Ann. Inst. Fourier, Grenoble
Working version – January 7, 2020

ON THE NON-VANISHING OF p-ADIC HEIGHTS ON
CM ABELIAN VARIETIES, AND THE ARITHMETIC OF
KATZ p-ADIC L-FUNCTIONS
by Ashay A. Burungale and Daniel Disegni

Abstract. Let B be a simple CM abelian variety over a CM fieldE, p a rational prime. Suppose that B has potentially ordinary reduction above p and is self-dual with root number −1. Under some further conditions, we prove the generic non-vanishing of (cyclotomic) padic heights on B along anticyclotomic Zp -extensions of E. This provides evidence towards Schneider’s conjecture on the non-vanishingof p-adic heights. For CM elliptic curves over Q, the result was previously known as a consequence of work of Bertrand, Gross–Zagier andRohrlich in the 1980s. Our proof is based on non-vanishing resultsfor Katz p-adic L-functions and a Gross–Zagier formula relating thelatter to families of rational points on B.

Contents
1. Introduction and statements of the main results
1.1. Non–vanishing of p-adic heights
1.2. Gross–Zagier formula for the Katz p-adic L-function
1.3. An arithmetic application
1.4. Context and strategy of proof
2. Proofs
2.1. CM theory
2.2. Theta lifts of Hecke characters
2.3. Rankin–Selberg p-adic L-function
2.4. Katz p-adic L-function
2.5. Factorisation of the Rankin–Selberg p-adic L-function
2.6. Construction of an auxiliary character
2.7. p-adic Gross–Zagier formula

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2.8. Non-vanishing of p-adic L-functions
2.9. Proofs of main theorems

20
21

1. Introduction and statements of the main results
Let B be an abelian variety over a number field E, and let B ∨ be its dual.
Let p be a prime and let L be a finite extension of Qp . The Néron–Tateheight pairing on B(E 0 )Q × B ∨ (E 0 )Q , where E 0 is a finite extension of E,
admits a p-adic analogue (see e.g. [34])(1.1)

B(E 0 )Q × B ∨ (E 0 )Q → L

×depending on the choices of a homomorphism ` : E × \EA∞ → L (“p-adic
1logarithm”) and on splittings of the Hodge filtration on HdR(B/Ev ) at
the primes v|p; in the potentially ordinary case under consideration in thispaper, there are canonical choices (the “unit root” subspaces) for the Hodge
splittings.
While it is a classical result that the Néron–Tate height pairing is nondegenerate, the pairing (1.1) can vanish for some choices of `. Suppose
however that ` = `Q ◦ NE/Q with `Q a p-adic logarithm of Q such that
`Q |1+pZp is nontrivial (we call such ` a cyclotomic logarithm). Then it isconjectured [39] that (1.1) is non-vanishing. This long-standing conjectureis only known in a few special cases: for CM elliptic curves, thanks to
Bertrand [5], and also for elliptic curves over Q at supersingular primes.(1)The stronger statement that (1.1) is non-degenerate is also conjectured to
be true; this is not known in any cases of rank higher than 1. The nondegeneracy conjecture has arithmetic consequences: it allows to formallydeduce the p-adic Birch and Swinnerton-Dyer conjecture from the Iwasawa
main conjecture for B.(2)The pairing (1.1) is equivariant for the actions of Gal(E 0 /E) and of
K := End0 (B) on both sides; for our purposes we can then assume that
B is simple, that the coefficient field L is sufficiently large and has the
structure of a K ⊗Qp -algebra, and then decompose the pairing into isotypiccomponents(1.2)

B(χ) ⊗L B ∨ (χ−1 ) → L(1) This was observed in [3]; see [31, §4.5] for the comparison between the definition of
the height pairing used in [3] and the ‘standard’ definition of Zarhin and Nekovář [34].
(2) See [39]. For further applications to the classical Birch and Swinnerton-Dyer formula,see [37, 31, 19].

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for the Gal(E 0 /E)-action. Here and in the rest of the paper, if R is a
K ⊗ Qp -algebra and χ is an R× -valued character of Gal(E ab /E), we define
B(χ) := (B(E ab ) ⊗K⊗Qp R(χ))Gal(E

ab

/E)

,

where R(χ) is a rank-1 R-module with Galois action by χ.
The most significant result of this paper is the proof that, under someassumptions, the non-vanishing conjecture for (1.2) is true “generically”
when B is a p-ordinary CM abelian variety over a CM field E and χ varies
among anticyclotomic characters of E unramified outside p.
The method of proof, different from that of previous results on this topic,
is automorphic. (In particular, the approach does not involve transcendence
arguments.) It combines two ingredients. The first is a pair of nonvanishing
results for Katz p-adic L-functions due to Hida, Hsieh, and the first author(in turn relying on Chai’s results on Hecke-stable subvarieties of a mod p
Shimura variety [15, 14, 16]). The second ingredient is a Gross–Zagier formula relating derivatives of Katz p-adic L-functions to families of rational
points on CM abelian varieties. We deduce this formula from work of thesecond author, by an argument employed by Bertolini–Darmon–Prasanna[4] in a different context.
In the rest of this section we describe the main results in more detail.

1.1. Non–vanishing of p-adic heights
Let B/E be a simple CM abelian variety over a CM field E, i.e. K :=End0 (B) is a CM field of degree [K : Q] = 2 dim B. Let F be the maximal
totally real subfield of E, and let η = ηE/F be the associated character of×
F × \FA.
Assumptions. The abelian variety B is associated with a Hecke character×λ = (λτ )(τ : K,→C) : E × \EA→ (K ⊗ C)×([42, §19]). Suppose that λ satisfies the condition(1.3)

λ|F × = η| · |−1A ;A

this holds in particular whenever B arises as the base-change of a realmultiplication abelian variety A/F [42, Theorem 20.15]. It implies that the

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ASHAY A. BURUNGALE AND DANIEL DISEGNI

for each τ there is a functional equation with sign w(λ) := ε(1, λτ ) ∈ {±1}(independent of τ ) relating L(s, λτ ) to L(2 − s, λτ ). We will assume thatw(λ) = −1.

