By definition, Λ k is a Q p -linear map. p -adic L -function of an ordinary weight k ≥ modular form In this section, we review how to use classical and overconvergent modular symbols to define the p -adic L -function of a weight k + 2 ≥ p -ordinary newform.Let R be a Q -algebra, and for k ∈ Z ≥ , let V k ( R ) = Sym k ( R ) be the R -module of homogeneouspolynomials of degree k in two variables X and Y with coefficients in R . Define an action of GL ( R )on V k ( R ) as follows: for γ = (cid:18) a bc d (cid:19) ∈ GL ( R ) and P ∈ V k ( R ), define( P | γ )( X, Y ) = P (( X, Y ) γ ∗ ) = P ( dX − cY, − bX + aY )where γ ∗ = (cid:18) d − b − c a (cid:19) . Since R is a Q -algebra, the space of modular symbols, Symb Γ ( V k ( R )),obtains an action of GL ( Q ), which determines a Hecke action of H .Let g ∈ S k +2 (Γ , C ). Define the standard modular symbol associated to g , denoted ψ g , to bethe function ψ g : △ −→ V k ( C ) ψ g ( { b } − { a } ) = 2 πi Z ba g ( z )( zX + Y ) k dz. It follows that ψ g ∈ Symb Γ ( V k ( C )) and the map S k +2 (Γ , C ) −→ Symb Γ ( V k ( C )) g ψ g is Hecke equivariant.Let f ∈ S ordk +2 ( N, ε, C ) be a p -ordinary newform, and let f α ∈ S ordk +2 (Γ , C ) be its p -ordinary p -stabilization. Shimura ([24]) showed that there exist complex periods Ω ± f α ∈ C × such that ψ ± f α / Ω ± f α ∈ Symb ± Γ ( V k ( Q )), and that the Hecke eigenspaces in Symb ± Γ ( V k ( Q )) with the sameeigenvalues as f α are one-dimensional over Q .The algebraicity result of Shimura allows one to view the modular symbol associated to f α p -adically in order to define the p -adic L -function of f α . Let ϕ ± f α = ψ ± f α / Ω ± f α ∈ Symb Γ ( V k ( Q )) forsome choice of complex periods Ω ± f α . (Each choice of period is determined up to a scalar in Q × .)Via ι p , view ϕ ± f α as an element of Symb Γ ( V k ( C p )).Mazur-Tate and Teitelbaum ([20]) proved that the function µ ± f α defined by the rule µ ± f α ( a + p m Z p ) = α − m ϕ ± f α (cid:18)(cid:26) ap m (cid:27) − {∞} (cid:19) | X =0 ,Y =1
11s a C p valued measure on Z × p . Given a finite order character ψ ∈ W ( C p ), we then define the p -adic L -function of f α twisted by ψ to be the analytic function of s ∈ Z p given by the formula L p ( f α , ψ, s ) = Z Z × p ψ − ( t ) h t i s − dµ sgn ( ψ ) f α ( t ) . We record here the interpolation property of L p ( f α , ψ, s ) for future reference. Theorem 3.7. ([20]) Let f α be the ordinary p -stabilization of a p -ordinary newform of level N andweight k + 2 ≥ . Let ψ ∈ W ( C p ) be a finite order character of conductor p m . Then L p ( f α , ψ, s ) is a p -adic analytic function on Z p with the interpolation property that for all integers j with < j < k + 2 , L p ( f α , ψ, j ) = 1 α m (cid:18) − ψ − ω − j ( p ) αp − j (cid:19) p m ( j − ( j − τ ( ψ − ω − j )(2 πi ) j − L ( f α , ψω j − , j )Ω sgn ( ψ ) f α . Here τ ( ψ − ω − j ) is the Gauss sum associated to ψ − ω − j .Remark . If f is a non-ordinary newform with Hecke polynomial x − a p ( f ) x + ε ( p ) p k +1 = ( x − α )( x − β )then one may define the p -adic L -function of either p -stabilization f α or f β of f in the same wayas above but a little more care is needed because the distribution µ f α (or µ f β ) is not a measure.For the critical p -stabilization f β when f is p -ordinary, even more care is needed. See [22] and [1]for more information about these cases.When f is a weight one modular form there is no modular symbol associated to f and so theabove constructions do not work. It is for this reason that we consider overconvergent modularsymbols of arbitrary integer weight k ∈ Z . We now explain the connection between overconvergentmodular symbols of weight k ∈ Z ≥ and the modular symbols just considered.Let k ∈ Z ≥ and define the map ρ k : D k −→ V k ( Q p ) ρ k ( µ ) = Z Z p ( Y − zX ) k dµ ( z ) . The integration in the definition of ρ k takes place coefficient by coefficient. The map ρ k is Σ ( p )-equivarient, so induces a Hecke equivariant map ρ ∗ k : Symb Γ ( D k ) −→ Symb Γ ( V k ( Q p )) . Let Symb Γ ( D k )
Fix a weight k ∈ Z ∩ W ( Q p ). Let Symb ± Γ ( D k ) o ⊂ Symb ± Γ ( D k ) (respectivelySymb ± Γ ( D ( R )) o ⊂ Symb ± Γ ( D ( R ))) be the subspace where U p acts with slope bounded by 0 in thesense of [1] Section 3.2.4. Let T ± k (respectively T ± W ) be the Q p -subalgebra of End Q p (Symb ± Γ ( D k ) o )(respectively the R -subalgebra of End R (Symb ± Γ ( D ( R )) o )) generated by the image of H . We callSymb ± Γ ( D k ) o (respectively Symb ± Γ ( D ( R )) o ) the ordinary subspace of Symb ± Γ ( D k ) (respectivelySymb ± Γ ( D ( R ))).We have ([1] Section 3.2.4) that Symb ± Γ ( D ( R )) o is a finite projective R -module. Then since T ± W is a finite R -algebra, T ± W is an affinoid algebra. Furthermore, T ± W is torsion-free as an R -moduleand since R is a principal ideal domain, T ± W is flat. Theorem 3.12. (Bella¨ıche’s specialization theorem (Corollary 3.12 in [1])) Let k ∈ Z ∩ W ( Q p ) .The specialization map restricted to the ordinary subspaces sp k : Symb ± Γ ( D ( R )) o −→ Symb ± Γ ( D k ) o (5) is surjective. Since sp k is an H -equivariant surjective map, it induces an H -equivariant map sp k : T ± W → T ± k ,which we use in the following definition. Definition 3.13.
Let x : T ± k → C p be a Q p -algebra homomorphism. The homomorphism x corresponds to a system of H -eigenvalues appearing in Symb ± Γ ( D k ) o . Let Symb ± Γ ( D k ) ( x ) denotethe corresponding generalized eigenspace and let Symb ± Γ ( D k )[ x ] denote the eigenspace.1. Let ( T ± k ) ( x ) be the localization of T ± k ⊗ Q p C p at the kernel of x . We have thatSymb ± Γ ( D k ) ( x ) = Symb ± Γ ( D k ) o ⊗ T ± k ( T ± k ) ( x ) .
2. Through the specialization map, x induces a Q p -algebra homomorphism which we also denoteby x : x = x ◦ sp k : T ± W −→ C p . Let ( T ± W ) ( x ) be the rigid analytic localization of T ± W ⊗ Q p C p at the kernel of x ◦ sp k , and letSymb ± Γ ( D ( R )) ( x ) = Symb ± Γ ( D ( R )) o ⊗ T ± W ( T ± W ) ( x ) . Let R ( k ) be the rigid analytic localization of R ⊗ Q p C p at the kernel of ev k . We can then localizethe specialization map sp k to get a map sp k : ( T ± W ) ( x ) ⊗ R ( k ) ,k C p −→ ( T ± k ) ( x ) . In ([1]), Bella¨ıche following Stevens uses these spaces of families of overconvergent modularsymbols to construct the eigencurve. Let C ± W = Sp T ± W . Then C ± W is the ordinary locus of theeigencurve above the open set W of weight space. The weight map κ ± : C ± W −→ W
14s the map of rigid analytic spaces induced by the Q p -algebra homomorphism R → T ± W . Since T ± W is a finite, flat R -module, the map κ ± is finite and flat. Given a point x ∈ C ± W ( C p ), we define theweight of x to be κ ± ( x ) ∈ W ( C p ). For any κ ∈ W ( C p ) we may consider the evaluation at κ map ev κ : R −→ C p ev κ ( F ) = F ( κ ( γ ) − . Define R ( κ ) to be the rigid analytic localization of R at the kernel of ev κ . Theorem 3.14. ([1]) Let U = W ( k ′ , /p d ) for some k ′ , d ∈ Z , d ≥ , and let x ∈ C ± U ( C p ) be asmooth point of weight k ′ . Then there exists a neighborhood, W = W ( k ′ , /p r ) of k ′ with r ≥ d ,such that the following hold. Let R be the ring of rigid analytic functions on W . Let T be the directfactor of T ± W corresponding to the connected component of C ± W that x lies in. (Note that T may bedefined over a finite extension of Q p .) Let T C p = T b ⊗ C p and R C p = R b ⊗ C p .1. The generalized eigenspace Symb ± Γ ( D k ′ ) ( x ) is free of rank one over the algebra ( T ± k ′ ) ( x ) , andthe eigenspace Symb ± Γ ( D k ′ )[ x ] is dimension one over C p .2. For all points y ∈ C ± W , except perhaps x , the algebra ( T ± W ) ( y ) is ´etale over R ( κ ± ( y )) .3. There exists u ∈ R C p such that ev κ ( u ) = 0 and κ is the only of u on W and an element t ∈ T such that x ( t ) = 0 as well as an isomorphism T C p −→ R C p [ X ] / ( X e − u ) sending t to X .4. The T C p -module Symb ± Γ ( D ( R )) o ⊗ T ± W T C p is free of rank one.5. For any point y ∈ C ± W ( C p ) of weight κ ± ( y ) ∈ Z , the H -equivariant map Symb ± Γ ( D ( R )) o ⊗ T ± W T C p −→ Symb ± Γ ( D κ ± ( y ) ) ( y ) sends any generator of Symb ± Γ ( D ( R )) o ⊗ T ± W T C p to a generator of Symb ± Γ ( D κ ± ( y ) ) ( y ) .Proof. This theorem is a combination of results from Section 4 of [1].
