Derived Beilinson–Flach elements and the arithmetic of the adjoint of a modular form

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Introduction
The purpose of this article is to prove two results conjectured by H. Darmon, A. Lauder and one of the present authors in [DLR4] and [DR1] respectively, concerning the interplay between the arithmetic of units in number fields, the theory of Coleman iterated integrals, and the Rankin p-adic L-function and Euler system of Beilinson-Flach elements associated to a pair (g, h) of Hida families of modular forms.
The first theorem of this article is Theorem A (together with its equivalent form given in Theorem A ), which yields a proof of the main conjecture of [DLR4] in the adjoint setting.Although Beilinson-Flach elements are certainly behind the scenes, we find it interesting to remark that this Euler system is involved neither in the statement nor in the proof we provide of Theorems A, A , and can be phrased in purely analytic terms by means of p-adic iterated integrals and p-adic L-functions.It is only later in the note that we explore the consequences that Theorem A has on the weight one specializations of the Euler system of Beilinson-Flach elements, as described in Theorems B and C.
Let χ be a Dirichlet character of level N ≥ 1 and let M k (N, χ ) (resp.S k (N, χ )) denote the space of (resp.cuspidal) modular forms of weight k, level N and nebentype χ.Let g = n≥1 a n q n ∈ S 1 (N, χ ) be a normalized newform and let g * = g ⊗ χ −1 denote its twist by the inverse of its nebentype.Let g : Gal(H g /Q) → GL(V g ) GL 2 (L), ad 0 (g) : Gal(H /Q) → GL(ad 0 (g)) GL 3 (L) denote the Artin representations associated to g and its adjoint, respectively.Here H g ⊇ H denote the finite Galois extensions of Q cut out by these representations, and L is a sufficiently large finite extension of Q containing their traces.The three-dimensional representation ad 0 (g) may be identified with the subspace of End(V g ) of null-trace endomorphisms, on which G Q acts by conjugation.There is a natural decomposition of L[G Q ]-modules ad(g) := End(V g ) = L ⊕ ad 0 (g), where L stands for the trivial representation.
Fix a prime p N and let S oc k (N, χ ) denote the space of overconvergent p-adic modular forms of weight k, tame level N and character χ.Fix an embedding Q ⊂ Qp and let L p denote the completion of L in Qp .
Label and order the roots of the p-th Hecke polynomial of g as X 2 −a p (g)X+χ (p) = (X − α)(X − β).We assume throughout that (H1) the reduction of g mod p is irreducible; (H2) g is p-distinguished, i.e. α = β (mod p), and (H3) g is not induced from a character of a real quadratic field in which p splits.

Let
g α (q) = g(q) − βg(q p ) denote the p-stabilization of g on which the Hecke operator U p acts with eigenvalue α.Enlarge L if necessary so that it contains all Fourier coefficients of g α .
As shown in [BeDi] and [DLR1], the above hypotheses ensure that any generalized overconvergent modular form with the same generalized eigenvalues as g α is classical, and hence simply a multiple of g α .This allows one to define a canonical projector e g α : S oc 1 (N, χ ) → L p which extracts from an overconvergent modular form its coefficient at g α with respect to an orthonormal basis: cf.[DLR1] for more details.
One can attach a p-adic invariant I p (g) ∈ L p to g α as follows. Let denote the models over Q of the (affine and projective, respectively) modular curves classifying pairs (A, P ) where A is a (generalized) elliptic curve and P is a point of order N on A.
For a ∈ (Z/NZ) × , let g a;N be Kato's Siegel unit whose q-expansion is given by where ζ N is a fixed primitive N -th root of unity.The unit g a;N can naturally be viewed as belonging to Q ⊗ O × Y , and its q-expansion is defined over Q(µ N ).Set and define E 0 = log p g.This is a locally analytic modular form of weight 0 on Y .It follows from ( 1) and (2) that the logarithmic derivative of g is where is the classical Eisenstein series of weight 2. Let E [p] 0 := p n a n (E 0 )q n denote the p-depletion of E 0 ; this is an overconvergent modular form and we may define I p (g) := e g α (E [p] 0 • g α ). (4) The invariant I p (g) can be recast as a p-adic iterated integral, denoted I p (E 2 , g, g * ) in [DLR4].The first main result of this note, conjectured in [DLR4], asserts that I p (g) is equal to the following motivic expression.As shown in [DLR4,Lemma 1.1], we have Fix a generator u of (O × H ⊗ ad 0 (g)) G Q and also an element v of (O × H [1/p] × ⊗ ad 0 (g)) G Q in such a way that {u, v} is a basis of (O H [1/p] × /p Z ⊗ ad 0 (g)) G Q .The element v may be chosen to have p-adic valuation ord p (v) = 1, and we do so.
Viewed as a G Q p -module, ad 0 (g) decomposes as ad 0 (g) = L ⊕ L α⊗ β ⊕ L β⊗ ᾱ , where each line is characterized by the property that the arithmetic Frobenius Fr p acts on it with eigenvalue 1, α/β and β/α, respectively1 .Let H p denote the completion of H in Qp and let denote the projection of the elements u and v in (H × p ⊗ ad 0 (g)) G Qp to the above lines: cf. ( 50 × det log p u 1 log p (u α⊗ β ) log p v 1 log p (v α⊗ β ) . (5) This is [DLR4,Conjecture 1.2] specialized to the pair (g, g * ): note that both statements are equivalent because Let L(ad 0 (g α )) be Hida's analytic L-invariant attached to the adjoint Galois representation of g α ; cf. ( 23).As we recall in Proposition 2.5, there are several equivalent definitions of this invariant.
These L-invariants are in general difficult to compute, and explicit expressions for them are rather rare.The proof of the above result, which we describe further below in this introduction, shows that Theorem A may be recast as a formula for L(ad 0 (g α )).Since this might be of independent interest, specially for the reader less acquainted with the Stark elliptic conjectures of [DLR1], [DLR4] and more familiar with the theory of L-invariants, we quote it below as a separate statement: Remark 1.1.The prototypical case where an explicit formula for the L-invariant is known arises of course when E/Q is an elliptic curve of split multiplicative reduction at p.In that case L p (E) = log p (q E ) ord p (q E ) where q E ∈ pZ p is a Tate period for E. Observe that this expression and the one in Theorem A are in fact very similar, as both may be recast as where κ is an element in H 1 (Q p , Q p (1)) arising from some global class in motivic cohomology.In the classical case one has κ = q E , which might be regarded as an element in the extended Mordell-Weil group Ẽ(Q), or Nekovář's extended Selmer group of E. In Theorem A one has, up to scalar, κ = log p (v α⊗ β )u 1 − log p (u α⊗ β )v 1 in the unit group of H ; note that recipe (7) indeed gives rise to (6) because ord p (u 1 ) = 0 and ord p (v 1 ) = 1.