(1.4)
Finally, we assume that

(1.5) B has potentially ordinary reduction at every prime of E above p.
+−) be the anticy(respectively E∞
Anticyclotomic regulators. Let E∞[F :Q]-extension (respectively cyclotomic Zp -extension) of E and,clotomic Zp−−
be the ℘-anticyclotomic subexfor a prime ℘ of F above p, let E℘,∞⊂ E∞
tension, i.e. the maximal subextension unramified outside the primes above− +−+. If • is any com℘ in E; finally, let E∞ := E∞
E∞ and E℘,∞ = E℘,∞E∞
bination of subscripts ∅, ℘ and superscripts ∅, +, − (we convene that ‘∅’
denotes no symbol), the corresponding infinite Galois group isΓ• := Gal(E•,∞ /E).
Let L be a finite extension of a p-adic completion Kw of K. If • is any
set of sub- and superscripts as above, we letΛ• := OL JΓ• K ⊗ L,Y• := Spec Λ• .

When we want to emphasise the role of a specific choice of L, we will writeΛ•,L , Y•,L .
For ◦ = ℘, ∅ we let−,×χuniv,◦ : Γ−
◦ → Λ◦

be the tautological anticyclotomic character. We then have a Λ−
◦ -module
Gal(E/E)
B(χuniv,◦ ) := (B(E) ⊗ Λ−,◦ (χuniv,◦ ))whose specialisation at any finite order character χ ∈ Y − is B(χ), andheight pairings ([36, §2.3], see also [35, §11])(1.6)−
B(χuniv,◦ ) ⊗Λ◦ B ∨ (χ−1univ,◦ ) → Λ◦

associated with choices of a p-adic logarithm ` and of Hodge splittings. We
suppose that ` : Gal(E ab /E) → L is the cyclotomic logarithm and that the
Hodge splittings are given by the unit root subspaces.
Theorem 1.1. — Let B be a simple CM abelian variety over the CM
field E with associated Hecke character λ satisfying (1.3), (1.4), and (1.5);
suppose that the extension E/F is ramified. Let p be a rational prime, and−suppose that p - 2DF h−
E , where hE = hE /hF is the relative class number

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of E/F and DF is the absolute discriminant of F . Let ℘ be a prime of F
above p, and let L ⊃ Kw ⊃ K be as above.
Then for almost all finite-order characters χ of Γ−℘ , the pairing (1.2)
B(χ) ⊗L B ∨ (χ−1 ) → L
is non-vanishing. Equivalently, the paring (1.6) for ◦ = ℘ (hence also for◦ = ∅) is nonzero.
Here “almost all” means that the set of finite-order characters χ ∈ Y℘−
which fail to satisfy the conclusion of the theorem is not Zariski dense inY℘− . When dim Y℘− = 1 (e.g. F = Q), this is equivalent to such set of
exceptions being finite.

1.2. Gross–Zagier formula for the Katz p-adic L-function
As recalled in §2.4, under our conditions and assuming that the extension
L ⊃ Kw splits E, the character λ (more precisely, its w-adic avatar) hasa p-adic CM type ΣE ⊂ Hom(E, L) of E. We also identify ΣE with (i) achoice, for each prime ℘|p of F , of one among the two primes of E above℘, and (ii) an element of Z[Hom(E, L)]. To the CM type ΣE is attached
the Katz p-adic L-function
LΣE ∈ Λ.
It interpolates the values L(0, λ0−1 ) for characters λ0 whose infinity typelies in a certain region of Z[Hom(E, Cp )]; this region is uniquely determinedby ΣE and contains in particular the infinity type ΣE .
Let λ∗ (x) := λ(xc ), where c denotes the complex conjugation of E/F .
The root number assumption (1.4) implies that LΣE (λ) = LΣE c (λ∗ ) =L(0, λ∗−1 ) = L(0, λ−1 ) = L(1, λ) = L(B, 1) = 0, and more generally thatthe function LΣE c,λ∗ : χ0 7→ LΣE c (λ∗ χ0 ) vanishes along Y − ⊂ Y . We maythen consider the cyclotomic derivatived
L0ΣE c,λ∗ : χ 7→LΣ c (λ∗ χ · χscyc )|s=0 ,ds E
×for χ ∈ Y − . (Here χcyc is the p-adic cyclotomic character of EA.)−−
For ◦ = ∅, ℘, let K◦ be the field of fractions of Λ◦ and let B(χuniv,◦ )K◦− :=B(χuniv,◦ ) ⊗Λ−K◦− ; similarly for B ∨ (χ−1univ )K◦− .
◦Theorem 1.2. — Let ◦ = ∅ or ◦ = ℘. Under the assumptions of Theorem 1.1, there is a ‘pair of points’P ⊗ P ∨ ∈ B(χ−1univ,◦ )K◦− ⊗ B(χuniv,◦ )K◦−

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satisfying
hP, P ∨ i◦ = L0ΣE c,λ∗ |Y◦−in K◦− , where h , i◦ is the height pairing (1.6), and we identify Γ+ ∼= Zp
via the cyclotomic logarithm.
The construction of the points depends on some choices, analogously
to how the construction of rational points on an elliptic curve over Q of
analytic rank one depends on the choice of an auxiliary imaginary quadratic
field. Like in that situation, it comes from Heegner points and relies on a
non-vanishing result for L-functions – in this case, the results of Hida andHsieh [25, 27] for anticyclotomic Katz p-adic L-functions. Nevertheless theauxiliary setup does not seem to be explored in regard to the cyclotomic
derivative.
The following conjecture, which can be regarded as analogous to theresults of Kolyvagin, would imply that the ambiguity is rather mild.
Conjecture 1.3. — Let ◦ = ∅ or ◦ = ℘ for a prime p of F . The∨
K◦− -vector spaces B(χ−1univ,◦ )K◦− , B (χuniv,◦ )K◦− have dimension one.
When E is an imaginary quadratic field and B is the base-change of anelliptic curve over Q, the conjecture is part of the main result of Agboolaand Howard in [2].
Remark. — It is natural to wonder about the arithmetic significance ofthe values
LΣ0E (λ∗ )(1.7)
for CM types Σ0E such thatδ(Σ0E ) := |Σ0E ∩ ΣE | > 1.
We would like to suggest that if δ(Σ0E ) 6 r := ords=1 L(B, s), the cyclotomic order of vanishing of LΣ0E at λ∗ (that is, the smallest k such that the(k)cyclotomic derivative LΣ0 (λ∗ ) 6= 0) should be
E

(1.8)

?

ordcyc LΣ0E ,λ∗ = r − δ(ΣE 0 )and that there should be an explicit formula relating(r−δ(ΣE 0 ))

LΣ0

(λ∗ )

E

to a p-adic regulator. When [E : Q] is a quadratic field and λ comes from anelliptic curve, (1.8) was conjectured by Rubin [38] together with a precise
formula, and proved by himself when r 6 1.