Remark . If k ∈ Z ≥ , then the eigencurve is ´etale over weight space at any weight k -point. Thepoint that we are interested in is when k = −
1, which corresponds to weight one modular forms.At weight k = − p -adic L -functions In this section we explain how to use Theorem 3.14 to construct a two-variable p -adic L -function.Let W = W ( k ′ , /p r ) = Sp R for some k ′ ∈ Z and r ≥
1. Let M = Symb ± Γ ( D ( R )) o . Define the R -linear map Λ : M −→ R b ⊗ Q p R to be the composition of evaluation at { } − {∞} and the map L from before. By construction,for all k ∈ Z ∩ W ( Q p ) we have the commutative diagram M R b ⊗R Symb ± Γ ( D k ) o R . Λ sp k Λ k ev k (6)which shows that for Φ ∈ M , Λ(Φ) interpolates the functions Λ k ( sp k (Φ)).15e now put ourselves in the situation of Theorem 3.14, and we extend scalars to C p . A C p in the subscript means completed tensor product over Q p with C p . Let x ∈ C ± U ( C p ) be a smoothpoint of weight k ′ ∈ Z for some U = W ( k ′ , /p d ). Let W = W ( k ′ , /p r ) = Sp R and T be as in theproposition. Let ǫ ∈ T ± W, C p be such that T C p = ǫ T ± W, C p . ThenSymb ± Γ ( D ( R )) o ⊗ T ± W T C p = ǫ Symb ± Γ ( D ( R )) o C p ⊂ Symb ± Γ ( D ( R )) o C p , so we let M = Symb ± Γ ( D ( R )) o ⊗ T ± W T C p = ǫ Symb ± Γ ( D ( R )) o C p . We first give a construction of a two-variable p -adic L -function, that we use when the weightmap κ ± : C ± W → W is ´etale. Assume κ ± : C ± W → W is ´etale.The module M is a rank one T C p -module, so let Φ be a generator. LetΛ(Φ , · , · ) : W × W −→ C p be the two-variable rigid analytic function that is the image of Φ in R b ⊗R under Λ. By thecommutative diagram (3), for all σ ∈ W and k ∈ Z ,Λ(Φ , k, σ ) = Λ k ( sp k (Φ) , σ ) . We now consider the non-´etale case. In the non-´etale case, if y ∈ C ± W is of weight k = k ′ , then sp k (Φ) ∈ Symb ± Γ ( D k ) o is not in the eigenspace corresponding to y . The construction that followsis due to Bella¨ıche ([1]). Let N = M ⊗ R C p T C p and let V = Sp T . DefineΛ T := Λ ⊗ Id T C p : N −→ ( R C p b ⊗ C p R C p ) ⊗ R C p T C p ∼ = T C p b ⊗ C p R C p . Then for Φ ∈ N , the function Λ T (Φ) ∈ T C p b ⊗R C p is a two-variable rigid analytic function on V C p × W C p . For each y ∈ V ( C p ) of weight κ ∈ W ( Q p ), define the specialization map sp y : N −→ Symb Γ ( D κ ) o C p as the natural map N −→ N ⊗ T C p ,y C p . We view N ⊗ T C p ,y C p as a subset of Symb ± Γ ( D κ ) o C p via N ⊗ T C p ,y C p = ( M ⊗ R C p T C p ) ⊗ T C p ,y C p = M ⊗ R C p ,ev κ C p ֒ → Symb ± Γ ( D κ ) o C p . By construction sp y is H -equivariant with respect to the action of H on the first component of N .Furthermore, if φ ∈ N and y ∈ V ( C p ) is of weight k ∈ Z , then ([1] Lemma 4.12)Λ T (Φ)( y, σ ) = Λ k ( sp y (Φ))( σ ) . We recall that we have an element t ∈ T C p and u ∈ R C p and an isomorphism T C p −→ R C p [ X ] / ( X e − u )sending t to X . Now let φ be a generator of M as a T C p module, and defineΦ = e − X i =0 t i φ ⊗ t e − − i ∈ N. Proposition 3.16.
1. Let T C p ⊗ R C p T C p act on N with the first factor acting on M and thesecond factor acting on T C p . Then ( t ⊗ − ⊗ t )Φ = 0 .
2. Let y ∈ C ± W ( C p ) be a point of weight κ ∈ W ( Q p ) . Then sp y (Φ) ∈ Symb ± Γ ( D κ )[ y ] . We note that if y = x , then Symb ± Γ ( D κ )[ y ] = Symb ± Γ ( D κ ) ( y ) , while if y = x and the ramifi-cation index is e , Symb ± Γ ( D κ ) ( y ) is an e -dimensional vector space. roof. The first part of the proposition is Lemma 4.13 of [1] and the second part is Proposition4.14 of [1].Define the two-variable p -adic L -function to beΛ T (Φ) : V C p × W C p −→ C p . To compare this second construction with the first construction when the ramification index is 1,we note that T C p ∼ = R C p [ X ] / ( X e − u ) = R C p , so N = M ⊗ R C p T C p ∼ = M and Φ = e − X i =0 t i φ ⊗ t e − − i = φ ⊗ e = 1. p -adic L -functions and p -adic Stark Conjecture We begin this section by introducing the objects we are working with and setting notation. Let F be a quadratic field of discriminant d F , and let χ : G F → Q × be a nontrivial ray class characterof F that is of mixed signature if F is real quadratic. Let K be the fixed field of the kernel of χ and let f be the conductor of K/F . Assume that ι ∞ ( K ) ⊂ R if F is real quadratic. Let v denotethe infinite place of K determined by ι ∞ . Let ρ = Ind G Q G F χ : G Q −→ GL ( Q ) be the induction of χ and let M be the fixed field of the kernel of ρ . Let f be the weight one modular form associated to ρ , so f has level N = N F/ Q ( f ) · | d F | and character ε = det ρ . The q -expansion of f is f = X a ⊂ O F ( a , f )=1 χ ( a ) q N a and we have that L ( f, s ) = L ( χ, s ) . Let x − a p ( f ) x + ε ( p ) = ( x − α )( x − β )be the Hecke polynomial of f at p . We note that when p splits in F , say p O F = pp , then α = χ ( p )and β = χ ( p ), and if p is inert, then α = p χ ( p O F ) and β = − p χ ( p O F ). Let k be the fieldobtained by adjoining the values of χ along with α and β to Q .We make some assumptions that will be fixed throughout. First we assume that p ∤ N , whichimplies in particular that p does not ramify in M . We further assume that p ∤ [ M : Q ], and weassume that α = β . With these assumptions, let f α ( z ) = f ( z ) − βf ( pz ) be a fixed p -stabilizationof f . p -adic L -function We use the constructions from the previous section to define our p -adic L -function. In order to dothat, we start with the following result of Bella¨ıche and Dmitrov about the eigencurve at weightone points. Theorem 4.1. ([2]) Let g be a classical weight one newform of level N , whose Hecke polynomialat p has distinct roots. Then the eigencurve is smooth at either p -stabilization of g . Moreover, theeigencurve is smooth but not ´etale over weight space if and only if the representation associated to g is obtained by induction from a mixed signature character of a real quadratic field in which p splits.
17y our assumption that α = β the above theorem implies that the eigencurve is smooth at thepoint corresponding to f α . We may break our situation into four cases, the cases when F is eitherimaginary or real quadratic and when p is either inert or split in F . In the case when F is realquadratic and p is split the eigencurve is smooth but not ´etale at f α . In the other three cases theeigencurve is ´etale at f α . We adopt the notation from the previous section except that we basechange everything to C p and we drop all the C p subscripts. Since we will conjecture the value at s = 0, we consider the minus subspace of modular symbols. Let T = T C p , M ⊂ Symb − Γ ( D ( R )) o C p , N , and R = R C p be as in Section 3.5 where the point of interest x is the point on the eigencurvecorresponding to f α . Let φ be a generator of M as a T -module and letΦ = e − X i =0 t i φ ⊗ t e − − i ∈ N. Let V = Sp T , W = W C p , W = Sp( R ), and let Λ(Φ) = Λ T (Φ) to make all the notation uniform.We record the interpolation formulas for our two-variable rigid analytic functionΛ(Φ , · , · ) : V × W −→ C p . For each classical point y ∈ V , let g y be the weight k ∈ Z ≥ p -stabilized newform corresponding to y . Let Ω ∞ ,g y ∈ C × be the complex period used to define the p -adic L -function associated to g y asin Section 3.3. Let ϕ g y ∈ Symb − Γ ( D k − ) ( y ) be the unique (by Theorem 3.9) modular symbol specializing under ρ ∗ k to ψ − g y / Ω ∞ ,g y ∈ Symb − Γ ( V k − ( Q )) . Let Ω p,g y ∈ C × p be the p -adic period such that sp y (Φ) / Ω p,g y = ϕ g y . For each y , the period pair(Ω ∞ ,g y , Ω p,g y ) viewed as an element of C × × C × p / Q × , where Q × is embedded diagonally, does notdepend on any choices. Proposition 4.2.