The second main result of this note may be regarded as a conceptual explanation of Theorem A, as it establishes a connection between the iterated integral I p (g), a special value of Hida's p-adic Rankin L-function associated to the pair (g, g * ), and the generalized Kato classes arising from the three-variable Euler system of Beilinson-Flach elements constructed in [KLZ1].
In order to describe it more precisely, let ] denote the Iwasawa algebra and denote W = Spf( ) the associated weight space.As shown by Wiles [Wi] (cf.also [KLZ1,§7.2]for the normalizations we adopt), associated to g α there are: (1) a finite flat extension g of , giving rise to a covering w : (2) a family of overconvergent p-adic ordinary modular forms g with coefficients in g specializing to g α at some point y 0 ∈ W g of weight w(y 0 ) = 1; our running assumptions imply that the above Hida family passing through g α is unique by [BeDi]; (3) a Galois representation g : G Q → GL 2 ( g ) characterized by the property that all its classical specializations coincide with Deligne's Galois representation associated to the corresponding specialization of the Hida family.
Let V g denote the rank two g -module realizing the Galois representation g as specified e.g. in [KLZ1,§7.2],where this is denoted M(g) * .Let g * := g⊗χ −1 denote the twist of g by the inverse of its tame nebentype.Note that g * specializes at y 0 to the eigenform g α ⊗χ −1 = g * 1/β , namely the p-stabilization of g * on which U p acts with eigenvalue 1/β.Let denote the p-adic cyclotomic character.Let ε cyc be the composition of ε cyc with the natural inclusion Z × p ⊂ × taking z to the group-like element [z] in × .Define the three-variable Iwasawa algebra gg * := g ⊗Z p g * ⊗Z p and the parametrized by the triple product of weight spaces W gg * := W g × W g × W, where the first two variables are afforded by the weight of the Hida families g and g * , and the third one is the cyclotomic variable.
The common key strategy in several recent works on the arithmetic of elliptic curves ( [BDR], [DR2], [LLZ], [KLZ1], [CH]) consists in proving a reciprocity law relating a suitable specialization of a -adic class as the one above to the critical value of the underlying classical L-function.The non-vanishing of the appropriate (classical or p-adic) L-value can then be invoked to show that the associated global cohomology class is not trivial, and one can derive striking arithmetic consequences from this (cf.e.g.[DR2, Theorems A and B], [KLZ1, Corollary C]).
Our setting differs in several key aspects from the previous ones.Namely, while it is also natural to consider the specialization of the Euler system of Beilinson-Flach elements at the point (y 0 , y 0 , 0), one can show that this global cohomology class is trivial and thus no arithmetic information can be extracted directly from it: cf.Theorem 3.5.The fact that κ(g α , g * 1/β ) = 0 might be regarded as an exceptional zero phenomenon, albeit quite different from the ones one typically encounters in e.g. the pioneering work [MTT] of Mazur, Tate and Teitelbaum, because our case corresponds to a point lying outside the classical region of interpolation of the associated Hida-Rankin p-adic Lfunction.But it still keeps the same flavor, because the vanishing of the global cohomology class ( 8) is caused by the cancellation of the Euler-like factor at p arising in the interpolation process of the construction of the Euler system.To be more precise, the situation is more delicate than that: for classical points y ∈ W g of weights > 1, the triples (y, y, − 1) do lie in the region of geometric interpolation of the Euler system of Beilinson-Flach elements, and the main theorem of [KLZ1] applies and implies that κ(g, g * )(y, y, − 1) = 0.The vanishing of κ(g α , g * 1/β ) then follows by a density argument exploiting various -adic Perrin-Riou regulators; we refer to §3.2 for more details.
In [DR1] a conjecture was laid proposing an explicit description of the generalized Kato classes arising from the Euler systems of diagonal cycles of [DR2].This conjecture, together with the expected (but so far also unproved) behavior of certain periods arising from Hida theory, is shown to imply the main conjecture of [DLR1] for twists of elliptic curves by Artin representations.This is seen to provide a conceptual interpretation of the numerical examples computed in [DLR1].
At the time of writing [DR1], the authors had in mind that a parallel formulation should also hold for the Euler system of Beilinson-Flach elements attached to pairs (g, h) of eigenforms of weights (1, 1) in connection with [DLR4].This is indeed the case for arbitrary pairs (g, h) such that h = g * , as there are no exceptional zero phenomena and the constructions and conjectures [DR1] can be adapted to the setting of units of number fields.We refer to [RR] for the details.
However, pairs (g, g * ) present a completely different scenario, as we already hinted at above.Not only κ(g α , g * 1/β ) = 0, but it also turns out that the derived class κ (g α , g * 1/β ) is not crystalline at p, as opposed to the set-up of [DR1] and [RR].Yet it is still possible to formulate an analogous conjecture, proposing an explicit description of κ (g α , g * 1/β ) in terms of the ad 0 (g)-isotypical component of the group of p-units of H .We can in fact prove it, giving rise to the second main result of this paper.
Let H 1 f,p (Q, ad 0 (g)( 1)) denote the subspace of H 1 (Q, ad 0 (g)( 1)) consisting of classes that are de Rham at p and unramified at all remaining places.Kummer theory (cf.Proposition 3.11 below) gives rise to a canonical isomorphism and we shall use (9) throughout to identify these two spaces.
Theorem B. Assume that L(ad 0 (g α )) = 0.The equality Here it again becomes apparent that there are notable differences between the phenomena occurring in this article and say [BDR], [DR2], [KLZ1].In the cited works, the nonvanishing of the central critical L-value is shown to imply the triviality of the Mordell-Weil group; conversely, the vanishing of the central L-value should imply the non-triviality of the Mordell-Weil group.In contrast, in our setting we have L(g, g * , 1) = 0 and nonetheless the associated motivic groups O × H [ad 0 (g)] and O H [1/p] × [ad 0 (g)] have positive rank.This is ultimately explained by the fact that L(g, g * , s) = L(ad 0 (g), s) • ζ (s) factors as the product of two zeta functions having a simple zero and a simple pole at s = 1, respectively.Remark 1.2.As we show in §5, it is actually possible to prove that L(ad 0 (g α )) = 0 in many dihedral cases, where g is induced from a finite order character of the Galois group of a (real or imaginary) quadratic field.We refer to the cited papers for more details.On the other hand, when g is exotic, we still expect the L-invariant not to vanish systematically, but proving this properly appears to be a less accessible question in the theory of transcendental p-adic numbers.
The main ingredients in the proof of Theorems A, A and B are the following: (I) Let L p (g, g * ) be the three-variable Hida-Rankin p-adic L-function as introduced in [Hi1] or [Das2,§3.6],and let L p (g α , g * 1/β , s) := L p (g, g * )(y 0 , y 0 , s) denote the restriction to {y 0 , y 0 } × W. This function satisfies a functional equation relating the values at s and 1 − s.