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Theorem 1.2 provides evidence for the general case of (1.8) in one of thecases with r 6 1, whereas the other such case is treated, when [E : Q] = 2,
by Rubin’s formula as generalised by Bertolini–Darmon–Prasanna [4].(3)The particular interest of the case of [E : Q] > 2 lies of course in thepossibility of having δ(Σ0E ) > 2: our speculation is also inspired from therecent work of Darmon–Rotger [18] on p-adic L-functions related to certain
Mordell–Weil groups of rank 2.

1.3. An arithmetic application
For an elliptic curve E/Q and a prime p, let us refer to the implication
corankZp Selp∞ (E/Q ) = 1 =⇒ ords=1 L(s, E/Q ) = 1
as the “p-converse theorem” (to the one of Gross–Zagier, Kolyvagin and
Rubin). In [12], the authors establish the p-converse theorem in the case of
p-ordinary CM elliptic curves (complementing the earlier works [43, 46]).
The approach crucially relies on the auxiliary setup introduced in the proofof the main results here, and also on the main results themselves (Theorem
1.1 and Theorem 1.2).

1.4. Context and strategy of proof
When E is an imaginary quadratic field and B = AE for a CM elliptic curve A/Q, Theorem 1.1 is a consequence (see Rubin [28]) of theaforementioned result of Bertrand together with Mazur’s conjecture on thegeneric non-vanishing of Heegner points along anticyclotomic extensions(which in that case is proved, via the Gross–Zagier formula, by the generic
non-vanishing of derivatives of L-functions established by Rohrlich). Our
method is rather different (although not distant in spirit): we first deduce
the formula of Theorem of 1.2 from the p-adic Gross–Zagier formula of thesecond author [20]. Theorem 1.1, or rather its more precise version Theorem 2.11 below, is then a consequence of Theorem 1.2 and non-vanishing
results of the first author [6] (or their refinement in [10]) for the derivativesof Katz p-adic L-functions. As a corollary we recover Mazur’s conjecture,
which in our case was proved by Aflalo–Nekovář [1] as a generalisation ofwork of Cornut and Vatsal. Note further that our method would readily(3) We hope to present a generalisation of this formula in a sequel to the present paper.

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ASHAY A. BURUNGALE AND DANIEL DISEGNI

adapt to cover the case of (generalised) Heegner cycles upon availability ofa suitably general p-adic Gross–Zagier formula for them. The second author expects to present such a formula as part of a forthcoming version of[21].(4)A parallel approach is followed by the first author in a series of works[8], [9] establishing, without assumptions of complex multiplication, thegeneric non-vanishing of Heegner points and cycles, or more precisely of(the reduction modulo p of) their images under the Abel–Jacobi map (alsosee [7] and [11]).(5) The strategy to prove Theorem 1.2 is inspired fromthe proof of Rubin’s formula in [4]. As in [4], we can remark that we haveestablished a result for a motive attached to the group U(1) by making useof p-adic L-functions for U(1) × U(2). Readers with a generous attitude
towards mathematical induction might find in this a good omen for future
progress.
Acknowledgments. We would like to thank D. Bertrand, K. Büyükboduk, F. Castella, M. Chida, H. Darmon, H. Hida, B. Howard, M.-L. Hsieh,
M. Kakde, J. Lin, J. Nekovář, D. Prasad, D. Rohrlich, K. Rubin, C. Skinnerand Y. Tian for useful conversations, correspondence or suggestions. We are
grateful to the referee for thorough comments and suggestions. During partof the writing of this paper, the second author was funded by a public grantfrom the Fondation Mathématique Jacques Hadamard.

2. Proofs
2.1. CM theory
We start by reviewing some basic results in the theory of Complex Multiplication. The classical reference is [42] (see especially §18-20); see also[17, §2.5].
Let B be an abelian variety of dimension d over a field E, such that
End0 (B) = K is a CM field of degree 2d. Denote by M ⊂ K the maximal
totally real subfield. The action of K on Lie B gives, after base-change
from E to an extension ι : E ,→ C which splits K, a CM type (K, Σ)
over C, namely Σ = Σ(B, ι) is a set of representatives for the action of(4) Note added in proof: this formula has now been proven.
(5) Note that, as p-adic heights factor through the p-adic Abel–Jacobi map, their non-vanishing is a finer statement.

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Gal(K/M ) on Hom(K, C). Finally, to the CM type (K, Σ) we can associate
its reflex CM type (K ∗ , Σ∗ ); the reflex field K ∗ = K ∗ (ι) (which depends onι) comes as a subfield K ∗ ⊂ E. The set ΣE := Inf E/K ∗ Σ∗ ⊂ Hom(E, C),consisting of those embeddings whose restriction to K ∗ belongs to Σ∗ , isa CM type of E. Finally, the CM type Σ(ι) gives rise to a homomorphismNΣ(ι)∗ : K ∗ (ι)× → K × called the reflex norm. The homomorphism(2.1)

NΣE := NΣ(ι)∗ ◦ NE/K ∗ (ι) : E × → K ×

is independent of choices.
A CM type Σ0 of K 0 with values in an extension C of Qp splitting E
is said to be a p-adic CM type (6) if its elements induce pairwise distinct
p-adic places of K 0 . This condition can only be satisfied if all primes p0+ of
K 0+ (the maximal totally real subfield of K 0 ) above p split in K 0 . We may
and will identify a p-adic CM type with a set of primes p0 of K 0 containing
exactly one prime above each p0+ |p of K 0+ .
Lemma 2.1. — Suppose that B has potentially ordinary reduction at
all primes of E above p. Then:(1) for each embedding ιp : E ,→ Qp , the set Σ(B, ιp ) is a p-adic CM
type of K;(2) the prime p is totally split in K ∗ ;(3) for each ιp : E ,→ Qp , the set Σ∗ (ιp ) := Σ(B, ιp )∗ is a p-adic CM
type of K ∗ .

Proof. Part (1), which is in fact equivalent to the hypothesis of thelemma, can be checked after base-change from E to a finite extension overwhich B acquires good reduction. There it becomes a well-known immediate consequence of the Shimura–Taniyama formula [17, (2.1.4.1)]. Part(1) implies part (2) by [29, Proposition 7.1]. Part (3) is implied by part(2).
The main theorem of Complex Multiplication attaches to B a character××
λ : EA∞ → K

such that(2.2)
×
for all τ : E ,→ C, λτ := τ ◦ λ · (NΣτ E ,∞ )−1 : EA
→ C×(6) In some of the literature this is called a p-ordinary CM type.