The two-variable rigid analytic function
Λ(Φ) on V × W is determined bythe following two interpolation properties. First, for all y ∈ V and all even characters σ ∈ W , Λ(Φ , y, σ ) = 0 . Second, for all y ∈ V corresponding to a p -stabilized newform g y of weight k ∈ Z ≥ ,and all odd characters ψ h·i j − ∈ W ( C p ) where ψ is a finite order character of conductor p m and ≤ j ≤ k − , Λ(Φ , y, ψ h·i j − )Ω p,g y = 1 a p ( g y ) m (cid:18) − ψω − j ( p ) a p ( g y ) p − j (cid:19) p m ( j − ( j − τ ( ψω − j )(2 πi ) j − ×× L ( g y , ψ − ω j − , j )Ω ∞ ,g y . (7) This equality takes place in Q . Here τ ( ψω − j ) is the Gauss sum associated to ψω − j .Proof. The first interpolation property follows from the fact that Φ is in the minus subspace forthe action of ι . For the second interpolation property, with the way everything is set up, it followsfrom the fact that Λ(Φ , y, σ )Ω p,g y = Λ k ( sp y (Φ) , σ )Ω p,g y = L p ( g y , ψ, j )where L p ( g y , ψ, s ) is defined using that complex period Ω ∞ ,g y . Remark . At this point, we would like to define the two-variable p -adic L -function associated to χ as L p ( χ, α, · , · ) : V × Z p −→ C p L p ( χ, α, y, s ) = Λ(Φ , y, ω − h·i s − ) . (8)18he p -adic L -function L p ( χ, α, y, s ) is determined by the above interpolation formula. The firstvariable is on the eigencurve varying through the p -adic family of modular forms passing through f α and the second variable is the usual cyclotomic variable. To get the one variable p -adic L -functionassociated to χ we would plug the point x ∈ V that corresponds to f α . It is then natural to makea conjecture for the value L p ( χ, α, x,
0) that is analogous to Conjectures 2.1 and 2.4, replacing thecomplex logarithm with the p -adic logarithm.The issue with making the conjecture this way is that the p -adic number L p ( χ, α, x,
0) is notcanonically defined because we made a choice for φ . The condition on the choice of φ is that φ isa generator of M as a T -module. If we choose a different generator of M as a T -module (changing φ by an element of T × ) that would change the value L p ( χ, α, x, L p ( χ, α, x, L p ( χ, α, x,
0) not being canonically defined is a question for furtherresearch. One way to approach the problem is to ask whether or not there is a way to canonicallychoose the periods (Ω p,g y , Ω ∞ ,g y ) so that they determine a two-variable modular symbol φ whichwould in turn define the function L p ( χ, α, x, s ) canonically. It is possible to do this in the case when F is imaginary quadratic and p is split in F (see Section 5.6). In this case when F is imaginaryquadratic and p is split in F the two-variable p -adic L -function L p ( χ, α, y, s ) is not canonicallydefined (it depends on the choice of canonical periods), but the one-variable p -adic L -function L p ( χ, α, x, s ) is.To get around these issues and make a precise conjecture we exploit the fact that in (7) thefunction Λ(Φ , y, σ ) interpolates the values of the complex L -function of g y twisted by p -powerconductor Dirichlet characters. Let ψ ∈ W ( C p ) be a p -power order character. We could thendefine, generalizing (8), the p -adic L -function of χ twisted by ψ to be L p ( χ, α, ψω, y, s ) = Λ(Φ , y, ψ − ω − h·i s − ) , and state a p -adic Stark conjecture for the value L p ( χ, α, ψω, x, L p ( χ, α, ψω, x, , y, σ ), but if it was in the range ofinterpolation it would be related to L ( f α , ψ,
0) at the point s = 0. We have the relation L ( f, ψ, s ) = L ( χψ, s ), and so a conjecture for the value L p ( χ, α, ψω, x,
0) should have the same shape as theconjecture for the value L ′ ( χψ,
0) with the complex logarithm replaced with the p -adic logarithm.Of course, the value L p ( χ, α, ψω, x,
0) has the same issue of not being canonically defined as L p ( χ, α, x, ψ ∈ W ( C p ) we canmake a function that is canonically defined. Fix two p -power order characters η, ψ ∈ W ( C p ) anddefine the function L p ( χ, α, ηω, ψω, y, s ) = Λ(Φ , y, η − ω − h·i s − )Λ(Φ , y, ψ − ω − h·i s − ) . Then L p ( χ, α, ηω, ψω, y, s ) does not depend on the choice of φ because the indeterminacy of theperiods in the interpolation formula (7) cancels out. The value L p ( χ, α, ηω, ψω, x,
0) is then canon-ically defined independent of any choices, and we formulate a conjecture for this value.
Definition 4.4.
Let η, ψ ∈ W ( C p ) be two p -power order characters. Define the two-variable p -adic L -function of χ with the auxiliary characters η and ψ as L p ( χ, α, ηω, ψω, · , · ) : V × Z p −→ C p ∪ {∞} L p ( χ, α, ηω, ψω, y, s ) = Λ(Φ , y, η − ω − h·i s − )Λ(Φ , y, ψ − ω − h·i s − ) . The function L p ( χ, α, ηω, ψω, y, s ) does not depend on the choice of Φ.Define the p -adic L -function of χ with the auxiliary characters η and ψ as L p ( χ, α, ηω, ψω, s ) = L p ( χ, α, ηω, ψω, x, s ) . emark . We may give the definition of L p ( χ, α, ηω, ψω, s ) without making reference to the two-variable p -adic L -function. The two-variable p -adic L -function is introduced for two reasons. Thefirst is that it satisfies an interpolation property, while the one-variable function L p ( χ, α, ηω, ψω, s )does not. The second is that we will use the two-variable p -adic L -function to prove our conjectureswhen F is imaginary quadratic and p is split in F .To define L p ( χ, α, ηω, ψω, s ) without referencing the two-variable p -adic L -function, we considerthe space, Symb ± Γ ( D − ) o , of weight negative one overconvergent modular symbols. Since theeigencurve is smooth at the point x corresponding to f α the eigenspace Symb ± Γ ( D − )[ x ] withthe same eigenvalues as f α is one-dimensional. If ϕ ± f α is a generator of this eigenspace, then L p ( χ, α, ηω, ψω, s ) may be defined as L p ( χ, α, ηω, ψω, s ) = Λ − ( ϕ − f α , η − ω − h·i s − )Λ − ( ϕ − f α , ψ − ω − h·i s − ) . Since Λ(Φ − , x, σ ) = Λ − ( sp x (Φ − ) , σ ) and 0 = sp x (Φ − ), this definition is the same as the firstdefinition. p -adic Conjecture For each n ∈ Z ≥ , let Q n be the n th layer of the cyclotomic Z p extension of Q , soGal( Q n / Q ) = 1 + p Z p / p n +1 Z p ∼ = Z /p n Z . Let Γ n = Gal( Q n / Q ). Let M n be the compositum of M and Q n . Let ∆ = Gal( M/ Q ), and for n ≥ n = Gal( M n / Q ). By our assumption that p does not ramify in M and p ∤ [ M : Q ],restriction gives an isomorphism ∆ n ∼ = ∆ × Γ n . For any n ≥
0, let v denote the infinite place of M n induced by ι ∞ . Let U n = O M n ⊂ M × n if M is not the Hilbert class field of F when F is imaginaryquadratic. If F is imaginary quadratic and M is the Hilbert class field of F , let U n = { u ∈ M × n : | u | w ′ = | u | w ′′ , ∀ w ′ , w ′′ | p, | u | w = 1 , ∀ w ∤ p, v } . Let k n be the field obtained by adjoining the p n +1 st roots of unity to k . For a character η of Γ n , let( ρη ) ∗ be the representation Ind G Q G F χ − ⊗ η − of ∆ n . Given a k n [∆ n ]-module A , let A ( ρ,η ) ∗ denotethe ( ρη ) ∗ -isotypic component of A .The following is how α is incorporated into our conjectures. It is an idea of Greenberg andVatsal ([15]), and is a key aspect to the conjecture. Let D p ⊂ ∆ be the decomposition group at p determined by ι p and let δ p be the geometric Frobenius. For a k [ D p ]-module A , let A δ p = α be thesubspace where δ p acts with eigenvalue α . Via the isomorphism ∆ n = ∆ × Γ n , we view D p as asubgroup of ∆ n for any n . Then the ∆ n -modules U n are also D p -modules.Let log p : C × p → C p denote Iwasawa’s p -adic logarithm. Extend log p to Q ⊗ Z C × p by Q -linearity. Conjecture 4.6.
Let ψ, η ∈ W ( C p ) be of orders p n and p m respectively with m, n ≥ . Then thereexists units u ∗ χψ,α ∈ ( k n ⊗ U n ) ( ρψ ) ∗ ,δ p = α and u ∗ χη,α ∈ ( k m ⊗ U m ) ( ρη ) ∗ ,δ p = α such that L p ( χ, α, ψω, ηω,
0) = (1 − βψ ( p )) (cid:16) − ψ − ( p ) αp (cid:17) τ ( ψ − ) p n +1 (1 − βη ( p )) (cid:16) − η − ( p ) αp (cid:17) τ ( η − ) p m +1 log p ( u ∗ χψ,α )log p ( u ∗ χη,α ) . (9) Remark .
1. This conjecture should be compared with Conjecture 2.4. We are relating the p -adic L -value L p ( χ, α, ψω, ηω,
0) to the spaces ( k n ⊗ U n ) ( ψρ ) ∗ ,δ p = α and ( k m ⊗ U m ) ( ηρ ) ∗ ,δ p = α via the p -adic logarithm. The spaces ( k n ⊗ U n ) ( ψρ ) ∗ ,δ p = α and ( k m ⊗ U m ) ( ηρ ) ∗ ,δ p = α have k n -and k m -dimension one respectively.Let K n be the fixed field of the kernel of χψ . At the outset of this project, it was expectedthat the unit u ∗ χη,α would be the projection of the unit u K n from definition 2.3 to the space20 k n ⊗ U n ) ( ψρ ) ∗ ,δ p = α ([13]). This is the case when F is imaginary and p is split in F (seeSection 5), while the numerical evidence suggests that this is not the case when F is imaginaryquadratic and p is inert in F (see Sections 6.2, 6.3). We do think that this is the case when F is real quadratic as we verify in the example in Section 6.1, but we do not have enoughevidence to conjecture it.2. It is also possible to state a conjecture for the p -adic value at s = 1 (see [13]), and thereshould be a functional equation relating the two conjectures.3. In ([15]), Greenberg and Vatsal define a Selmer group associated to the representation ρ andprove that the characteristic ideal of the Selmer satisfies an interpolation property that issimilar to the statement of our conjecture. Proving a main conjecture relating the character-istic ideal of the Selmer group associated to ρ to the analytic p -adic L -functions defined herewould allow one to prove this conjecture using Greenberg and Vatsal’s result. F is imaginary quadratic and p splits in F p -adic L -function In this section we state relevant facts that are needed about Katz’s two variable p -adic L -function.Let F be an imaginary quadratic field of discriminant d F , and assume p splits in F . Let p factoras p O F = pp , where p is the prime induced by the embedding ι p . Let O p = { x ∈ C p : | x | ≤ } be the closed unit ball in C p . Let f be an integral ideal of F such that ( f , p ) = 1. Let f factor as f = Q v | f f v . Let A F denote the adeles of F .The domain of Katz’s p -adic L -function is the set of all p -adic Hecke characters of modulus f , sowe begin by giving our conventions for Hecke characters. Define the subgroups U f , U f ,p , U f , ∞ ⊂ A × F as U f = (cid:26) ( x v ) v ∈ A × F : x v ≡ f v if v | f x v ∈ O × F v if v ∤ f and is finite (cid:27) U f ,p = ( x v ) v ∈ A × F : x v ≡ f v if v | f x v ∈ O × F v if v ∤ f p and is finite x v = 1 if v | p U f , ∞ = ( x v ) v ∈ A × F : x v ≡ f v if v | f x v ∈ O × F v if v ∤ f and is finite x v = 1 if v | ∞ . Let σ , σ be the two embeddings of F into Q . Order σ , σ so that σ is how we view F as a subsetof Q . Definition 5.1.