As explained in [DLR4], the p-adic iterated integral introduced in (4) may be recast as a special value of a p-adic L-function, namely [DLR4,Lemma 4.2] asserts that under our running assumptions H1 to H3 that are in force throughout the article, we have Define S Hida ⊂ W g × W g × W as the surface for which the collection of crystalline points (y, z, s) of weights ( , m, m) is Zariski dense.The Euler-like factor at p showing up in the interpolation formula of Hida-Rankin's p-adic L-function turns out to be a rigidanalytic Iwasawa function when restricted to S Hida .In §2.3 we recall how this observation allows one to conclude thanks to [Das2] that Computing the latter derivative is a question which can be reduced to a problem in Galois deformation theory as in [DLR2], although the methods there do not apply directly and need to be adapted in order to cover our setting.This is carried out in §4, and allows us to prove Theorems A and A , as we spell out in detail in §5.
(II) The main result of §3 is Theorem 3.7, which we state here in slightly rough terms (as the Bloch-Kato logarithm map appearing below has not been specified; cf.loc.cit.for more details): Theorem C (Theorem 3.7).The derived Beilinson-Flach element satisfies We may see this as an exceptional zero formula, reminiscent of the main theorem of [GS], which asserts that for an elliptic curve E/Q with split multiplicative reduction at p, we have We may thus invoke Theorem A, which combined with (12) implies that In light of the general properties satisfied by Beilinson-Flach elements established in [KLZ1], this allows us to prove in §5 that the following equality holds in up to a factor in L × , as claimed.
We close the paper in §6 by pointing out the connection between our derived Beilinson -Flach classes, Darmon-Dasgupta units and the Artin p-adic L-functions associated to the motives in play.We prove a factorization theorem in the CM case, and explain how our main results shed some light on the much more intriguing RM setting.
As a concluding remark, we hope the approach introduced in this paper may be adapted to prove other instances of variants of the Elliptic Stark Conjecture in the presence of exceptional zeroes.We also hope that the derived global cohomology classes appearing in this work may also be useful for the on-going research project [DHRV] of Darmon, Harris, Rotger and Venkatesh on derived Hecke algebras in weight one.

Hida families
Fix an algebraic closure Q of Q.For a number field K, let G K := Gal( Q/K) denote its absolute Galois group.Fix a prime p and an embedding Q → Qp , and let ord p denote the resulting p-adic valuation on Q× , normalized in such a way that ord p (p) = 1.
Let g be a newform of weight k ≥ 1, level N and character χ, with Fourier coefficients in a finite extension L of Q. Label the roots of the p-th Hecke polynomial of g as α g , β g with ord p (α g ) ≤ ord p (β g ).By enlarging L if necessary, we shall assume throughout that L contains α g , β g , the N-th roots of unity and the pseudo-eigenvalue λ N (g) with respect to the Atkin-Lehner operator W N (cf.[AL]).
Let L p denote the completion of L in Qp and let V g denote the two-dimensional representation of G Q with coefficients in L p associated to g as defined e.g. in [KLZ1,§2.8],where it is denoted M L p (g) * .If g is ordinary at p, there is an exact sequence of where ψ g is the unramified Galois character of ] is called the weight space attached to , and their A-valued points over a p-adic ring A are given by W(A) = Hom alg ( , A) = Hom grp (Z × p , A × ).The weight space is equipped with a distinguished class of arithmetic points ν s,ε indexed by integers s and Dirichlet characters ε : We let W cl denote the subset of W formed by such arithmetic points.When ε = 1 is the trivial character, we denote the point ν s,1 simply as ν s or even s by a slight abuse of notation.
If ˜ is a finite flat algebra over , there is a natural finite map W := Spf( W) w → W, and we say that a point x ∈ W is arithmetic of weight s and character ε if w(x) = ν s,ε .As in the introduction, let ε cyc denote the p-adic cyclotomic character and ε cyc be the composition of ε cyc with the natural inclusion Z × p ⊂ × taking z to the group-like element [z] in × .
Let N ≥ 1 be an integer not divisible by p, and let χ : (Z/NZ) × → C × p be a Dirichlet character.Let g be a Hida family of tame level N and tame character χ as defined and normalized in [KLZ1, §7.2].Let g be the associated Iwasawa algebra (cf.Def.7.2.5 of loc.cit.), which is finite and flat over .As in [KLZ1, §7.3], we may specialize g at any arithmetic point x ∈ W g = Spf( g ) of weight k ≥ 2 and character ε : and obtain a classical p-ordinary eigenform g x in the space S k (Np r , χ ε) of cusp forms of weight k, level Np r and nebentype χ ε.
Definition 2.1.Let x ∈ W g be an arithmetic point of weight k ≥ 1 and character ε.We say x is crystalline if ε = 1 and there exists an eigenform g • x of level N such that g x is the ordinary p-stabilization of g • x (which given the previous condition, is automatic if k > 2, but not necessarily if k ≤ 2).We denote by W • g the set of crystalline arithmetic points of W g .
As shown by Wiles [Wi], [KLZ1, §7.2] and already recalled in the introduction, a Hida family g as above comes equipped with a free g [G Q ]-module V g of rank two, yielding a Galois representation g : G Q → GL 2 ( g ), all whose classical specializations recover the p-adic Galois representation associated to the classical specialization of g.We assume throughout that the mod p residual representation ¯ g associated to any of its classical specializations is irreducible, and that the semi-simplification of ¯ |G Qp is non-scalar, as in hypotheses (H1)-(H2) in the introduction.
Similarly to (13), the restriction to where V + g and V − g are flat g [G Q p ]-modules, free of rank one over g .If we let ψ g denote the unramified character of G Q p taking the arithmetic Frobenius element Fr p to a p (g), then Let Z ur p be the ring of integers of the completion of the maximal unramified extension of Q p .Given a finite-dimensional G Q p -module V with coefficients over a finite extension L p of Q p , the de Rham Dieudonné module associated to V is defined as where B dR is Fontaine's field of de Rham periods.If V is unramified there is a further canonical isomorphism where V int is an integral lattice in V .Similarly, if V is a free module over an Iwasawa algebra equipped with an unramified action of G Q p , we may define Given a newform g ∈ S k (N, χ ) as at the beginning of this section, D(V g ) is an L pfiltered vector space of rank 2. Set g * = g ⊗ χ ∈ S k (N, χ ).Poincaré duality induces a perfect pairing If g is ordinary at p, then (13) gives rise to an exact sequence of Dieudonné modules where D(V + g ) and D(V − g ) have rank 1 over L p .If k ≥ 2, Faltings' comparison theorem allows one to associate to g a regular differential form ω g ∈ Fil(D(V g )), which induces a linear form There is also the differential form η g , characterized by the properties that it spans the line D(V + g ) and η g , ω g * = 1.It again gives rise to a linear functional [KLZ1], the differential forms (or linear functionals) ω g and η g vary in families.In order to recall this more precisely, let g be a Hida family of tame level N and tame character χ as above.Set g * = g ⊗ χ .Let Q g denote the fraction field of g , and set . By [KLZ1, Proposition 10.1.1],there exist: such that for every y ∈ W • g , the specialization of ω g at y is the linear form where Pr α * is the p-stabilization pull-back isomorphism defined in loc.cit.Note that this makes sense: since L p is assumed to contain the N-th roots of unity, D(U + g * y ) and D(V + g * y ) are the same module up to a shift in their filtration.
such that for every y ∈ W • g we have where λ N (g • y ) stands for the pseudo-eigenvalue of g • y , and 2.2.Hida-Rankin's three-variable p-adic L-function ] be a pair of Hida families of tame level N and tame characters χ g , χ h respectively.As in the introduction, set gh := g ⊗Z p h ⊗Z p and W gh := Spf( gh ).Let Q gh denote the fraction field of gh , and set Definition 2.2.The critical range is the set of points (y, z, σ ) ∈ W • gh of weights ( , m, s) such that , m ≥ 2 and m ≤ s < .
The geometric range is defined to be the set of points (y, z, σ ) ∈ W • gh of weights ( , m, s) such that , m ≥ 2 and 1 ≤ s < min( , m).
Hida [Hi1], [Hi2] constructed a three-variable p-adic Rankin L-function L p (g, h) on W gh , interpolating the algebraic parts of the critical values L(g • y , h • z , s) for every triple of classical points (y, z, s) in W • gh lying in the critical range.More precisely, [Hi2, Theorem 5.1d] asserts the following.
Theorem 2.3 (Hida).There exists a unique element L p (g, h) ∈ Q gh whose value at any (y, z, s) ∈ W • gh of weights ( , m, s) in the critical range is well-defined and equal to where C is a non-zero algebraic number in the finite extension Q(g Let g and h be classical specializations of the families g and h at some point (y 0 , z 0 ) ∈ W cl g × W cl h of weight ( , m).We denote by L p (g, h, s) the restriction of L p (g, h)(y, z, s) to the line (y 0 , z 0 , s).As stated e.g. in [Das2, §9.2], this p-adic L-function satisfies the functional equation where

Improved p-adic L-functions
As before, let g ∈ g [[q]] be a Hida family of tame level N and tame character χ = χ g , and set h = g * := g ⊗ χ −1 .As in the introduction, define the surface Note that this is a subvariety of W gg * all of whose crystalline arithmetic points have weights ( , m, m) for some , m ≥ 1.
The restriction to S Hida of the second multiplier in the Euler-like factor (20) appearing in the interpolation formula for Hida-Rankin's p-adic L-function is This expression interpolates to an Iwasawa function in g × g , which by abuse of notation we continue to denote with the same symbol.One naturally expects 1 − α g • z /α g • y should divide the two-variable p-adic L-function L p (g, g * )(y, z, s), where (y, z) vary in W g ×W g and s = w(z) is determined by the weight of z.Hida [Hi2] proved the following stronger statement: Theorem 2.4 (Hida).Let Lp (g, g * ) be the unique element in the fraction field of g ⊗ g such that Then, for fixed y 0 ∈ W • g , the one-variable meromorphic function on W g has a simple pole at z = y 0 whose residue is a non-zero explicit rational number.
Starting from the adjoint representation attached to any classical specialization g = g y 0 of the Hida family g at some arithmetic point y 0 ∈ W cl g , Hida defined an analytic Linvariant, which can be recast in several equivalent ways (cf. the works of Hida, Harron, Citro and Dasgupta (cf.[Hi3], [Ci], [Das2]).We may define it for instance as where recall α g = a p (g) ∈ g is the Iwasawa function given by the eigenvalue of the Hecke operator U p acting on g, and α g is its derivative.Let L p (ad 0 (g y 0 ), s) denote Hida-Schmidt's p-adic L-function associated to the adjoint of the ordinary eigenform g y 0 (cf.[Sc], [Hi3]).The argument below is mainly due to Citro and Dasgupta, but since in their works they often assume that ≥ 2, we include it in order to ensure that it holds as well at weight 1, which is the case we mostly focus on.The main point is that the objects in play all vary in Hida families.
Proposition 2.5.For a crystalline classical point y 0 ∈ W • g of weight ≥ 1, we have up to a non-zero rational constant.
Proof.The first equality follows from Theorem 2.4, which amounts to saying that L p (g, g * )(y 0 , y 0 , ) = lim vanishes at z = y 0 and Lp (g, g * )(y 0 , z) has a pole at z = y 0 given by a non-zero rational number, the value of the previous limit agrees, modulo L × , with the derivative of the first factor, i.e., L p (g, g * )(y 0 , y 0 , ) = −α g (y 0 ) α g (y 0 ) .

Derived Beilinson-Flach elements
3.1.The three-variable Euler system of Kings, Lei, Loeffler and Zerbes Let g and h be a pair of p-adic cuspidal Hida families of tame conductor N and tame nebentype χ g and χ h as in §2.1.As in the Introduction, and keeping the notations of the previous section, define the gh -module This -adic Galois representation is characterized by the property that for any (y, z, σ ) ∈ W • gh with w(σ ) = ν s with s ∈ Z, (24) specializes to the (1 − s)-th Tate twist of the tensor product of the Galois representations attached to g y and h z .Fix c ∈ Z >1 such that (c, 6pN g N h ) = 1.[KLZ1, Theorem A] yields a three-variable -adic global Galois cohomology class that is referred to as the Euler system of Beilinson-Flach elements associated to g and h.We denote by κ c p (g, h) ∈ H 1 (Q p , V gh ) the image of κ c (g, h) under the restriction map.Since c is fixed throughout, we may sometimes drop it from the notation.This constant does make an appearance in fudge factors accounting for the interpolation properties satisfied by the Euler system, but in all cases we are interested in these fudge factors do not vanish and hence do not pose any problem for our purposes.
Given a crystalline arithmetic point (y, z, s) ∈ W • gh of weights ( , m, s), set for notational simplicity throughout this section g = g • y , h = h • z .With these notations, g y (resp.h z ) is the p-stabilization of g (resp.h) with U p -eigenvalue α g (resp.α h ). Define as the specialization of κ(g, h) at (y, z, s).
If one further assumes that (y, z, s) lies in the geometric range, Kings, Loeffler and Zerbes [KLZ2] showed that the cohomology group appearing in (25) also hosts a canonical Rankin-Eisenstein class, denoted Eis This class is attached to the classical pair (g, h) and can be constructed purely by geometric methods, without appealing to the variation of (g y , h z ) in p-adic families.It is for this reason that in fact the classes Eis [g,h,s] et lie in the Bloch-Kato Selmer subgroup We refer to [KLZ2,§5] and [KLZ1,Definition 3.3.2]for the precise statements.