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satisfies λτ|E × = 1,

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ASHAY A. BURUNGALE AND DANIEL DISEGNI(2.3)

λ(x)λ(x)ρ = |x|−1A∞

×for all x ∈ EA∞.

××
Here ρ is the complex conjugation in K and NΣτ E ,∞ : E∞
→ K∞
→ Kτ×
is the continuous extension of NΣE . We say that λτ is an an algebraic
Hecke character of infinity type ΣτE , where ΣτE ⊂ Hom(E, C) is defined byQ×NΣτ E,∞ (x) = ι∈Στ ι(x) for all x ∈ E∞.
E
The L-function of B is(2.4) L(B, s) = L(s, λ) := (L(s, λτ ))τ ∈Hom(K,C) ∈ K ⊗ C ∼= CHom(K,C) ;it satisfies a functional equation with centre at s = 1.
Suppose now that B = AE for an abelian variety A/F with End0 (A) =M (the maximal totally real subfield of K). Then E = K ∗ F [42, Remark
20.5].
Suppose conversely that λ is a Hecke character of E satisfying the conditions (2.2), (2.3) for a CM type (K, Σ). Then by a theorem of Casselman(e.g. [17, Theorem 2.5.2]), there is an abelian variety B = Bλ /E, unique up
to E-isogeny, satisfying (2.4). The abelian variety B is simple if and only
if the CM type (K, Σ) is not induced from a CM type of a subfield of K.

2.2. Theta lifts of Hecke characters××→ Q be finite
Let Q denote an algebraic closure of K, let χ0 : E × \EA
order character, and let ψ be a Hecke character of E with the same CM
type as the character λ from the Introduction. We suppose that

(2.5)

χ0 |F × = ω := η · ψ|F × · | |AF .
A

A

Let K 0 ⊂ Q be a CM extension of K containing the values of χ0 andψ|E ×∞ .
A

The abelian variety associated with ψ. By construction, ψ satisfies
conditions (2.2), (2.3) for the CM type (K 0 , Inf K 0 /K Σ), and in particular
it is associated with an abelian variety Bψ,K 0 /E of dimension [K 0 : Q]/2,
which is uniquely determined up to E-isogeny, has CM by K 0 , and is Klinearly isogenous to a sum of copies of a simple CM abelian variety. (Notethat Bψ,K 0 depends on the choice of K 0 : if K 0 ⊂ K 00 is a finite extension of\⊕d0
CM fields of degree d0 , then Bψ,K 00 ∼ Bψ,K
0 .) We denote by Bψ any one of
the simple E-isogeny factors of Bψ,K 0 .

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On the other hand, the theta correspondence attaches to ψ a Gal(M/Q)conjugacy class (for some totally real field M ⊂ K 0 ) of cuspidal automorphic representations σ = (σ τM )(τM ∈Hom(M,C)) of ResF/Q GL2 of parallel0
weight 2; namely σ τM = θ(ψ τ ) if τ 0 : K 0 ,→ C satisfies τ 0 |M = τM .(7)Let A := Aσ /F be the simple abelian variety associated with σ = θ(ψ),
which is determined uniquely up to F -isogeny. Suppose that
(2.6)

ε(1/2, σE ⊗ χ−10 ) = −1

As discussed in the introductions to [45] or [20], the condition (2.6) guarantees that A can be found as an isogeny factor of the Jacobian of a Shimura
curve over F ; its endomorphism algebra End0 A is a totally real field ofdimension d = dim A, which can be identified with M .
0
For any embedding τM : M ,→ C, we have L(s, σ τM ) = L(s, ψ τ ) forτ 0 : K ,→ C such that τ 0 |M = τM . The following consequences of this0
identity are proved in [23] (note that the CM type of ψ τ is ΣE ): (1) A
acquires complex multiplication by K over some finite extension of F , andin fact, as remarked before, a minimal such extension is K ∗ F = E; (2) AE
is isogenous to a sum of copies of the abelian variety Bψ\ defined above.
As M is contained in the maximal real subfield of K 0 , the dimension[M : Q] of AE divides the dimension [K 0 : Q]/2 of Bψ,K 0 ; hence in fact
Bψ,K 0 is K-linearly isogenous to a sum of copies of AE , i.e. for some r > 1,
Bψ,K 0 ∼ A⊕r
E .

(2.7)

2.3. Rankin–Selberg p-adic L-function
Moving to a more general context for this subsection only, let F be a
totally real field, E a CM quadratic extension of F . Let M be a number
field and let σ be an M -rational automorphic representation of ResF/Q GL2of parallel weight 2, with central character ω. Let Mv be a p-adic completionof M , let L be a finite extension of Mv , and let
×χ0 : E × \EA→ L×

be a finite-order character satisfying χ0 |F × = ω. (We will later specialiseA
to the situation σ = θ(ψ) considered in §2.2). We recall the definition of ap-adic Rankin–Selberg L-function on YL attached to the base-change σEof σ to E twisted by χ−1
0 .
(7) In the terminology of [45, 20], σ is an M -rational representation.

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ASHAY A. BURUNGALE AND DANIEL DISEGNIGiven a place ℘|p of F , we say that σ is nearly ordinary at ℘ withunit characters α℘ if, after possibly enlarging L, there exist charactersα℘ : F℘× → OL× , such that σ℘ ⊗ L is either special α℘ · St with character α℘ ,or irreducible principal series Ind(|·|℘ α℘ , β℘ ) (un-normalised induction) forsome other character β℘ .
Theorem 2.2. — Suppose that for all ℘|p, σ is nearly ordinary at ℘.
Then there is a function
Lp (σE ⊗ χ−10 )∈Λcharacterised by the interpolation property0ι
0 ι,−1Lp (σE ⊗ χ−1)·0 )(χ ) = ep (σE ⊗ (χ0 χ )ι⊗ (χ0 χ0 )ι,−1 )L(p) (1/2, σEΩισ×for all sufficiently p-ramified(8) finite order characters χ0 : Γ → Q andQιι⊗(χ0 χ0 )ι,−1 ) = ℘|p e℘ (σE⊗ι : Q ,→ C. Here Ωισ := L(1, σ ι , ad) and ep (σE(χ0 χ0 )ι,−1 ) with
(2.8)

ι
−1
−1
e℘ (σE⊗ (χ0 χ0 )ι,−1 ) = ε(0, ι(α℘
χ0,p χ0p )) · ε(0, ι(α℘χ0,pc χ0pc )).