1. Let ( a , a ) ∈ Z . An algebraic Hecke character of F of infinity type ( a, b ) and modulus f is a group homomorphism χ : A × F −→ Q × such that the image of χ is contained in a finite extension of Q , U f ⊂ ker( χ ), and for all x ∈ F × , χ ( x ) = σ ( x ) a σ ( x ) a . The smallest f with respect to divisibility such that U f ⊂ ker( χ ) iscalled the conductor of χ . 21f χ is an algebraic Hecke character of modulus f and a an ideal of F such that ( a , f ) = 1 andthat factors as a = Y ( p , a )=1 p a p , then we define χ ( a ) as χ ( a ) := Y ( p , f )=1 χ ( π p ) a p where π p denotes a uniformizer of F p .2. A p -adic Hecke character of F is a continuous group homomorphism χ : A × F /F × −→ C × p . By continuity, there exists an integral ideal f ′ of F such that ( f ′ , p ) = 1 and U f ′ ,p ⊂ ker( χ ).Any f ′ for which this is true is called a modulus of χ and we say that χ is a p -adic Heckecharacter of modulus f ′ .3. A complex Hecke character of F is a continuous group homomorphism χ : A × F /F × −→ C × . By continuity, there exists an integral ideal f ′ of F such that U f ′ , ∞ ⊂ ker( χ ). Any f ′ for whichthis is true is called a modulus of χ and we say χ is a complex Hecke character of modulus f ′ .If χ is an algebraic, p -adic, or complex Hecke character and v is a place of F , then we let χ v denote χ restricted to F × v ⊂ A × F . Remark . In the literature, these notions of Hecke characters go by various names and definitions.We introduce and use the definitions given to avoid confusion.We will also need the following alternative definition of an algebraic Hecke character in termsof ideals. Let f be an ideal of O F and let α ∈ F × be an element such that (( α ) , m ) = 1 and saythat f factors as f = Y i p f i i . Define α ≡ f to mean that α ≡ p f i i in O F p i for all i .Let I ( f ) denote the group of fractional ideals of F that are coprime with f . Let P ( f ) = { ( α ) ∈ I ( f ) : α ∈ K × , α ≡ f } . The second definition of an algebraic Hecke character is, an algebraic Hecke character of F ofmodulus f and infinity type ( a , a ) ∈ Z is a group homomorphism χ : I ( f ) → Q × such that theimage of χ is contained in a finite extension of Q , and for all a ∈ P ( f ) such that a = ( α ) with α ≡ f , χ (( α )) = σ ( α ) a σ ( α ) a . Given an algebraic Hecke character, χ , of modulus f and infinity type ( a , a ), using the seconddefinition, we get an algebraic Hecke character of the same modulus and infinity type, χ A using thefirst definition by defining χ A to be the unique group homomorphism χ A : A × F −→ Q × such that:(i) For all primes p ∈ I ( f ), χ A | O × F p = 1 and χ A ( π p ) = χ ( p ) for any uniformizer in F p .(ii) For all x ∈ F × , χ A ( x ) = σ ( x ) a σ ( x ) a . (iii) U f ⊂ ker( χ A ).This gives a one-to-one correspondence between algebraic Hecke characters of modulus f andinfinity type ( a , a ) using the first and second definitions.Given an algebraic Hecke character, χ , of F of infinity type ( a , a ) and modulus f we obtain p -adic and complex Hecke characters χ p and χ ∞ which are defined as follows. Define χ p : A × F /F × −→ C × p at places v of F not dividing p as χ , so χ p | F × v = χ | F × v . At places above p we define χ p to be thegroup homomorphism χ p : ( F ⊗ Q p ) × −→ C × p p ( α ⊗
1) = χ ( α ) /ι p ( σ ( α ) a σ ( α ) a ) . Since the image of F × in ( F ⊗ Q p ) × is dense this defines χ p on ( F ⊗ Q p ) × . We do something similarfor χ ∞ . Define χ ∞ : A × F /F × −→ C × at the places v of F not dividing ∞ as χ , so χ ∞ | F × v = χ | F × v . At the place above ∞ we define χ ∞ to be the group homomorphism χ ∞ : ( F ⊗ R ) × −→ E ×∞ ⊂ C × χ ∞ ( α ⊗
1) = χ ( α ) /ι ∞ ( σ ( α ) a σ ( α ) a ) . Since the image of F × in ( F ⊗ R ) × is dense this defines χ ∞ on ( F ⊗ R ) × .Given an algebraic Hecke character χ when we consider χ p or χ ∞ , we will drop the subscripts p and ∞ . It will be clear from context when we are considering χ as a p -adic of complex Heckecharacter. Furthermore, given a p -adic (or complex) Hecke character ψ we may abuse of languageand say that ψ is an algebraic Hecke character of infinity type ( a , a ) if there exists an algebraicHecke character χ of infinity type ( a , a ) such that ψ = χ p (or ψ = χ ∞ ).Let ψ be an algebraic Hecke character of F of infinity type ( a, b ) and conductor f ′ p a p p a p where f ′ divides f . Define the p -adic local root number associated to ψ to be the complex number W p ( ψ ) = ψ p ( π − a p p ) p a p X u ∈ ( O F p / p a p ) × ψ p ( u ) exp( − πi ( T r F p / Q p ( u/π a p p ))) (10)where ψ p denotes ψ restricted to F × p and π p is a uniformizer for F p . Since F p = Q p we could take π p = p .Let G ( f p ∞ ) = A × F /F × U f ,p , so the space of p -adic Hecke characters of F of modulus f isHom cont ( G ( f p ∞ ) , C × p ) . In [4], Buzzard explains how to view Hom cont ( G ( f p ∞ ) , C × p ) as the C p -points of a rigid-analyticvariety. When we say rigid analytic function in the following theorem it is this rigid analyticstructure that we are referring to.Let S be the set of places containing the infinite places of F and the places of F dividing f . Theorem 5.3. ([18], [9]) There exists a p -adic rigid analytic function L p = L p,Katz : Hom cont ( G ( f p ∞ ) , C × p ) −→ C p as well as complex and p -adic periods Ω ∞ ∈ C × , Ω p ∈ C × p such that for all algebraic Hecke character ψ of F of conductor f ′ p a p p a p where f ′ divides f and infinity type ( a, b ) with a < and b ≥ , wehave L p ( ψ )Ω b − ap = ( − a − π ) b √ d F b W p ( ψ ) (cid:18) − ψ − ( p ) p (cid:19) (1 − ψ ( p )) L S ( ψ, b − a ∞ . (11) Remark .
1. Katz originally proved this theorem in [18] for imaginary quadratic fields andthen a similar theorem in [19] for CM fields. The above statement is taken from [9] with thecorrection from [3] and with a slight modifications in order to state everything adelically.2. The interpolation property (11) uniquely determines Katz’s p -adic L -function.We now state Katz’s p -adic Kronecker’s second limit theorem. Let ζ n = ι − ∞ ( e πi/n ) ∈ Q for n ∈ Z ≥ be a collection of primitive n th roots of unity in Q . Theorem 5.5. ([18], [9]) Let χ be an algebraic Hecke character of conductor f and trivial infinitytype and let ψ be a Dirichlet character of conductor p n . Let K be the fixed field of the kernel of χψ hen χψ is viewed as a Galois character via the Artin isomorphism G ( f p ∞ ) ∼ = Gal ( F ( f p ∞ ) /F ) . Let u K be the Stark unit for K/F , G = Gal ( K/F ) , and e be the number of roots of unity in K . Then L p ( χψ ) = − e ψ ( − τ ( ψ − ) χ ( p n ) p n (cid:18) − ( χψ ) − ( p ) p (cid:19) (1 − χψ ( p )) X σ ∈ G χψ ( σ ) log p ( σ ( u K )) Remark . A version of this was proved in Katz’s original paper. The formulas for this theoremare taken from [9] with a minor correction so the 1 − χψ ( p ) factor is correct (see [16]). (Ω ∞ , Ω p ) In this section, we explain how to define the period pair (Ω ∞ , Ω p ). The pair (Ω ∞ , Ω p ) viewed as anelement of C × × C × p / Q × where Q × is embedded diagonally, is a canonical element associated to F .Let K be a finite extension of F that contains the Hilbert class field of F . Let P be the primeof K determined by ι p . Let E be an elliptic curve with CM by O F defined over K and with goodreduction at P . Let ω ∈ Ω ( E/K ) be an invariant differential of E defined over K . Attached tothe pair ( E, ω ), we let x and y be coordinates on E such that ι : E −→ P P ( x, y, K , which embeds E as the zero set of y = 4 x − g x + g and suchthat ι ∗ ( dxy ) = ω . Let E ω denote the image of E under ι . Let E ω ( C ) ⊂ P ( C ) denote the complexmanifold which consists of the complex points of E ω . Let γ ∈ H ( E ω ( C ) , Q ) and define the complexperiod Ω ∞ = 12 πi Z γ ω. Let L = (cid:26) πi Z η ω : η ∈ H ( E ω ( C ) , Z ) (cid:27) be the period lattice of E ω . We have the complex uniformizationΦ : C / L −→ E ω ( C ) z ( P ( L , z ) , P ′ ( L , z ) , P is the Weierstrass function. We consider the element( p − n Ω ∞ ) ∞ n =1 ∈ lim ←− n ( p − n Ω ∞ F/ Ω ∞ F ) = (lim ←− n p − n L / L ) ⊗ Q p which is in the Tate module of C / L tensored with Q p . Let V p E ω = T p E ω ⊗ Q p , V p E ω = T p E ω ⊗ Q p , V p E ω = T p E ω ⊗ Q p , and let ξ = ( ξ n ) ∞ n =1 be the image of ( p − n Ω ∞ ) ∞ n =1 under the compositionlim ←− n p − n Ω ∞ F/ Ω ∞ F Φ p −−→ V p E ω −→ V p E ω where the second map is the projection corresponding to T p E ω = T p E ω × T p E ω .The coordinates x and y on E ω determine a formal group of E over K P , b E ω . Let V p b E ω = T p b E ω ⊗ Q p . Since p splits in F and p is the prime of F determined by ι p , T p b E ω = T p E ω . Let ξ nowdenote the corresponding elemet of V p b E ω . Since V p b E is a rank one Q p -module, ξ is a basis element.Let ζ = ( ζ p n ) ∞ n =1 = ( ι − p (exp(2 πi/p n ))) ∞ n =1 so ζ is a basis element of V p b G m := T p b G m ⊗ Q p . Define ϕ p : V p b E ω −→ V p b G m ϕ p ( ξ ) = ζ . It is a result of Tate ([33]) that the mapHom O C p ( b E ω , b G m ) −→ Hom Z p ( T p b E ω , T p b G m )is a bijection. We note thatHom Q p ( V p b E ω , V p b G m ) = Hom Z p ( T p b E ω , T p b G m ) ⊗ Q p and let ϕ ∈ Hom O C p ( b E, b G m ) ⊗ Q p be the element corresponding to ϕ p . Define Ω p by the rule ω = Ω p ϕ ∗ ( dT / (1 + T )) . This defines a pair (Ω ∞ , Ω p ) ∈ C × × C × p . The definition depends on the choice of E , ω , and γ , but is canonically defined as an element of C × × C × p / Q × . That is, if we make different choicesfor E , ω , or γ , then Ω ∞ and Ω p are both scaled by the same element of Q × (see [13] for furtherexplanation of the dependence). For the remainder of Section 5, fix a nontrivial ray class character χ of conductor f such that( f , p ) = 1, and let f = P a ⊂ O F χ ( a ) q N a be the weight one modular form associated to χ . Let f α be a p -stabilization of f , so α is either χ ( p ) of χ ( p ). Recall that the character of f is ε :( Z /N Z ) × → Q determined by the rule ε ( ℓ ) = χ ( ℓ O F ) for primes ℓ ∤ N p . The goal of this sectionis to explicitly describe the rigid analytic functions T ℓ for ℓ ∤ N p and U p on a neighborhood of thepoint corresponding to f α on the eigencurve.For k ∈ Z , let ν k ∈ W ( Q p ) denote the character t t k − . By Theorem 4.1), the eigencurveis ´etale at the point corresponding to f α . Let w = ν ∈ W ( Q p ) and let W = W ( w, /p r ) = Sp R be a neighborhood of w such that the weight map C − W → W is ´etale at all points in the connectedcomponent containing the point corresponding to f α . Let x ∈ C − W ( C p ) be the point correspondingto f α and let V C p = Sp T C p ⊂ C − W, C p be the connected component of C − W, C p containing x . Then V C p → W C p is ´etale, and we take W to be as in Proposition 3.14. Then the weight map on the levelof rings R C p → T C p is an isomorphism, and we use this map to identify T C p with R C p .Fix a choice of topological generator γ of 1 + p Z p , so R = nX a n ( t − ( w ( γ ) − n ∈ Q p [[ t − ( w ( γ ) − | a n p rn | → n → ∞ o . Let z = t − ( w ( γ ) − R is the set of all F ( z ) ∈ Q p [[ z ]] that converge on the closed around0 disk of radius 1 /p r in C p . By the Weierstrass preparation theorem, any F ( z ) ∈ R is determinedby its values ev ν k ( F ( z )) = F ( ν k ( γ ) − w ( γ )) = F ( γ k − − γ − )at the integers k ∈ Z such that ν k ∈ W . For an integer k , ν k is in W = W ( w, /p r ) if and only if k ≡ p r − ( p − V is ´etale over weight space, the Hecke operators T ℓ for ℓ ∤ N p U p , and [ a ] for a ∈ ( Z /N Z ) × as rigid analytic functions in R C p are deterined by the following two properties:1. At the weight w , ev w ( T ℓ ) = a ℓ ( f α ) = ( χ ( q ) + χ ( q ) if ℓ O F = qq ℓ is inert in Fev w ( U p ) = α , and ev w ([ a ]) = ε ( a ) for all a ∈ ( Z /N Z ) × .2. For all k ∈ Z ≥ such that ν k ∈ W , ev ν k ( T ℓ ) , ev ν k ( U p ) are the T ℓ and U p Hecke eigenvalues ofan eigenform g of weight k , level Γ , and character ε which is new at level N .The second condition implies that the functions [ a ] ∈ R C p are the constant function [ a ] = ε ( a ).We exhibit explicit elements of R C p with the above two properties as T ℓ for ℓ ∤ N p and U p .25n the interest of clarity of composition and space, we assume for the rest of Section 5 that α = χ ( p ). The case α = χ ( p ) is similar (see [13] for more details).To begin we define an algebraic Hecke character of F . Since p ≥
3, the only root of unity (andso the only unit of F ) congruent to 1 mod p in F is 1. Therefore we may identify the group P ( p )with a subgroup of F × : P ( p ) = { α ∈ F × : (( α ) , p ) = 1 , α ≡ p } ⊂ F × . Define λ as λ : P ( p ) −→ F × ⊂ Q × λ ( α ) = α = σ ( α ) . Since Q × is divisible, we may extend λ to I ( p ) to define an algebraic Hecke character λ of infinitytype (1 ,
0) and modulus p . The choice of extension of λ is determined up to multiplication bea character of I ( p ) /P ( p ). We impose a condition on the extension λ we choose. Recall that C × p may be written as C × p = p Q × W × U , where W is the group of roots of unity of order prime to p and U = { u ∈ C × p : | − u | < } . By construction, after composing with ι p the image of λ iscontained in U . Since U is a divisible group, we may choose our extension λ so that the image of λ after composing with ι p is also contained in U , which we do. Since the only torsion elements in U are the p -power roots of unity, any two extensions λ and λ ′ of λ that have image in U differby a character of I ( p ) /P ( p )[ p ∞ ] where the [ p ∞ ] denotes the maximal quotient of I ( p ) /P ( p ) with p -power order.Let p n = | I ( p ) /P ( p )[ p ∞ ] | . If p r ≤ p n , then we shrink W so that W = W ( w, p n +1 ). We maydo this without changing anything we have assumed previously, and the reason for doing this willbecome clear momentarily.Let M = | I ( p ) /P ( p ) | and note that | M | p = 1 /p n . For each prime q of F such that q = p definethe power series G q ( z ) = exp p ( z log p ( λ ( q ))) = ∞ X n =0 z n log p ( λ ( q )) n n !as an element of C p [[ z ]]. The power series G q ( z ) converges if | z | < p / ( p − | log p ( λ ( q )) | . Since M = | I ( p ) /P ( p ) | , q M = ( q ) for some q ∈ O F such that q ≡ p . Hence by definition of λ λ ( q ) M = λ (( q )) ≡ p so | − λ ( q ) M | < p − / ( p − . Then by properties of the p -adic logarithm,1 p / ( p − > | − λ ( q ) M | = | log p ( λ ( q ) M ) | = | M || log p ( λ ( q )) | = | log p ( λ ( q )) | p n so 1 p n < p / ( p − | log p ( λ ( q )) | . Therefore G q ( z ) converges for | z | ≤ p n , which is independent of q .Recall that log γ ( z ) := log p ( z )log p ( γ ) , and define F q ( z ) = G q ◦ log γ (1 + γz ) . By construction, if | z | ≤ p n +1 then F q ( z ) converges. This implies that F q ( z ) ∈ R C p . The function F q ( z ) is the unique element of R C p with the property that for all k ∈ Z such that ν k ∈ W ,26 v ν k ( F q ( z )) = ( λ ( q )) k − . Furthermore, since k ∈ Z is such that ν k ∈ W if and only if k ≡ p r − ( p −
1) and r > n , F q ( z ) does not depend on the choice of extension λ of λ since p n divides k − k − I ( p ) /P ( p )[ p ∞ ].Now let a ⊂ O F be a nontrivial ideal of O F such that ( a , p ) = 1, and define F a ( z ) = Y q F q ( z ) val q ( a ) if ( a , p ) = 10 else.Further, define A ( z ) = 1 and for n ≥ A n ( z ) = X a ⊂ O F N F/ Q a = n χ ( a ) F a ( z ) . Define the formal q -expansion F = ∞ X n =1 A n ( z ) q n ∈ R C p [[ q ]] . This formal q -expansion is the CM Hida family specializing to f α in weight one. Proposition 5.7.