Since both κ(g y , h z , s) and Eis [g,h,s] et live in the same space, it makes sense to ask whether they are related.This is the content of [KLZ1,Theorem A (8.1.3)]: gh lies in the geometric range.Then where In particular κ(g y , h z , s) lies in The following proposition recalls the existence of the so-called Perrin-Riou big logarithm, interpolating the Bloch-Kato logarithm log BK and dual exponential map exp * BK associated to the classical specializations of a -adic representation of G Q p .We refer to [BK] and [Bel] for an introduction to p-adic Hodge theory and the definitions of these maps.
Recall the unramified character ψ g of G Q p taking a Frobenius element Fr p to a p (g), and as before, let ε g be the composition of the cyclotomic character ε cyc with the natural inclusion of rank one over the two-variable Iwasawa algebra g ⊗ h .In light of (15), the Galois action on V −+ gh is given by the character It follows that module and we can thus invoke its -adic Dieudonné module as defined in (16).

Proposition 3.2 ([KLZ1, Theorem 8.2.8]).
There is an injective morphism of gh -modules ) and all (y, z, s) ∈ W • gh of weights ( , m, s): Here, log BK and exp * BK stand for the Bloch-Kato logarithm (resp.dual exponential) associated to the Dieudonné module of the p-adic representation V − g y ⊗ U + h z (1 − s).Remark 3.3.In the case where α g β h = p s , we implicitly understand that the Euler factor in the denominator appears on the left hand side of the equality.
As shown in [KLZ1,Theorem 8.1.7],there is an injection As a consequence, the projection of κ p (g, h) to ) and we may hence denote it κ −+ p (g, h).Let λ N (g) denote the -adic pseudo-eigenvalue of g as defined in [KLZ1,§10], interpolating the Atkin-Lehner pseudo-eigenvalues of the classical specializations of g.Recall from ( 18) and (19) Ohta's families of differential forms η g ∈ D(U + g ) and ω h ∈ D(V − h ).It follows from the properties recalled in loc.cit.that there exists a homomorphism of gh -modules where, recall again, The following explicit reciprocity law is [KLZ1, Theorem 10.2.2].

The self-dual case
Let g be a Hida family of tame conductor N and tame nebentype χ, and set again h = g * = g ⊗ χ.Define the curve Note that C is a finite cover of the line in W 3 given as the set of regular points in the Zariski closure of the set of points of weights ( , , − 1) for some ≥ 1.
Theorem 3.5.The restriction of κ p (g, g * ) to C is zero.
Finally, note that there is an exact sequence The first map above is injective since as follows again from the description of V −+ gg * given in (29) (and similarly for V +− gg * and V −− gg * ).Since we have already shown that κ −− p (g, g * ) |C = κ −+ p (g, g * ) |C = κ +− p (g, g * ) |C = 0, this implies that the image of κ(g, g * ) |C in the rightmost term of (37) vanishes.Hence, κ(g, It follows from [KLZ1,Theorem 8.2.3,Remark 8.2.4] that the latter space is isomorphic to g , and thus κ(g, g * ) |C is zero if and only if infinitely many of its specializations are, which is the case as already argued above.

A derived system of Beilinson-Flach elements
Keep the notations and assumptions as in previous sections.Theorem 3.5 above establishes the vanishing of the local cohomology class κ p (g, g * ) along C and it is thus natural to ask about the existence of a derived cohomology class κ p (g, g * ) on a proper subspace of three-dimensional weight space W gg * containing C, bearing a reciprocity law with Hida-Rankin's improved p-adic L-function.
The purpose of this section is to make this construction explicit.Consider the surface which is a finite cover of the plane in W 3 arising as the Zariski closure of points of weights ( , m, m − 1) for some , m ≥ 1.Note that S obviously contains the curve C. Let V gg * |S denote the restriction of V gg * to the surface S, and let V gg * |C denote its restriction to C. The following proposition establishes the existence of a class κ p (g, g * ) ∈ H 1 (Q p , V gg * |S ) that may be regarded as the derivative of κ p (g, g * ) along the z-direction.
We shrink the weight space W to a rigid-analytic open disk U ⊂ W centered at 1 at which the finite cover w : W g → W restricts to an isomorphism w : U g ∼ → U with y 0 ∈ U g .Let U g = O(U g ) denote the Iwasawa algebra of analytic functions on U g whose supremum norm is bounded by 1. Shrink likewise C and S so that projection to the weight space restricts to an isomorphism with U and U × U respectively.Then the associated Iwasawa algebras are respectively is not canonical and depends on the choice of an element γ ∈ × U g which is sent to 1 + Z.Consider the short exact sequence of Z p -modules Under the above identifications the sequence may be recast as Proposition 3.6.There exists a unique local class κ p,γ (g, Proof.The short exact sequence of gives rise to the long exact sequence Since H 0 (Q p , V gg * |C ) = 0 as already argued in the proof of Theorem 3.5, the vanishing of κ p (g, g * ) |C implies the existence of a unique element κ p,γ (g, g * ) ∈ H 1 (Q p , V gg * |S ) satisfying the claim.
We are interested in the restriction of κ p,γ (g, g * ) to C; although it depends on the choice of the topological generator γ , the class κ p (g, g * ) ∈ H 1 (Q p , V gg * |C ) defined by is independent of γ .Since in our setting the Euler factor 1 − α g it is natural to introduce a modified p-adic L-function on the surface S, defined as ) and let L p denote the p-adic completion of L. Define where the first map is the projection onto and the last one is the composition of the Bloch-Kato logarithm with the pairing with the differential η g • y ⊗ ω h • z .It follows from Theorems 3.2 and 3.4 that for all (y, z, s) ∈ W • gh ∩ S, up to multiplication by the c-factor we have described in Theorem 3.4 and which does not affect our discussion since we always work modulo L × .Observe that the function Lp (g, g * ) vanishes along the curve C, so the restriction of its derivative to that line is exactly zero.Recall that Lp (g, g * ) is a two-variable function, determined by the values of y and z, since the third variable σ is automatically determined by z.Hence, Recall that α g is an analytic function defined over g ; we denote by α g its derivative.We may consider as before the L-invariant attached to the adjoint representation of g, L(ad 0 (g)) = −α g /α g .
Observe that its specializations at classical points agree with the definitions given before in ( 23).Now, we can compute the partial derivatives of Lp (g, h) at a crystalline point (y, y, − 1) of the curve C. Using (38), one gets

Weight one modular forms
Let g ∈ S 1 (N, χ g ) and h ∈ S 1 (N, χ h ) be two cuspidal eigenforms of weight one.Let V g and V h denote the Artin representations over a finite extension L of Q attached to g and h.
Let α g , β g (resp.α h , β h ) denote the roots of the p-th Hecke polynomial of g (resp. of h).
We assume throughout that α g = β g and α h = β h .We also assume L is large enough as specified in §2.1.
Definition 3.8.Let g and h be Hida families passing through p-stabilizations g α , h α of g, h at some point (y as the specialization of κ(g, h) at the point (y 0 , z 0 , 0).