Proof. This is essentially [20, Theorem A]. Our Lp (σE ⊗χ−10 ) differs fromthe one of loc. cit. by the involution χ0 7→ χ0−1 , the shift χ−1
0 , and somealgebraic constants. Regarding the interpolation factors, note that if χ0p , χ0pc
are sufficiently ramified then all local L-values in the interpolation formulafrom [20] are equal to 1. The relation between the Gauß sums used in [20]and our epsilon factors (2.8) follows from e.g. [13, (23.6.2)].


2.4. Katz p-adic L-function
From now until the end of this paper, let K, E, ΣE be as in §2.1. Fix ap-adic place w of K, and denote by v be the induced place of the maximal
real subfield M ⊂ K. We let Q ⊂ Qp ⊃ Kw be algebraic closures of K andof Kw , and let Cp be the completion of Qp . We also let L be a sufficiently
large finite extension of Kw inside Qp as in the introduction.
(8) See the proof for the precise meaning.

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(w)

×
p-adic CM type associated with w. Let NΣE ,p : Ep× → Kp× → Kw
be the continuous extension of NΣE . We let
(w)

(w)

ΣE := {p|p prime of E : | · |w ◦ NΣE ,p |Ep× is a non-trivial norm on Ep× },which, by Lemma 2.1 and the construction of NΣE , has the property that
(w)
(w)
(w)
ΣE t ΣE c is the set of all primes p of E above p. Hence ΣE is a p-adic
CM type of E, identified with a set of embeddings E ,→ Qp .
Lemma 2.3. — Every prime of F above p splits in E, and the CM type(w)
ΣE is a p-adic CM type of E.

Proof. The first assertion is part (2) of Lemma 2.1. The second asser(w)tion follows from part (3) of the same lemma, after noting that ΣE =∗Inf E/K ∗ Σ (ιp ) for any ιp : E ,→ Qp inducing a prime in ΣE .
p-adic Hecke characters. Assume that L splits E, and let
××χ : E × \EA∞ → L

be a locally algebraic p-adic character. We say that χ is a Hecke characterof p-adic infinity type k ∈ Z[Hom(E, L)] if there exists an open subgroup×U ⊂ EA∞ such thatYσχ(t) =t−kσσ∈Hom(E,L)

for t ∈ U . Note that this definition differs from the one in some of theliterature (e.g. [22] and [25]) by a sign in the exponents; it agrees with [4].
If χ is a locally algebraic character of p-adic infinity type k and ι : L → C
×is an embedding, we can define the ι-avatar χι : EA
→ C× similarly to[32, Def. 1.5]. The embedding ι induces an isomorphism Hom(E, L) →Hom(E, C) by σ 7→ ι ◦ σ; the infinity type k ι ∈ Z[Hom(E, C)] of χι corresponds to k under this bijection. The association χ → χι defines a bijection
between locally algebraic Hecke characters over E with values in L andarithmetic Hecke characters over E in the usual sense.
For example, if λ is as in §2.1, the character(w)××
λ(w) (x) := λ(x)NΣE ,p (xp )−1 : E × \EA∞ → KwSUBMITTED ARTICLE : HEIGHT.TEX

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ASHAY A. BURUNGALE AND DANIEL DISEGNI(w)

is a Hecke character of p-adic infinity type ΣE .(9) The place w being fixed,in the rest of this paper we will often abuse notation by simply writing λin place of λ(w) .
Katz p-adic L-function. Let now L be an extension of Kw splitting]
E, and let ΣE be a p-adic CM type of E over some L. Let E∞
be a
ab
]
]
finite extension of E∞ contained in E , and let Γ := Gal(E∞ /E). Let
Λ] := Zp JΓ] KL , Y ] := Spec Λ] .
By [30, 22] and the first assertion of Lemma 2.3, there is an element
LΣE ∈ Λ]uniquely characterised by the interpolation property that we now describe.
The domain of interpolation consists of locally algebraic p-adic Hecke char×acters λ0 : Γ] → Qp with infinity type(w)kΣE + κ(1 − c)(w)

for k ∈ Z, κ ∈ Z[ΣE ] such that(i) k > 1, or(w)(w)(ii) k 6 1 and kΣE + κ ∈ Z>0 [ΣE ].
The interpolation property is then the following. There exist explicit p-adic
periods
ΣEΩΣE = (ΩΣE ,p )p∈ΣE ∈ (Z̄×p)

and, for each complex CM type ΣE,∞ of E, complex periodsΩΣE,∞ = (ΩΣE,∞ ,τ )τ ∈ΣE,∞ ∈ (C× )ΣE,∞×(both defined in [22, (4.4)]) such that for any character λ0 : Γ → Qp in thedomain of interpolation, and any ι : L(λ0 ) ,→ C, we have(10)!
×: OF× ]
LΣE (λ0 )L(p) (0, ((λ0 )−1 )ι ) π κ ΓΣ (kΣE,ι + κ) [OE
0 −1 ι
p
ι
=
e
((λ
)
)
)
·
·
·
p
k+2κ
(Im ϑ)κ
Ωk+2κ
ΩΣ
|DF |
Σ
,
E

E,ι

In the interpolation formula, ΣE,ι is the complex CM type induced from(w)
ΣE via ι, and we then identify κ with κι ∈ Z>0 [ΣE,ι ]; when k ∈ Z appears(9) Strictly speaking this assertion holds after considering λ(w) as valued in some extension L ⊃ Kw splitting E.
(10) Note that we are ignoring interpolation factors at places away from p appearing
elsewhere in the literature, since those, while non-integral, can be interpolated by polynomial functions on Y ] .

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ON THE NON-VANISHING OF p-ADIC HEIGHTS

in the exponent of one of the periods ΩΣ it is considered as k ·χ = (λ0 )−1 is ramified at all p|p, the p-Euler factor is given byYe(χιp )
ep (χι ) =Pτ ∈ΣιE τ . If

(w)

p∈ΣE

for (dropping all superscripts ι)(2.9)

e(χp ) =L(0, χp )ε(0, χp )L(1, χ−1p )QFinally, ΓΣ (kΣE,ι + κ) = τ ∈ΣE,ι Γ(k + κτ ) for the usual Γ-function, andϑ ∈ E as in [27, §3.1]. All local epsilon factors in this paper are understoodwith respect to some uniform choice of additive characters of Ep of level
one for all p|p.
For a later consideration, we fix a sufficiently large extension E ] as above
and consider the restriction of LΣE to certain open subsets of Y ] : if λ0 isa p-adic Hecke character factoring through Γ] with values in L, and ΣE isa p-adic CM type, we defineLΣE ,λ0 ∈ ΛbyLΣE ,λ0 (χ0 ) := LΣE (λ0 χ0 ).
We will henceforth drop the superscript w from the notation for thecharacter λ(w) .