For all k ∈ Z ≥ , ν k ∈ W F k := ∞ X n =1 ev ν k ( A n ( z )) q n = X a ⊂ O F χλ k − ( a ) q N a is the q -expansion of a weight- k cusp form of level Γ and character ε that is new at level N .Proof. By definition of A n ( z ) we have that ∞ X n =1 ev ν k ( A n ( z )) q n = X a ⊂ O F χλ k − ( a ) q N a . Shimura ([25]) showed that X a ⊂ O F χλ k − ( a ) q N a is the q -expansion of a weight- k cusp form of level Γ which is new at level N and has characterdefined by ℓ χ (( ℓ )) λ k − (( ℓ )) ℓ k − = χ (( ℓ )) (cid:18) λ (( ℓ )) ℓ (cid:19) k − for ℓ ∈ ( Z /N Z ) × a prime not equal to p . A simple calculation shows that this is the character ε . By the proposition, the functions A ℓ ( z ) ∈ R C p for ℓ ∤ N p and A p ( z ) ∈ R C p satisfy the twoproperties that uniquely determine T ℓ , U p ∈ R C p . Hence T ℓ = A ℓ for ℓ ∤ N p and U p = A p . p -adic L -function of the CM family Keeping the notation of the previous section, let Φ be a generator for the rank one T C p -moduleSymb − Γ ( D ( R )) o ⊗ T − W T C p ⊂ Symb − Γ ( D ( R )) o and let Λ(Φ , · , · ) : W × W −→ C p
27e the two-variable p -adic L -function associated to Φ as in Section 3.5. In order to prove Conjecture4.6 we restrict Λ(Φ , · , · ) to a particular subset of W × W . Let U = { t ∈ Z p : t ≡ p r − } where W = W ( w, p r ) and r was chosen in the previous section. Let η be a p -power order characterand let ψ = ηω . Let p m be the conductor of ψ . Define the two-variable restriction of Λ(Φ , · , · ): L p ( χηω, α, · , · ) : U × Z p −→ C p L p ( χηω, α, t, s ) = Λ(Φ , ω − h·i t − , ( ηω ) − h·i s − ) . For all k ∈ Z ≥ , k ≡ p r − , let (Ω ∞ ,k , Ω p,k ) be the periods for ν k ∈ W that appear in the in-terpolation formula for Λ(Φ , · , · ). Then L p ( χηω, α, t, s ) is determined by the following interpolationproperty: for all k ∈ Z ≥ , k ≡ p r − , and j ∈ Z , 1 ≤ j ≤ k − j ≡ p − L p ( χηω, α, k, j )Ω p,ν k = E p ( α, ηω, k, j ) L ( χλ k − ηω, j )Ω ∞ ,ν k where E p ( α, ηω, k, j ) = 1 χλ k − ( p ) m (cid:18) − ( ηω ) − ( p ) p j − χλ k − ( p ) (cid:19) ×× p m ( j − ( j − τ (( ηω ) − )(2 π ) j − and L ( χλ k − ηω, s ) is the complex Hecke L -function associated to χλ k − ηω . L p,Katz In this section we define a two-variable specialization of Katz’s p -adic L -function that we compareto the two-variable p -adic L -function defined in the previous section.Observe that the complex L -value appearing the interpolation formula in the previous sectionis L ( χλ k − ηω, j ) = L ( χλ k − ηωN − j , . By our choice of λ , the algebraic Hecke character χλ k − ηωN − j has infinity type ( k − − j, − j ),which is not in the range of interpolation of Katz’s p -adic L -function.From here on, let c denote complex conjugation, so c is an automorphism of C . Via ourembedding ι ∞ , c acts on ideals of F , and there is the relation of complex L -functions L ( χλ k − ηωN − j , s ) = L ( χλ k − ηωN − j ◦ c, s )that changes the infinity type. Therefore, χλ k − ηωN − j ◦ c has infinity type ( − j, k − − j ), whichis in the range of interpolation of Katz’s p -adic L -function.Let κ = λ ◦ c viewed as an algebraic Hecke character. By our choice of λ , κ has infinity type(0 ,
1) and conductor p . Further when we view κ as a p -adic Hecke character, since λ takes valuesin U = { u ∈ C × p : | − u | < } ⊂ C × p we may consider the p -adic Hecke character κ s for any p -adicnumber s ∈ Z p .Let κ be the algebraic Hecke character κ = ω − N where N is the norm character N : A F −→ Q × N (( x v ) v ) = Y v − finite | x v | − . Viewing κ as a p -adic Hecke character, κ has image 1 + p Z p in C × p . It therefore makes senseto consider κ s as a p -adic Hecke character for any s ∈ Z p . Let e χ = χ ◦ c and note that e χ hasconductor f . Let L p,Katz be Katz’s p -adic L -function with respect to the ideal m where as in the28otation of Section 4.1, m is the conductor of M/F . The ideal m is divisible by all the primes thatdivide f and f . Let (Ω ∞ , Ω p ) be the period pair used to define L p,Katz .Define L p,Katz ( χη, α, · , · ) : U × Z p −→ C p L p,Katz ( χη, α, s , s ) := L p,Katz ( e χηκ s − κ − s ) . Proposition 5.8. L p,Katz ( χη, α, s , s ) is determined by the following interpolation property: forall k ∈ Z ≥ , k ≡ p r − , j ∈ Z , ≤ j ≤ k − , j ≡ p − , L p,Katz ( χη, α, k, j )Ω k − p = E p ( α, ηω, k, j ) − (2 π ) k − √ d F k − − j L ( χλ k − ψω j − , j )Ω k − ∞ where E p ( α, ηω, k, j ) is defined as in the previous section.Proof. That L p,Katz ( χη, α, s , s ) is determined by the interpolation property follows from the con-tinuity of L p,Katz ( χη, α, s , s ) and that the set of k ’s and j ’s is dense in U × Z p . Let k ∈ Z ≥ , k ≡ p r − and j ∈ Z , 1 ≤ j ≤ k − j ≡ p −
1. By our definitions e χηκ k − κ − j = χηωλ k − N − j ◦ c (12)so e χηκ k − κ − j has infinity type ( − j, k − − j ) which is in the range of interpolation for L p,Katz .By the interpolation formula for L p,Katz , L p,Katz ( χη, α, k, j )Ω k − p = ( j − π ) k − − j √ d F k − − j W p ( χηωλ k − N − j ◦ c ) ×× (cid:18) − ( χηω ) − λ − k N j ( p ) p (cid:19) (1 − χηωλ k − N − j ( p )) ×× L ( χηωλ k − N − j , k − ∞ Since λ has modulus p , 1 − χηωλ k − N − j ( p ) = 1. We also have that ( ηω ) − ( p ) = ( ηω ) − ( p ), N j ( p ) = p j , and a calculation shows that, W p ( χηωλ k − N − j ◦ c ) = − p m ( j − τ (( ηω ) − ) χλ k − ( p ) m . Therefore the formula becomes L p,Katz ( χη, α, k, j )Ω k − p = ( j − π ) k − − j √ d F k − − j − p m ( j − τ (( ηω ) − ) χλ k − ( p ) m ×× (cid:18) − ( ηω ) − ( p ) p j − χλ k − ( p ) m (cid:19) L ( χλ k − ηω, j )Ω k − ∞ = E p ( α, ηω, k, j ) − (2 π ) k − √ d F k − − j L ( χλ k − ψω j − , j )Ω k − ∞ . Let S C p be the fraction field of T C p = R C p . Proposition 5.9.
There exists Ψ ∈ Symb − Γ ( D ( R )) ⊗ T ± W S C p such that the p -adic L -function L p ( χηω, α, t, s ) := Λ(Ψ , ω − h·i t − , ( ηω ) − h·i s − )29 s calculated with the p -adic and complex periods (Ω p,k , Ω ∞ ,k ) = Ω k − p , Ω k − ∞ (cid:18) √ d F π (cid:19) k − ! where (Ω p , Ω ∞ ) is the period pair used to define Katz’s p -adic L -function. We note that the domainof L p ( χηω, α, t, s ) is as in the previous section.Proof. Let L p ( χηω, α, t, s ) = Λ(Φ , ω − h·i t − , ( ηω ) − h·i s − ) be as in Section 5.4. We determine ameromorphic function P ( t ) on U such that P ( t ) L p ( χηω, α, t, s ) has interpolation formula with theperiods Ω k − p , Ω k − ∞ (cid:18) √ d F π (cid:19) k − ! . Let P : U × Z p −→ C p ∪ {∞} be the p -adic meromorphic function defined by the ratio P ( t, s ) = L p,Katz ( χη, α, t, s ) L p ( χηω, α, t, s ) . Then P ( t, s ) has the interpolation property: P ( k, j )Ω p,k Ω k − p = Ω ∞ ,k Ω k − ∞ − (2 π ) k − √ d F k − − j for k ’s and j ’s as in the previous section.When defining the periods for L p ( χηω, α, t, s ), we choose Φ which we’ve done, and we choosethe Ω ∞ ,k . These choices determine the Ω p,k . The condition on the choice of Ω ∞ ,k is that thecomplex values in the interpolation formula L p ( F k , · , · ) are algebraic. These values are for all oddfinite order characters ψ ∈ W ( C p ), k ∈ Z ≥ , 1 ≤ j ≤ k − C alg ( α, k, j ) L ( χλ k − ψω j − , j )(2 π ) j − Ω ∞ ,k where C alg ( α, k, j ) = p m ( j − ( j − τ ( ψ − ω − j ) χλ k − ( p ) m (cid:18) − ψ − ω − j ( p ) χλ k − ( p ) p − j (cid:19) i j − and m is the power of p in the conductor of ψ .It is clear then that we may defineΩ ∞ ,k = Ω k − ∞ (cid:18) √ d F π (cid:19) k − since by the interpolation property of Katz’s p -adic L -function, the values C alg ( α, k, j ) (2 π ) k − − j L ( χλ k − ψω j − , j ) √ d F k − Ω k − ∞ are algebraic.If we consider P ( t, s ) with this choice of complex periods, then P ( t, s ) satisfies the interpolationformula for k ∈ Z ≥ k ≡ p r − , j ∈ Z , 1 ≤ j ≤ k − j ≡ p − P ( k, j )Ω p,k Ω k − p = − p d F j − . Now we separate variables for the function P ( t, s ). Since p splits in F , √ d F ∈ Q p = F p . Define theanalytic function Q ( s ) as Q ( s ) = −h√ d F i s − , and let P ( t ) = P ( t, s ) /Q ( s ). The function P ( t ) is30 p -adic meromorphic function on U satisfying the relation that for all k ∈ Z ≥ , k ≡ p r − , P ( k )Ω p,k = Ω k − p . Since P ( t ) is a p -adic meromorphic function on U , there exists an element e P ∈ S C p such that for all t ∈ U , e P ( γ t − − γ − ) = P ( t ).If we define Ψ = e P Φ − and redefine the function L p ( χηω, α, t, s ) = Λ(Ψ , ω − h·i t − , ( ηω ) − h·i s − )then L p ( χηω, α, t, s ) satisfies the interpolation property that for all k, j as above, L p ( χηω, α, t, s )Ω k − p = E p ( α, ηω, k, j ) (2 π ) k − L ( χλ k − ψω j − , j ) √ d F k − Ω k − ∞ . Remark . If P ( t ) in the proof of the previous proposition does not have any zeros or poles,then Ψ is a generator for the free rank one T C p -module Symb Γ ( D ( R )) o ⊗ T − W T C p and so Ψ wouldbe a valid choice to define the p -adic L -function as in Section 3.5.We record the precise comparison of the p -adic L -function defined in the previous two sectionsthat appeared in the proof of the previous proposition. Corollary 5.11.
Let L p,Katz ( χη, α, t, s ) and L p ( χηω, α, t, s ) be defined as in the previous two sec-tions, so L p ( χηω, α, t, s ) = Λ(Φ , ω − h·i t − , ( ηω ) − h·i s − ) where Φ is a generator of Symb − Γ ( D ( R )) o ⊗ T ± W T C p as a T C p -module. Then L p,Katz ( χη, α, t, s ) = P ( η, t, s ) L p ( χηω, α, t, s ) where P ( η, t, s ) is a p -adic meromorphic function determined by the interpolation property that forall k ∈ Z ≥ , k ≡ p r − , j ∈ Z , ≤ j ≤ k − , j ≡ p − , P ( η, k, j )Ω p,k Ω k − p = Ω ∞ ,k Ω k − ∞ − (2 π ) k − √ d F k − − j . Remark . We remark that P ( η, t, s ) a priori depends on η and α , but as is clear from theinterpolation formula does not actually depend on η or α . The reason for putting η in the notationwill become clear in the next section. In this section we prove Conjecture 4.6 for χ . We adopt the notation of Section 4. For each r ≥ u r = u M r be the Stark unit for M r /F from Definition 2.10. For ϕ ∈ W ( C p ) a character oforder p r , the unit u ∗ χϕ,α is obtained from u r by first mapping u r to the ( ρϕ ) ∗ -isotypic component of k r ⊗ U r and then projecting to the subspace where δ p acts with eigenvalue α . Let π ∗ ρϕ be the map π ∗ ρϕ : U r −→ ( k r ⊗ U r ) ( ρϕ ) ∗ π ∗ ρϕ ( u ) = X σ ∈ ∆ n T r (( ρϕ ) ∗ ( σ )) ⊗ σ ( u ) . The idea to project to the subspace where δ p acts with eigenvalue α is of Greenberg and Vatsal([15]) and we adopt their notation. Let | · | α denote the map | · | α : ( k ⊗ U r ) ( ρϕ ) ∗ −→ ( k r ⊗ U r ) ( ρϕ ) ∗ ,δ p = α | u | α = 1 | ∆ p | | ∆ p |− X i =0 α − i δ ip ( u ) . Then | π ∗ ρϕ ( u r ) | α ∈ ( k r ⊗ U r ) ( ρϕ ) ∗ ,δ p = α and so the following theorem implies Conjecture 4.6.31 heorem 5.13. Let η, ψ ∈ W ( C p ) be of orders p m and p n respectively. Then L p ( χ, α, ηω, ψω,
0) = τ ( η − ) p m +1 (cid:16) − η − ( p ) αp (cid:17) (1 − βη ( p )) τ ( ψ − ) p n +1 (cid:16) − ψ − ( p ) αp (cid:17) (1 − βψ ( p )) log p | π ∗ ρη ( u m ) | α log p | π ∗ ρψ ( u n ) | α . Proof.