This procedure yields four a priori different global cohomology classes: one for each choice of pair of roots of the p-th Hecke polynomials of g and h.
Given a p-adic representation V of G Q p with coefficients in Q p , Bloch and Kato introduced in [BK] a collection of subspaces of the local Galois cohomology group ). Definition 3.9.Let V be a representation of G Q with coefficients in Q p .The group of classes that are de Rham at p (i.e. the restriction to Q p lies in H 1 g (Q p , V )) and unramified at all primes = p is denoted as H 1 f,p (Q, V ).The group of classes that are crystalline at p and unramified at any other prime q = p is denoted as Proposition 3.10.The four classes in (43) lie in H 1 f,p (Q, V gh ⊗ L p (1)).
Proof.The dimensions of H 1 g (Q p , V gh (1)) and H 1 (Q p , V gh (1)) are equal, according to the discussion of [Da,Section 1.4]; hence, the classes are de Rham at p. Furthermore, the restriction of these classes to Q q is 0 for q = p.In fact a stronger fact holds true: the local -adic classes κ q (g, h) ∈ H 1 (Q q , V gh ) are 0 at all q = p.This can be argued for instance by fixing weights ( , m, s) large enough so that for every triple ( 1 , 2 , 3 ) of characters of arbitrary p-power conductor and for any point (x, y, z) above (ν , 1 , ν m, 2 , ν s, 3 ), V(g, h)(x, y, z) contains no subquotient isomorphic to either Q p or Q p (1).It then follows from Tate's local Euler characteristic formula (cf.[Nek,2.5])that H 1 (Q q , V gh (x, y, z)) = 0. From this it follows that H 1 (Q q , V gh ) = 0 arguing as in e.g.[KLZ1,Prop. 8.1.7 or Lemma 8.2.6].
Recall that we let H denote the Galois extension of Q cut out by V g ⊗ V h .Proposition 3.11.There are natural identifications Proof.This follows from the same arguments as in e.g.[Bel,Prop. 2.9]. Since g , where V α g g and V β g g are the G Q p -invariant lines on which Fr p acts with eigenvalue α g and β g respectively.We similarly have h , and we may define Note that these four G Q p -invariant lines in V gh are linearly independent even though some of the eigenvalues α g α h , α g β h , β g α h , β g β h might be equal.Hence there is a decomposition of G Q p -modules It follows that the local class κ p (g α , h α ) may be decomposed as where As before, we are specially interested in the case where h = g * = g ⊗ χ −1 , and we impose on g the assumptions (H1)-(H3) listed in the introduction; if we denote by {α, β} the p-th Hecke eigenvalues of g, the p-th Hecke eigenvalues of h are {1/β, 1/α}.Let g and h = g * := g ⊗ χ −1 denote the Hida families over W g = W h passing through g α and (g α ) * = (g * ) 1/β respectively at some point y 0 ∈ W g in the weight space.Thanks to our running assumptions, the main theorem of [BeDi] ensures that the weight map W g → W is étale at the point associated to g α .It is therefore possible to fix an open subset in W g around g α on which the weight map is an isomorphism.This way we are entitled to work under the simplifying assumptions posed in §3.2 and the results in loc.cit.and §3.3 may be applied.
Define the curve Proposition 3.13.There exists a unique global class κ γ (g, Proof.The short exact sequence of gives rise to the long exact sequence Since H 0 (Q, V gh ⊗ L p (1)) = 0, the vanishing of κ(g α , g * 1/β ) proved in Proposition 3.12 implies the existence of a unique element κ γ (g, g * ) ∈ H 1 (Q, V gg * |D ) satisfying the claim.
Remark 3.14.The global cohomology class in Proposition 3.13 only makes sense along the curve D. However, the local cohomology class constructed in Proposition 3.6 exists along the whole surface S. Hence one cannot define the latter on the surface S as the restriction at p of the former, although this is indeed true after restricting to D, a fact that we shall apply right below.

Set
κ (g α , g * 1/β ) = 1 log p (γ ) κ γ (g, g * )(y 0 , y 0 , 0). ( 47) Since the construction of this class coincides with the one performed in the previous section once we localize at p and restrict to the curve D, Theorem 3.7 applies and we deduce that Recall from (21) that L p (g, g * ) satisfies a functional equation relating the values at s = 0 and s = 1 up to a simple non-zero rational constant.Together with Proposition 2.5 this implies that This formula is the key input for deriving Theorem B, the second main result of this note.

Derivatives of Fourier coefficients via Galois deformation theory
As in previous sections, let g ∈ S 1 (N, χ ) satisfy the hypothesis of the introduction.Let denote the Artin representations associated to g and its adjoint, respectively.Here L is a finite extension of Q, and H g ⊇ H denote the finite Galois extensions of Q cut out by these representations.Let P denote the set of primes of H lying above p, and fix once for all a prime ℘ ∈ P, thus determining an embedding H ⊂ H p ⊂ Qp of H into its completion H p at ℘, and an arithmetic Frobenius Fr p ∈ Gal(H p /Q p ).
As in [DLR4], the regularity assumptions we have imposed on g imply by e.g.[Das1,Prop. 3 Fix two linearly independent global cohomology classes ) that under the identifications provided by Proposition 3.11 project to a basis of the twodimensional space (O H [1/p] × /p Z ⊗ ad 0 (g)) G Q .By a slight abuse of notation, we continue to denote the resulting elements Recall from (45) that V gh admits a natural decomposition as G Q p -module as the direct sum of the four different lines V αα gg * , . . ., V ββ gg * .Since ad 0 (V g ) is the quotient of V gg * by the trivial representation, (45) descends to a decomposition of ad 0 (V g ) as G Q p -module as ad 0 (g) = ad 0 (g) 1 ⊕ ad 0 (g) α⊗ β ⊕ ad 0 (g where Fr p (e 1 ) = e 1 , Fr p (e α⊗ β ) = α β • e α⊗ β , Fr p (e β⊗ ᾱ ) = β α • e β⊗ ᾱ .Note that α/β = 1 thanks to the regularity assumption.It could be that α = −β and hence α/β = β/α = −1, but the above decomposition is still available as explained in (45).
Restriction to the decomposition group at p allows us to regard u and v as elements in H 1 (Q p , ad 0 (g) ⊗ L p (1)) = (H × p ⊗ ad 0 (g)) G Qp , and as such we may write them as where and similarly for v.
We can now provide the last step in our proof of Theorems A, A and B in the introduction.As shown in Corollary 2.5, Theorem A may be reduced to the computation of the derivative of the Fourier coefficient a p (g) at y 0 , where g stands for the unique Hida family passing through g α .
Let ρg : ) be the unique first order α-ordinary deformation of ρ g such that det ρg = χ g (1 This representation, whose existence follows from [BeDi], satisfies for some cohomology class κ g : G Q → ad(ρ).Considering a diagonal basis for the Frobenius action (where we take the first vector to have eigenvalue α), the matrix form of κ(σ ) can be expressed as We denote by ρg,p the restriction of ρg to the decomposition group at p; in the same way, the restriction of κ to the decomposition group at p is denoted by κ p , and similarly we denote by κ i,p the restriction of κ i to G Q p .