2.5. Factorisation of the Rankin–Selberg p-adic L-function
Let σ := θ(ψ) as in §2.2. There is a factorisation(2.10)L(s − 1/2, σE ⊗ (χ0 χ0 )−1 ) = L(s, ψ(χ0 χ0 )−1 )L(s, ψ ∗ (χ0 χ0 )−1 ))of complex (more precisely K ⊗ C-valued) L-functions, valid for algebraic
Hecke characters χ0 over E. It implies the following factorisation of p-adic
L-functions.
Lemma 2.4. — Let Lp (σE ⊗ χ−10 ) be the p-adic L-function associatedwith σ = θ(ψ) and the embedding M ⊂ Mv ⊂ Kw , where w is the placeof K fixed above and v its restriction to M . Let L be a finite extension of
Kw splitting E. We haveLΣE c,ψ∗ χ∗−1
. LΣE ,ψχ∗−1
0
0(2.11)·Lp (σE ⊗ χ−1
0 )=
Ωp,ΣE
Ωp,ΣE cSUBMITTED ARTICLE : HEIGHT.TEX

16

ASHAY A. BURUNGALE AND DANIEL DISEGNI.
in Λ = ΛL , where we use the symbol = to signify an equality which holds
×
up to multiplication by a constant in Q .

Proof.
We evaluate both sides of the proposed equality at finite order characters
χ0 of Γ which are sufficiently ramified in the sense that, for all primes p|pof E, χp has conductor larger than the conductors of ψp and χ0,p . This issufficient as such characters are dense in Y − (cf. [20, Lemma 10.2.1]).
Note first the self-duality relation
λ0∗ = λ0−1 | · |−1AE(2.12)∗−10(where | · | = | · |AE ), valid for both λ0 = ψχ−1
(this follows
0 and λ = ψχ0from (2.5)). Evaluating (2.10) at s = 1, we findL(1/2, σE ⊗ (χ0 χ0 )−1 ) = L(0, ψ| · |(χ0 χ0 )−1 )L(0, ψ ∗ | · |(χ0 χ0 )−1 )= L(0, (ψ ∗ χ∗−1χ0 )−1 )L(0, (ψχ∗−1χ0 )−1 ).
00
Therefore the L-values agree with the ones being interpolated by thep-adic L-functions in (2.11). We now compare the local Euler-like interpolation factors and the complex periods.
Recall from e.g. [26, p. 119] that, for ℘OE = ppc , we haveσ℘ ' π(ψp , ψpc )(w)

By the construction of ΣE and the Shimura–Taniyama formula for Hecke
characters (see [17, Proposition A.4.7.4 (ii)], cf. also the paragraph after
Example A.4.8.3 ibid.), the character ψp has values in w-adic units if and(w)(w)
only if p ∈/ ΣE , equivalently p ∈ ΣE c.
Denoting by ℘ a fixed prime of F above p and by p the unique prime in(w)
ΣE above ℘, it follows that α℘ = ψpc under the identification Epc = F℘ .
Then, under our assumption on the ramification of χ0 , we have
(2.13)0−10−10e℘ (σE ⊗ χ−1
0 , χ ) = ε(0, ψpc χ0,p χp ) · ε(0, ψpc χ0,pc χpc )whereas the Katz interpolation factors above ℘ areε(0, ψp−1 χ0,pc χ0p −1 )−1 · ε(0, ψp−1 χ0,p χ0pc −1 )−1 .
By the functional equation
ε(s, χ)−1 = ε(1 − s, χ−1 )χ(−1)

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valid for any character χ of a local field (e.g. [13, (23.4.2)]) and the selfdualities (2.12), these equal−1 00
ε(0, | · |ψp χ−10,pc χp ) · ε(0, | · |ψp χ0,p χpc )0−10= ε(0, ψp−1
c χ0,p χp ) · ε(0, ψpc χ0,pc χpc )matching (2.13).
In regard to periods, we first note that the periods in the asserted equality
are independent of χ0 . As
∗ −1),Ad(σ) ' η ⊕ IndF
E (ψ(ψ )we haveL(1, σ ι , ad) = L(1, η)L(1, ψ(ψ ∗ )−1 ) = L(1, η)L(0, (ψ(ψ ∗ )−1 )D )The infinity type of ι(ψ(ψ ∗ )−1 )D is 2ΣιE . From the algebraicity of Hecke
L-values due to Shimura, we have.
L(0, (ψ(ψ ∗ )−1 )D ) = Ω2ΣιE(see [40] and [22, §4], especially [22, pp. 215-16]). Moreover, we have aperiod relation.
Ω1ΣιE = Ω1ΣιE ◦c(see [41] and [42, Thm. 32.5]).
It follows that.
.
Ωισ = Ω2ΣιE = ΩΣιE ΩΣιE ◦c .


2.6. Construction of an auxiliary character
We will now look for a character χ0 suitable for our purposes. If a Hecke×
character λ0 of EA
satisfies (2.12), then its functional equation relates0L(s, λ ) with L(1 − s, λ0−1 ) = L(2 − s, λ0∗ ) = L(2 − s, λ0 ); the sign ofthis functional equation is the root number
w(λ0 ) := ε(1, λ0 ) ∈ {±1}.
××is anticyclotomic
We say that finite-order character χ : E × \EA∞ → Q
if it satisfies the following two conditions, which are equivalent by [24,Lemma 5.31]:(1) χ∗ = χ−1 ;SUBMITTED ARTICLE : HEIGHT.TEX

18

ASHAY A. BURUNGALE AND DANIEL DISEGNI

×(2) there exists a finite-order character χ0 : E × \EA∞ → Q(2.14)

×such thatχ = χ0 /χ∗0 .

Lemma 2.5. — Let λ be a Hecke character satisfying (2.12). Supposethat the extension E/F is ramified. Then there exist an anticyclotomic
××
finite-order character χ = χ0 /χ∗0 : E × \EA→ Q such that the root numberw(λχ) = +1.

Proof.
Suppose first that λ satisfies the following condition. (We will later reduceto this case.)(∗) there is a prime ℘ of E, ramified over F , such that ord℘ (C) is odd,
where C is the conductor of λ. In particular, the norm ideal NE/F (C) is
not a square. For a quadratic character χ0 over F , let χ0E = χ0 ◦ NE/F be
the corresponding Hecke character over E. By definition, χ0E is a quadratic
Hecke character over E and also anticyclotomic. For the latter, note that0002
χ0E (a)χ0∗
E (a) = χ (NE/F (a))χ (NE/F (c(a))) = χ (NE/F (a)) = 1.