To begin, we simplify the expression | π ∗ ρη ( u m ) | α . Since ( ρη ) ∗ = Ind ∆ m H m ( χη ) − , for all σ ∈ ∆ m − H m , T r (( ρη ) ∗ ( σ )) = 0. Since c ∈ ∆ m − H m , for all σ ∈ H m , T r (( ρη ) ∗ ( σ )) = χη ( σ ) + χη ( cσc ).Let χ c denote the character χ c ( σ ) = χ ( cσc ) and note that since Q n is totally real, η ( cσc ) = η ( σ )for all σ . Therefore, π ∗ ρη ( u m ) = X σ ∈ H m χη ( σ ) ⊗ σ ( u m ) + χ c η ( σ ) ⊗ σ ( u m ) . Since α = χ ( p ), we have that | X σ ∈ H m χη ( σ ) ⊗ σ ( u m ) | α = 0 and | X σ ∈ H m χ c η ( σ ) ⊗ σ ( u m ) | α = X σ ∈ H m χ c η ( σ ) ⊗ σ ( u m ) . Therefore | π ∗ ρη ( u m ) | α = X σ ∈ H m χ c η ( σ ) ⊗ σ ( u m ) . A similar formula holds for | π ∗ ρψ ( u n ) | α .Let L p ( χηω, α, t, s ) and L p ( χψω, α, t, s ) be as defined in Section 5.4. By construction L p ( χ, α, ηω, ψω, s ) = L p ( χηω, α, , s ) L p ( χψω, α, , s ) . Then by Corollary 5.11, L p ( χ, α, ηω, ψω, s ) = P ( η, , s ) L p,Katz ( χηω, α, , s ) P ( ψ, , s ) L p,Katz ( χψω, α, , s )= L p,Katz ( χηω, α, , s ) L p,Katz ( χψω, α, , s ) . Plugging in 0, we get L p ( χ, α, ηω, ψω,
0) = L p,Katz ( χη ◦ c ) L p,Katz ( χψ ◦ c ) . We now use Theorem 5.5. By the above simplifications of | π ∗ ρη ( u m ) | α and | π ∗ ρψ ( u n ) | α , L p,Katz ( χη ◦ c ) L p,Katz ( χψ ◦ c ) = τ ( η − ) p m +1 (cid:18) − ( χη ) − ( p ) p (cid:19) (1 − χη ( p )) log p | π ∗ ρη ( u m ) | α τ ( ψ − ) p n +1 (cid:18) − ( χψ ) − ( p ) p (cid:19) (1 − χψ ( p )) log p | π ∗ ρψ ( u n ) | α . To finish, we just note that since α = χ ( p ), β = χ ( p ), so ( χη ) − ( p ) = η − ( p ) /α and ( χψ ) − ( p ) = ψ − ( p ) /α , as well as χη ( p ) = βη ( p ) and χψ ( p ) = βψ ( p ). The programming for the examples consisted of three basic parts: computing the minimal poly-nomial of the Stark units, viewing the Stark units p -adically to take their p -adic logarithm, and com-puting the p -adic L -values. The code used for the examples can be found at https://github.com/Joe-Ferrara/p-adicStarkExamples .We briefly explain the basis of the code. 32n the case where F is real quadratic, the minimal polynomial of the Stark units was computedin SAGE combining the strategies of Stark in [30] and Dummit, Sands, and Tangedal in [11]. In thecases where F is imaginary quadratic, the minimal polynomials of the Stark units were computedin pari/gp using the formulas from Section 2.1. To view the Stark units p -adically and take the p -adic logarithm we wrote a class in SAGE to represent the extension of Q p the Stark units arein and to take their p -adic logarithm. To compute the p -adic L -values, we used code written inSAGE by Rob Harron and Rob Pollack to compute overconvergent modular symbols (their code isbased off the algorithms described in [21]). We computed the weight negative one overconvergentmodular symbol associated to f α to get the p -adic L -values as described in Remark 4.5.An important reason for these examples is that we expected the units appearing in Conjecture4.6 to be related to the Stark units in definition 2.3 in the way that they are related in Section5.7, when F in imaginary quadratic and p is split in F (see [13] for what we expected). As theexamples show this may be the case when F is real quadratic. When F is imaginary quadratic and p is inert in F , we can verify the conjecture, but it is not clear how or if the units in Conjecture4.6 are related to the Stark units in 2.10. In the case when F is imaginary quadratic and p is inertin F , the expected formulas conjectured in [13] are not correct.We adopt all the notation of Section 4. All three examples are of the following form which wedescribe before specifying the exact examples.Let ψ ∈ W ( C p ) be the character ψ : ( Z /p Z ) × → Q × that sends the generator of ( Z /p Z ) × withminimal positive integer coset representative, to ζ p . For α = ± α = − L p ( χ, α, ψ i ω, ψ j ω,
0) when 1 ≤ i < j ≤ p − K be the compositum of K and Q . We computed the minimal polynomial of the Starkunit for K over F . Let u be a root of the minimal polynomial, so u is a Stark unit for K over F . In all three examples, the Hecke polynomial of f at p is x −
1, so α = ± δ p has order two. For a ∆ p -module A and a ∈ A , let | a | α = ( aδ p ( a ) if α = 1 aδ p ( a ) if α = − | · | α : A → A δ p = α . (Note that the definition of | · | α appearing here differs from the one appearingin Section 5.7 by the scalar | ∆ p | .) Let u ∗ χψ i ,α = X σ ∈ Gal( K /F ) χψ i ( σ ) ⊗ | σ ( u ) | α ∈ ( k ⊗ O × M ) ( ρψ i ) ∗ ,δ p = α . We computed each example to two levels of precision. First to check the results we computedwith 60 p -adic digits of precision. Then to reproduce and reaffirm the results we computed eachexample at a higher level of precision. Let prec be the number of p -adic digits that each computationwas done with. We computed each of the p -adic numbers L p ( χ, α, ψ i ω, ψ j ω,
0) and log p ( u ∗ χψ i ,α )log p ( u ∗ χψ j ,α ) ,which lie in the p -adic field Q p ( ζ p ). The field Q p ( ζ p ) has ramification index p ( p −
1) over Q p andwas represented in the computer with respect to the uniformizer π = ζ p −
1. Computing with precp -adic digits in Q p ( ζ p ) is prec · p ( p − π -adic digits. To verify the conjecture, we calculated the π -adic valuation of the difference L p ( χ, α, ψ i ω, ψ j ω, − τ ( ψ − i ) τ ( ψ − j ) log p ( u ∗ χψ i ,α )log p ( u ∗ χψ j ,α ) . (13)A number in our computer representation of Q p ( ζ p ) is 0 if it has π -adic valuation prec · p ( p − .1 F = Q ( √ , K = Q ( p √ , p = 5 In this example, Conjecture 4.6 is true because ρ is also the induction of a ray class character χ ′ of F ′ = Q ( i ) where p = 5 splits (and Conjecture 4.6 only depends on ρ ). To see this, define χ ′ sothat the fixed field of the kernel of χ ′ is K ′ = Q ( √ i ). Then since the fixed field of the kernelof ρ is M = K ( p − √
17) and we have the relation p √
17 + p − √
17 = √ i , a simplecalculation shows that Ind χ = ρ = Ind χ ′ . For a further analysis of this situation where there is aray class character of a real quadratic field and of an imaginary quadratic field where p splits, andsuch that the induction of the two ray class characters is the same, see chapter 5 of [13].We include this example because the units appearing are the Stark units from 2.3 associated tothe real quadratic field F = Q ( √ ψ is defined by ψ (2) = ζ . Let a = 1 + √