In [BeDi,Lemmas 2.3 and 2.5] the authors determine the tangent space to a deformation problem which can be seen to be equivalent to ours.They conclude that there is a natural bijection between this tangent space and a certain subspace of H 1 (Q, ad(ρ g )); in this case, it consists of those classes κ whose matrix representation satisfies where I p is the inertia group at p.In [DLR3], the restriction of κ to the inertia group at p was determined, and the identifications of class field theory allow us to extend this to the whole decomposition group.A similar setting, where also a related deformation problem arises, is exploited in [BDP] to treat the case of weight one Eisenstein points.Let V α g be the étale subspace on which the action of the Frobenius is unramified for the -adic representation.Restricting to the decomposition group at p, we find that If g α stands for the derivative of g evaluated at y 0 , then a p (g α ) and consequently we have the following.
Proposition 4.1.Let g be the Hida family through g α .Then Taking into account the identifications provided by class field theory, one can make κ 1 (Fr p ) explicit.
From [BeDi,Section 3.2], there is an exact sequence Similarly, one has another exact sequence Consequently, we have identifications and this corresponds to the subspace of homomorphisms of We now recall some results which allow us to determine each of the κ i,p .Since there are isomorphisms we may write elements in From class field theory, one has two distinguished elements in H 1 (Q p , Qp ): (1) The class κ nr , which is the unique homomorphism taking Fr p to 1.
(2) The restriction to G Q p of the logarithm of the cyclotomic character κ cyc := log p (ε cyc ), which gives a ramified element of H 1 (Q p , Qp ).
Furthermore, H 1 (Q p , Qp ) is identified with Hom(G H p , Qp ) Gal(H p /Q p ) as explained above.In the latter space, we denote by κ nr the morphism that takes Fr p to 1, and by κ cyc the ramified element defined by log p (ε cyc ).
Now we can explicitly determine the element κ ∈ H 1 (Q, ad(ρ)), by constructing a homomorphism g (x) : (H ⊗ Q p ) × → H p ⊗ ad(ρ) corresponding to κ via the previous identifications.In particular, it vanishes when evaluated at the basis {u, v} of units for the adjoint.Moreover, g (π −1 p ) leaves invariant the one-dimensional space V α g , where π −1 p is the idèle which is equal to the inverse of a local uniformizer of H p for the fixed prime ℘ above p, and to 1 everywhere else.The eigenvalue for the action of g (π −1 p ) on that subspace is precisely κ 1 (Fr p ).Let u × g be any generator of (O × H ⊗ ad(ρ)) G Q , and let v × g be any element of the space Consider the element A g ∈ (H p ⊗ ad(ρ)) G Qp , which on an eigenbasis for the Frobenius takes the form .
Consider also J ∈ (H p ⊗ ad(ρ)) G Qp , which in the same basis is expressed as where a, b, c ∈ H p .These choices give rise to the G Q p -equivariant homomorphism g : H × → H p ⊗ ad(ρ) given by the rule Here, σ −1 • A g denotes the action of σ −1 by conjugation on the second factor of the tensor product, ad(ρ); then, this homomorphism can be extended to the whole (H ⊗Q p ) × .The aim is to determine a suitable J such that g (x) corresponds to κ via class field theory.The following result explains the behavior of the terms coming from the log-part.Proposition 4.2.Let ad(ρ) ord := Hom(V g /V α g , V g ).The homomorphism g vanishes on O × H ⊗ Q p , and g (x) ⊂ H p ⊗ ad(ρ) ord for any x of the form (x p , 1, . . ., 1), where x p ∈ O × H p .Moreover, g (x) fixes V α g for all x of the form (x p , 1, . . ., 1) with x p ∈ H × p .
Proof.This follows from [DLR3,Lemma 1.6], where g is defined in the same way but without the ord-terms.It is clear that the behavior at the O H -units is not affected by the presence of these extra terms.The last part of the statement follows from the definition of A g and J .
To determine the remaining parameters of the matrix J (a, b and c), we first observe that the homomorphism g must vanish identically on O H [1/p] × ⊗ Q p .The endomorphism v g is represented by a matrix of the form and hence Furthermore, the restrictions of both κ 1,p and κ 4,p to the decomposition group at p belong to this space is two-dimensional and it is generated by κ cyc and κ nr ; in the same way, κ 2,p is a cohomology class in the one-dimensional space It is clear that Then, to determine the restriction of the cohomology classes to the whole decomposition group, we impose these three conditions: (1) The fact that the trace of the representation is prescribed (and is equal to κ cyc ) forces that (2) The fact that H 1 (H p , Qp (β/α)) G Qp is one-dimensional (since α = β) yields κ 2 = log p (u 1 ) log p (u α⊗ β ) κ cyc (i.e., there is no contribution coming from the Frobenius).
(3) The remaining parameter λ can be determined by considering the associated matrix J giving rise to the endomorphism g (x).This is the content of the following proposition.
Observe that w × g can be expressed in terms of the basis u × g and v × g .In particular, let Observe that J = −λ 0 0 λ , and from (53) one sees that Hence, we conclude that and evaluating at Fr p we finally obtain the formula we anticipated below: as claimed.
The derivative of the Fourier coefficient was computed in the previous section, and it follows from (54) that This proves Theorems A and A .

Proof of Theorem B
We may combine (49) with the above result to deduce that Recall that in (39) we defined the map log −+ as the composition of the Perrin-Riou big logarithm of Proposition 3.2 specialized at weight (y 0 , y 0 , 0) and the pairing with the class η g α ⊗ ω g * 1/β introduced in §2.1.These differential classes satisfy under the perfect pairing D(V g ) × D(V g * ) → L p .We may take decompositions of V g and V g * as G Q p -modules 1/β } are bases of V g and V g * , dual to each other, and compatible with the choice of the basis for the tensor product V gg * considered in the previous section.As explained in [DR1,§2], one may define p-adic periods The natural pairing between D(V g ) = (H p ⊗ V g ) G Qp and D(V g * ) = (H p ⊗ V g * ) G Qp induces a duality between V g and V g * , and hence the quantity ).We may write the cohomology class κ (g α , g * 1/β ) as a linear combination where a, b, c ∈ L p .The condition for an element to lie in the kernel of the map log −+ is Consider now the map log −− : H 1 (Q p , V gh (1)) According to (30), the class κ p (g α , g * 1/β ) also lies in the kernel of log −− .Hence, taking equalities up to periods, The assumption L(ad 0 (g α )) = 0 in Theorem B implies that L p (g, g * )(y 0 , y 0 , 1) = 0 by the first display in this section.Hence λ = 0 and Theorem B follows.