We consider twists of λ by characters of the form χ0E with the conductorof χ0E prime to C. Recall that the twist of a self-dual character by an
anticyclotomic character is again self-dual. To prove existence of twist withchange in the root number, it thus suffices to show that χ0 can be chosen
so that χ0 (NE/F (C)) takes value 1 or −1. The sufficiency follows from theexplicit root number formula for the twist λχ0E in [44, (3.4.6)]. As the norm
ideal NE/F (C) is not a square, the existence of desired χ0 follows readily.
To reduce to condition (∗), it suffices, given λ, to find λ0 = λχ1 /χ∗1
satisfying (∗). Let ℘ be a prime of E ramified in E/F and let δ := ord℘ (2).
Let r be an odd integer greater than δ + 2 and the exponent of ℘ in the×conductor of λ. Let χ1 be a finite order character of EA/E × such that theexponent of ℘ in the conductor of χ1 is exactly r − δ. As (1 + ℘t−δ OE )2 =1 + ℘t OE for all t > δ + 1, the conductor of χ21,℘ is exactly ℘r . Lettingχ = χ1 /χ∗1 , and denoting by $ a uniformiser at ℘, for any t > δ + 1 wehave for any t > (r − δ)/2:χ℘ (1 + $t a) = χ1,℘ (1 + $t a)χ1,℘ (1 − $t a)−1 = χ1,℘ (1 + $t a)2 .

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It follows that the conductor of χ℘ is the same as the conductor of χ21 ,that is, ℘r ; by our choice of r the same is true of λχ, and in particular λ0
satisfies (∗).

Remark 2.6. — Let χ = χ0 χ∗−1
be as in the Lemma and let ψ :=0
λχ0 , where λ is as fixed in the Introduction. Then ψ satisfies (2.5); by theanalogous formula to (2.10), σ := θ(ψ) has root number −1 (i.e., it satisfies(2.6)).

2.7. p-adic Gross–Zagier formula
We recall a formula relating Rankin–Selberg p-adic L-functions to points
on abelian varieties; we will later deduce from it an analogous formula for
Katz p-adic L-functions. Let σ, A = Aσ , and χ0 be as in Theorem 2.2, andassume that E/F splits at each prime above p. and thatε(1/2, σE ⊗ χ−10 ) = −1.

(2.15)

Theorem 2.7. — Let ◦ = ∅ or ◦ = ℘ for a prime ℘|p of F . There is a‘pair of points’−1Pχ0 ⊗ Pχ∨0 ∈ AE (χ−1A∨K◦− ,E (χ0 χuniv,◦ ) ⊗Λ−
0 χuniv,◦ ) ⊗Λ−◦◦

such thathPχ0 ⊗ Pχ∨0 i = L0p (σE ⊗ χ−1
0 )in K◦− . Here hx ⊗ yi := hx, yi is the big height pairing relative to thecyclotomic logarithm as in (1.6).

Proof. This follows from [20, Theorem C.4]. Consider the scheme YV p /Lcorresponding to the rigid space with that name in loc. cit. (in the sensethat our YV p is the spectrum of the ring of bounded functions on the space×
YV p of [20]), for a choice of level V p ⊂ EA
p∞ ; it parametrises continuous×× p
p-adic characters χ
e of EA∞ /E V satisfying ω χ
e|F ×∞ = 1. We denote
A

×
× p
by χ
euniv : EA
→ O(YV )× the universal character. Assume that
∞ /E V
ω|V p = 1. Then we may identify Y − with the connected component Yχ◦0 ⊂YV p containing the character χ−1
0 via(2.16)SUBMITTED ARTICLE : HEIGHT.TEX−1
χ 7→ χ
e = χ−1.
0 χ

20

ASHAY A. BURUNGALE AND DANIEL DISEGNI

Using the notation of loc. cit. with the addition of a tilde, the p-adic
Gross–Zagier formula proved there has the formf+ (f +,p ), P
f− (f −,p )ι i = L
e +,p , f −,p )|Ye 0p (σE ) · Q(f
hP
χ0

in Λ−

up to an explicit and nonzero rational constant. Here ι is the involution
χe 7→ χ
e−1 on YV p , and thef± (f ±,p ) ∈ A± ((ePχuniv )±1 )are families of Heegner points associated with E/F and (the limits of certain sequences of) parametrisations f ±,p of A and A∨ by a (tower of)e ·) ∈ Λ− is a product of local terms at
Shimura curves. The term Q(·,
primes not dividing p.
By results of Tunnell and Saito explained in [20, Introduction], under theassumption (2.15) for each χ ∈ Yχ0 we may find families of Shimura curve
e +,p , f −,p )(χ) 6= 0. Applying this result
parametrisations f ±,p such that Q(fto a character χ in the image Yχ0 ,◦ of Y◦− under the isomorphism Y − →e +,p , f −,p )|YYχ0 , we find f ±,p such that Q(fχ0 ,◦ 6= 0. Up to constants in
−1
−1 −1
×
0
0
e
L , we have Lp (σE ⊗ χ0 )(χ) = −Lp (σE )(χ0 χ ). Then we may choose,using the identification (2.16)
e +,p , f −,p )|−1 · P(ff + )|Yf − )ι |YPχ0 ⊗ Pχ∨0 := −Q(f
⊗ P(f
.
χ0 ,◦
χ0 ,◦
Yχ ,◦
0

There are four conditions to be verified in order to be able to invoke theresult of [20].(11) The first one is (weaker than) the potential ordinarinessof A, which can be verified after base-change to E where it becomes theconverse to part (1) of Lemma 2.1. The second one is that all primes of F
above p split in E: this is satisfied by Lemma 2.3. Finally, the conditions
on the central character and root number are satisfied by Remark 2.6. 
2.8. Non-vanishing of p-adic L-functions
Let λ be the character fixed in the Introduction.
Theorem 2.8. — Let χ = χ0 /χ∗0 be as in Lemma 2.5. For every ℘|p,
the restriction of the anticyclotomic Katz p-adic L-function
LΣE ,λχ∗ χ−1 |Y℘−00does not vanish.

(11) It is crucial here that in [20] the sets of ramified primes of E/F and of bad-reduction

primes for A are not required to be disjoint.