172 . Then the minimal polynomial ofthe Stark unit for K /F is x + ( − a − x ++(101815525268417913200 a + 158990319870506526445) x ++( − a − x ++(1212779745101402982169172133826675 a + 1893819622280672026587959027568110) x ++( − a − x ++(1212779745101402982169172133826675 a + 1893819622280672026587959027568110) x ++( − a − x ++(101815525268417913200 a + 158990319870506526445) x ++( − a − x + 1 . The data for this example is in the following table. α (i,j) π -adic valuation of (13)when prec =60 π -adic valuation of (13)when prec =63 F = Q ( √− , K = Hilbert class field of F , p = 5 The character ψ is defined by ψ (2) = ζ . The minimal polynomial of the Stark unit for K /F is34 − x + 65231675 x − x ++15533478425 x − x − x − x + − x − x + 3771011381950 x − x ++99067277500 x − x + 466875 x − . The data for this example is in the following table. α (i,j) π -adic valuation of (13)when prec =60 π -adic valuation of (13)when prec =72 -1 (1,2) 1135 1436-1 (1,3) 1135 1436-1 (1,4) 1135 1435-1 (2,3) 1136 1436-1 (2,4) 1135 1435-1 (3,4) 1135 1435When α = 1, we made the same calculation and got for (13) a p -adic number that is not closeto 0. This indicates that when F is imaginary quadratic and p is inert in F , the units that appearin Conjecture 4.6 may not come from the elliptic units from definition 2.10. For reference we givethe first 100 π -adic digits of the quantities in (13) for this example when α = 1: τ ( ψ − ) τ ( ψ − ) log p ( u ∗ χψ ,α )log p ( u ∗ χψ ,α ) =2 + π + 4 π + 3 π + 3 π + 4 π + π + 2 π + 4 π + 4 π + 2 π + 2 π + 3 π + π + π +3 π +3 π + π + π +3 π +3 π +3 π +3 π +4 π +3 π +2 π +2 π +2 π +2 π + π + π + π +2 π +3 π +4 π +3 π + π + π +2 π +2 π +4 π +3 π +2 π +3 π + π +2 π +2 π + π +3 π +2 π +2 π + π +2 π +3 π + π +3 π +2 π + π +2 π +4 π +4 π +2 π +4 π +2 π + 3 π + 3 π + 3 π + 2 π + 4 π + 2 π + π + 4 π + 2 π + O ( π ) (14) τ ( ψ − ) τ ( ψ − ) log p ( u ∗ χψ ,α )log p ( u ∗ χψ ,α ) =3 + 3 π + π + 2 π + 4 π + 4 π + 2 π + 3 π + 4 π + 3 π + 3 π + 4 π + 4 π + 3 π + π + π +3 π +4 π + π + π +4 π +4 π +3 π +3 π +4 π +4 π +2 π +2 π +4 π +4 π +3 π +3 π + π +4 π +2 π +2 π +2 π +3 π +3 π +2 π +4 π +3 π +3 π +4 π + π + π +2 π +2 π +2 π +3 π +3 π + π +2 π +4 π +4 π +2 π +3 π +3 π +3 π +2 π +3 π +4 π +4 π +2 π + 4 π + 4 π + π + π + 2 π + 2 π + 2 π + O ( π ) (15)35 ( ψ − ) τ ( ψ − ) log p ( u ∗ χψ ,α )log p ( u ∗ χψ ,α ) =4 + π + 4 π + π + 4 π + 3 π + 3 π + 4 π + 4 π + 3 π + 3 π + 4 π + π + π + 2 π +2 π + π + 3 π + π + π + 3 π + 3 π + 2 π + 4 π + π + 2 π + π + π + 2 π + 3 π + 2 π + π + π +3 π +4 π +2 π +4 π +3 π +4 π +2 π +4 π +4 π +3 π +3 π + π +4 π +3 π + π + π +4 π +2 π +2 π + π +4 π +4 π + π +3 π + π +2 π +3 π +2 π +2 π +3 π + π + π + 2 π + π + 2 π + 4 π + 2 π O ( π ) (16) τ ( ψ − ) τ ( ψ − ) log p ( u ∗ χψ ,α )log p ( u ∗ χψ ,α ) =4 + 2 π + 2 π + 4 π + π + 3 π + π + π + 3 π + π + 2 π + 2 π + 4 π + 4 π + 2 π +3 π +4 π + π +3 π +3 π + π +3 π + π + π +4 π +4 π +3 π +2 π +2 π +3 π + π +2 π +4 π +2 π + π +4 π +3 π +2 π +3 π +3 π + π + π +3 π +2 π + π +4 π +3 π +3 π +2 π +4 π +3 π + π +4 π +3 π +4 π +2 π + π +4 π +4 π +4 π +3 π +2 π +2 π +4 π + π + 3 π + 3 π + 2 π + 4 π + 2 π + O ( π ) (17) τ ( ψ − ) τ ( ψ − ) log p ( u ∗ χψ ,α )log p ( u ∗ χψ ,α ) =2 + 2 π + π + 3 π + π + π + 3 π + 2 π + 2 π + 4 π + 4 π + 2 π + 3 π + 3 π + 3 π +4 π +3 π +2 π +2 π +2 π +2 π + π +4 π +4 π + π + π + π + π +2 π +2 π +4 π + π +4 π + π +4 π +2 π + π +3 π +4 π +4 π + π +3 π +2 π +2 π +4 π +2 π +2 π +4 π + π + π + 3 π + 4 π + 3 π + 4 π + 2 π + 2 π + 2 π + 2 π + O ( π ) (18) τ ( ψ − ) τ ( ψ − ) log p ( u ∗ χψ ,α )log p ( u ∗ χψ ,α ) =3 + 4 π + 3 π + π + π + 2 π + 2 π + π + 4 π + 2 π + 4 π + 4 π + 2 π + π + π +2 π + 2 π + π + π + 3 π + 2 π + π + 3 π + 4 π + π + 3 π + π + 3 π + 4 π + 4 π + π +4 π + π + π +3 π +4 π + π +4 π +4 π +4 π +2 π +3 π +4 π +2 π +3 π +4 π +4 π +3 π + π + π + 3 π + 4 π + π + 3 π + 2 π + π + π + π + 4 π + 2 π + π + 2 π + 2 π +2 π + 3 π + 2 π + 4 π + O ( π ) (19)36 p ( χ, α, ψ ω, ψ ω,
0) =1 + 3 π + 3 π + π + 2 π + 4 π + 4 π + 2 π + 2 π + π + 3 π + π + π +3 π +2 π +2 π + π +2 π +3 π +4 π +2 π +3 π + π + π +4 π +2 π +3 π + π +2 π + π +4 π +3 π +3 π +4 π +2 π +3 π +4 π + π +3 π +2 π +3 π +3 π +3 π +4 π +2 π + π +4 π +4 π +3 π +3 π +3 π + π + π + π +2 π +3 π + π +3 π +4 π +4 π + π + 2 π + 3 π + 4 π + 3 π + π + O ( π ) (20) L p ( χ, α, ψ ω, ψ ω,
0) =1 + π + 4 π + 3 π + 3 π + 4 π + 4 π + 2 π + 4 π + 4 π + 2 π + 4 π +3 π + π + π +3 π +3 π + π + π +3 π +3 π +3 π +4 π +3 π +2 π +3 π +2 π +4 π +3 π +4 π +3 π +2 π +4 π +3 π + π + π +2 π +3 π +2 π +2 π + π +2 π +3 π + π +2 π + π + π +4 π +2 π + π +3 π +4 π + π +4 π +3 π +3 π +4 π +4 π +4 π +3 π +3 π + 4 π + π + 2 π + π + 2 π + 4 π + 3 π + 2 π + 2 π + O ( π ) (21) L p ( χ, α, ψ ω, ψ ω,
0) =1 + 4 π + π + 4 π + π + π + 2 π + 2 π + π + π + π + 2 π +2 π + π +3 π +4 π +3 π +3 π +4 π +2 π + π +4 π +3 π +4 π +3 π +3 π +3 π +4 π +4 π +3 π +3 π +4 π +4 π +4 π +3 π +2 π +2 π +2 π +4 π + π +4 π +2 π + π +2 π +3 π +2 π +2 π +4 π + π +2 π +4 π +3 π + π +4 π +2 π +4 π +2 π +2 π +4 π + π +2 π + π + 3 π + π + 4 π + π + π + π + 4 π + 4 π + O ( π ) (22) L p ( χ, α, ψ ω, ψ ω,
0) =1 + 3 π + 3 π + π + 2 π + 4 π + 4 π + 2 π + 2 π + π + 3 π + π + π +3 π +3 π +2 π + π +2 π +3 π + π +2 π +3 π + π +2 π +4 π +2 π +3 π + π +2 π +3 π + π + π +4 π +2 π +3 π +2 π +2 π +4 π +2 π +4 π +3 π +3 π + π + π +3 π +3 π + π +2 π +4 π +4 π + π + π +4 π +4 π +4 π +3 π +2 π + π + π +2 π +4 π + 2 π + 3 π + O ( π ) (23) L p ( χ, α, ψ ω, ψ ω,
0) =1 + π + 4 π + 3 π + 3 π + 4 π + 4 π + 2 π + 4 π + 4 π + 2 π + 4 π +3 π + π + π +3 π +3 π + π + π +3 π +3 π +3 π +4 π +3 π +2 π +3 π +2 π +4 π +3 π +4 π +3 π +2 π +4 π +3 π + π + π +2 π +3 π +2 π +2 π + π +2 π +3 π + π + π + π +4 π +2 π +3 π +3 π +4 π + π +4 π +3 π +3 π +2 π +3 π + π +4 π +2 π +2 π + 4 π + π + 2 π + 3 π + π + 3 π + O ( π ) (24)37 p ( χ, α, ψ ω, ψ ω,
0) =1 + 3 π + 3 π + π + 2 π + 4 π + 4 π + 2 π + 2 π + π + 3 π + π + π +3 π +2 π +2 π + π +2 π +3 π +4 π +2 π +3 π + π + π +4 π +2 π +3 π + π +2 π + π +4 π +3 π +3 π +4 π +2 π +3 π +4 π + π +3 π +2 π +3 π +3 π + π +4 π + π + π + 4 π + 3 π + 3 π + 3 π + π + 4 π + 2 π + 3 π + π + π + 2 π + 3 π + 2 π + π + π + 2 π + 2 π + 3 π + O ( π ) . (25) F = Q ( √− , K = Hilbert class filed of F , p = 3 This example is interesting because it does not satisfy the assumption, p ∤ [ M : Q ] (in this example M = K ). In this example p = 3 which divides [ M : Q ] = 6. The example does satisfy the condition∆ = Gal( M / Q ) ∼ = Gal( M/ Q ) × Gal( Q / Q ) = ∆ × Γ . The character ψ is defined by ψ (2) = ζ . The minimal polynomial of the Stark unit for K /F is x − x − x − x + 60156 x + 117180 x + 25704 x − x + 5022 x − . The data for this example is in the following table. α (i,j) π -adic valuation of (13)when prec =60 π -adic valuation of (13)when prec =77 -1 (1,2) 352 441When α = 1, as in the previous example, we made the same calculation and got for (13) a p -adic number that is not close to 0. Again, this indicates that when F is imaginary quadratic and p is inert in F , the units that appear in Conjecture 4.6 may not come from the elliptic units fromdefinition 2.10. For reference we give the first 100 π -adic digits of the quantities in (13) for thisexample when α = 1: τ ( ψ − ) τ ( ψ − ) log p ( u ∗ χψ ,α )log p ( u ∗ χψ ,α ) =2 + 2 ∗ π + π + π + π + π + π + 2 ∗ π + 2 ∗ π + 2 ∗ π + 2 ∗ π + π + 2 ∗ π + 2 ∗ π + π + π +2 ∗ π + π +2 ∗ π + π +2 ∗ π +2 ∗ π + π + π +2 ∗ π + π +2 ∗ π +2 ∗ π +2 ∗ π +2 ∗ π + π + 2 ∗ π + π + 2 ∗ π + 2 ∗ π + π + 2 ∗ π + 2 ∗ π + π + 2 ∗ π + 2 ∗ π + π + 2 ∗ π + 2 ∗ π + π +2 ∗ π +2 ∗ π +2 ∗ π + π +2 ∗ π + π +2 ∗ π + π +2 ∗ π +2 ∗ π +2 ∗ π + π + π +2 ∗ π +2 ∗ π + 2 ∗ π + 2 ∗ π + O ( π ) (26) L p ( χ, α, ψ ω, ψ ω,
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Joseph Ferrara,
Department of Mathematics, University of California, San Diego, 9500 Gilman Drive
E-mail address: [email protected]@ucsd.edu