Non-vanishing of the L-invariant
We conclude this article by noting that the non-vanishing of the L-invariant can be proved in most dihedral cases, because the expression (55) simplifies considerably.Indeed, let K be a real or imaginary quadratic field of discriminant D and let ψ : G K → L × be a finite order character of conductor c ⊂ O K (and of mixed signature at the two archimedean places if K is real).Then the theta series g = θ (ψ) attached to ψ is an eigenform of weight 1, level N g = |D|•N K/Q (c) and nebentype χ g = χ K χ ψ , where χ K is the quadratic character associated to K/Q and χ ψ is the central character of ψ.
Let ψ denote the Gal(K/Q)-conjugate of ψ defined by the rule ψ Let p N g be a prime number and fix an embedding Q ⊂ Qp .In line with the introduction, suppose that hypotheses (H1)-(H3) are fulfilled.This amounts to requiring that: (i) ψ = ψ (mod p), so that g is cuspidal even residually at p. (ii) If K is imaginary and p = ℘ ℘ splits in K, then ψ(℘) = ψ (℘) (mod p).(iii) If K is real, p does not split.If K is imaginary and the field H = Kker(ψ) cut out by ψ has Galois group Gal(H /Q) = D 4 , the dihedral group of order 8, then p does not split in the single real quadratic field contained in H .
Proof.If K is real (and thus p remains inert in it) then u 1 is the norm of the fundamental unit of K and hence its p-adic logarithm vanishes.If K is imaginary and p splits, then v is a p-unit in K × and hence log p (v α⊗ β ) = 0.
Note that Theorem B simplifies considerably in the setting of the above proposition.The simplest scenario where it is not a priori obvious that the L-invariant is non-zero arises when K is imaginary but p remains inert.Fix a prime ℘ in H above p and set ψ ad = ψ/ψ .Note that ψ ad is a ring class character, regardless of whether ψ is so or not.Let u ψ ad (resp.v ψ ad ) denote any element spanning the 1-dimensional L-vector space . Set also u ψ ad = Fr ℘ u ψ ad and v ψ ad = Fr ℘ v ψ ad .Then it is a straightforward computation to check that the L-invariant appearing in Theorems A and A is The combination of [Park,Theorem 4.4] and Darmon-Dasgupta-Pollack's [DDP,Theorem 2] provides strong evidence for this conjecture, as putting these results together it follows that, in our notations, log p (N K p /Q p (v DD [ψ ad ])) = log p (v 1 ) (mod L × ). (59) The above equality together with Theorems B and C yields a formula relating the derived Beilinson-Flach elements of this note to Darmon-Dasgupta units, much in the spirit of [BSV, Theorem A] and [Da, Theorem C] for diagonal cycles versus Stark-Heegner points.On account of the decomposition introduced in (50), the element κ (g α , g * 1/β ) 1 belongs to Q × p ⊗ L p .
Corollary 6.1.Let g = θ (ψ) be the theta series associated to a finite order character ψ of mixed signature of a real quadratic field K. Let p be a prime that remains inert in K. Then In spite of the ostensible parallelism between the above formula and [BSV, Theorem A] and [Da, Theorem C], the proof of Corollary 6.1 follows quite a different route from [BSV] and [Da].The main reason is that in the last two references a factorization of padic L-functions was crucially exploited, which follows from a comparison of critical values.
In our setting one still expects to have an analogous factorization, but proving it appears to be far less trivial.Since this issue poses intriguing questions, and our results shed some light on them, we discuss it in more detail below.

Artin p-adic L-functions
Let g = θ (ψ) ∈ S 1 (N, χ ) be a theta series of a quadratic field K and p be a prime which is split (resp.inert) if K is imaginary (resp.real).Keep the assumptions of Proposition 5.1, and set h = g * = θ (ψ −1 ) and ψ ad = ψ/ψ as usual.Let L p (g α , g * 1/β , s) denote the cyclotomic p-adic Rankin-Hida L-function associated to the pair (g α , g * 1/β ) of p-stabilizations of g and h, as in §2.2.Note that this p-adic Lfunction has no critical points.Nevertheless, since one may still wonder whether L p (g α , g * 1/β , s) admits a factorization mirroring the one satisfied by its classical counterpart: L(g, h, s) = ζ (s) • L(χ K , s) • L(K, ψ ad , s). (61) While (61) follows directly from (60) by Artin formalism, a putative analogous factorization of L p (g α , g * 1/β , s) is far less trivial.In the CM case one can prove it by a method which is nowadays standard, but rather deep as one needs firstly to extend L p (g α , g * 1/β , s) If we now restrict to = 1 and invoke Katz's p-adic analogue of Kronecker's limit formula which asserts that L Katz p (ψ) = log p (u ψ ad ) (mod L × ), it follows that L p (g, g * , s) = f(s) log p (u ψ ad ) • L Katz p (N s ) • L Katz p (ψ ad , s). (63) Finally, Gross's main theorem in [Gr] together with the functional equation for Kubota-Leopoldt's p-adic L-function asserts that up to a rational fudge factor.This yields the theorem.
Assume now that g = θ (ψ) is the theta series of a character of a real quadratic field in which p is inert.In light of Theorem 6.2 it is natural to pose the following question: Question 6.3.Assume K is real and p remains inert in K. Let u K be a fundamental unit of K and let L p (ψ ad ω, s) denote the Deligne-Ribet p-adic L-function attached to ψ ad ω.
Is it true that L p (g, g * , s) up to an L-rational fudge factor?
Note that the results of this paper, combined with Darmon-Dasgupta-Pollack's [DDP, Theorem 2] prove that the above factorization holds when evaluated at s = 0 and s = 1.Indeed, Theorem A in this setting takes the simple form L p (g, g * , 1) = log p (v 1 ) (mod L × ), while [DDP,Theorem 2] asserts that L p (ψ ad ω, s) vanishes at s = 0 and L p (ψ ad ω, 0) = log p (v 1 ) (mod L × ).
Moreover, ζ p (s) has a simple pole at s = 0 whose residue is a non-zero rational number, and Leopoldt's formula asserts that L p (χ K , 0) = log p (u K ) (mod L × ).Putting all together shows that ( 65) is true at s = 0.The functional equations satisfied by each of the p-adic L-functions in play ensure that the same is true at s = 1.This of course falls short from establishing (65).
as tuples (x i ) i∈P .The action of the Galois group G = Gal(H /Q) is transitive on P, so any Galois equivariant homomorphism from(H ⊗ Q p ) × (resp.(OH ⊗ Q p ) × )to Qp is completely determined by its values on H × p (resp.O × H p ).
Proposition 4.3.There exists a unique λ such that the homomorphism g (x) vanishes identically on O H [1/p] × ⊗ Q p .Proof.Picking w ∈ O H [1/p] × and an arbitrary B ∈ ad(ρ), it is enough to see that Tr( g (w) • B) = 0, due to the non-degeneracy of the H p -valued trace pairing on H ℘ ⊗ ad(ρ).Set