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Proof. By construction, w(λχ∗0 χ−10 ) = +1. The theorem thus follows from[10, Thm. B] combined with the main results of [25] and [27]. Here we only
use the hypothesis p - 2DF .

Theorem 2.9. — For every ℘|p, the restriction of the cyclotomic derivative
L0ΣE c,λ∗ |Y℘−does not vanish.

Proof. Recall from the Introduction that λ∗ is self-dual with infinity typeΣE c and root number −1. The theorem thus follows from [10, Thm. C]
combined with the main result of [6]. Here we use the hypothesis p - 2DF h−E.
2.9. Proofs of main theorems
We introduce the useful category
CME,(K,Σ) ,described as follows. The objects are abelian varieties B over E of dimension
equal to d2 [K : Q] for some d > 1, together with an inclusion i : R ,→End0 (B) of a K-algebra R of dimension d such that the type of i is (K, dΣ).
For two objects B = (B, R, i), B 0 = (B, R0 , i0 ) of CME,(K,Σ) , let R◦ ⊂ R
0and R◦ ⊂ R0 be finite-index subrings in the integral closure of OK in R,
R0 whose image by i, i0 is contained in End(B), End(B 0 ) respectively. Let
preHomCME,(K,Σ) (B, B 0 )be the set of pairs of morphisms (f, γ) with f : B → B 0 , γ : R → R0 suchthat i0 (γ(r)) ◦ f = f ◦ i(r) for any r ∈ R. This is a module over a sufficiently
small order O in K, and we let
HomCME,(K,Σ) (B, B 0 ) := preHomCME,(K,Σ) (B, B 0 ) ⊗O K.
If (B, i, R) is an object of CME,(K,Σ) and T is a finite-dimensional Kalgebra, then Serre’s construction provides a well-defined isomorphism class(B ⊗K T, i ⊗ idT , R ⊗K T ) of objects in CME,(K,Σ) , with action by the Kalgebra R ⊗K T .
Note that in CME,(K,Σ) any object (B, i, R) is isomorphic to one suchthat End(B) contains the integral closure Rint of OK in R: namely, if R◦ ⊂SUBMITTED ARTICLE : HEIGHT.TEX

22

ASHAY A. BURUNGALE AND DANIEL DISEGNIRint is an order conatined in End(B) we may take B 0 := HomR◦ (Rint , B).
Given an object (B, i, R) of CME,(K,Σ) and a finite-order character χ : Gal(Hχ /E) →K × we define the twist
B ⊗K K(χ)(an object of CME,(K,Σ) with the R-action of induced from the one onB) as follows. Assume that R ⊃ OK , which as noted above is not re×strictive. Then we may regard χ as an element in H 1 (Gal(Hχ /E), OK)⊂
H 1 (Gal(H/E), AutK (B)); the abelian variety B⊗K K(χ) is the corresponding inner twist (denoted by B χ in [4]), so that for any finite characterχ0 : Gal(E/E) → K × we have (B ⊗ K(χ)(χ0 ) = B(χχ0 ).
Let us now return to our usual setting, so that B = Bλ . It is an object ofCME,(K,Σ) with R = K. (Note that, as the validity of the statements we areinterested in is invariant under K-linear isogenies, it is appropriate to workin this category.) Let χ0 : Gal(E/E) → K 0× be a finite-order character,
where K 0 is the CM extension of K fixed above. Let ψ := λχ−1and let0
A = Aσ be the abelian variety associated with σ = θ(ψ) as in 2.2. The
abelian varieties Aσ and Bψ,K 0 have CM by K 0 .
Lemma 2.10. — There is an isomorphism in CME,(K 0 ,Σ)
−1
0
∼ ⊕r
f : Bλ ⊗K K 0 → Bψ,K 0 ⊗K 0 K 0 (χ−10 ) = AE ⊗K K (χ0 ).

Proof. The second isomorphism is (2.7). The proof of the first one, basedon Casselman’s theorem, is entirely analogous to the proof of [4, Lemma
2.9].

Proof of Theorem 1.2. — We prove the p-adic Gross–Zagier formula ofTheorem 1.2. Let χ0 be as in Lemma 2.5 for λ0 := λ∗ , let A = Aθ(λχ−1 )0and let−1Pχ0 ⊗ Pχ∨0 ∈ AE (χ−1
A∨
K◦−
E (χ0 χuniv,◦ ) ⊗Λ−
0 χuniv,◦ ) ⊗Λ−◦◦be as in Theorem 2.7. Let f be as in Lemma 2.10 and let
P ⊗ P ∨ := LΣE c,λ∗ χ0 χ∗−1 |−1· (f −1 ⊗ f ∨ )(i1 (Pχ∨0 ⊗ Pχ∨0 ))|Y◦− ,Y−
0◦which is an element of B(χ−1univ,◦ )K◦− ⊗ B(χuniv,◦ )K◦− . by Theorem 2.8.
By the projection formula for heights [33], for any P1 ∈ B(E), P2 ∈B (E) we have∨hf −1 (P1 ), f ∨ (P2 )iB = hP1 , P2 iBψ,K 0 = hP1 , P2 iA⊕r .
E

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As maps of finitely generated Λ−◦ -modules are determined by their specialisations at finite order characters, this implies the analogous result for
big height pairings. Then Theorem 1.2 follows from Theorem 2.7 and thefactorisation (2.10).

Proof of Theorem 1.1. We state and prove the following slightly more
precise version of Theorem 1.1. Recall from the Introduction that we saythat a property P holds for almost all finite-order characters in Y◦− if theset of those χ not satisfying P is not Zariski dense in Y◦− .
We keep the assumption of Theorem 1.1.
Theorem 2.11. — For almost all finite-order χ ∈ Y◦− , we have
L0ΣE (λχ) 6= 0,
the specialisation P ⊗ P ∨ (χ) is a well-defined and non-zero element ofB(χ−1 ) ⊗ B ∨ (χ), andhP ⊗ P ∨ (χ)i =6 0.

Proof. The first assertion is equivalent to Theorem 2.9. That the points
are generically well-defined at χ amounts to the assertion thatLΣE ,λχ∗ χ−1 |Y℘− 6= 0,00which is Theorem 2.8. Finally the non-vanishing of p-adic heights follows
from the other assertions and the Gross–Zagier formula of Theorem 1.2. 
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Ashay A. Burungale
California Institute of Technology, 1200 E California
Blvd, Pasadena CA 91125
ashayburungale@gmail.com
Daniel Disegni
Department of Mathematics, Ben-Gurion Universityof the Negev, Be’er Sheva 84105, Israel
disegni@bgu.ac.il

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