THETA CYCLES AND THE BEILINSON–BLOCH–KATO CONJECTURES
byDaniel Disegni

Abstract. — We introduce ‘canonical’ classes in the Selmer groups of certain Galois representations with aconjugate-symplectic symmetry. They are images of special cycles in unitary Shimura varieties, and defined
uniquely up to a scalar. The construction is a slight refinement of one of Y. Liu, based on the conjecturalmodularity of Kudla’s theta series of special cycles. For 2-dimensional representations, Theta cycles are(the Selmer images of) Heegner points. In general, they conjecturally exhibit an analogous strong relation
with the Beilinson–Bloch–Kato conjectures in rank 1, for which we gather the available evidence.

Contents
1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. The conjecture of Beı̆linson–Bloch–Kato–Perrin-Riou. . . . . . . . . . . . . . . . . . . . . . . . . 3
3. Descent and theta correspondence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4. Theta cycles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5. Relation to L-functions and Selmer groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1. Introduction
The purpose of this largely expository note is to introduce certain Selmer classes of algebraic cycles, and discuss their relation to the Beilinson–Bloch–Kato (BBK) conjectures. These classes, called
Theta cycles, should play an analogous role to Heegner points on elliptic curves, in that the Bloch–
Kato Selmer group H f1 (E, ρ) of a relevant Galois representation ρ should be 1-dimensional precisely

when its Theta cycle is nonzero (cf. [BST21, Kim23] and references therein for the case elliptic
curves). Moreover, the BBK conjectures, reviewed in § 2, predict that the 1-dimensionality of theSelmer group is equivalent to the (complex or, for suitable primes, p-adic) L-function of ρ vanishing
too̧rder 1 at the center, and Theta cycles allow to approach this conjecture.
The following theorem summarises the state of our knowledge on the topic. Unexplained notions
or loose formulations will be defined and made precise in the main body of the paper.
We fix a rational prime p and denote by Q◦ ⊂ Q p the extension of Q generated by all roots ofunity, and we fix an embedding ι◦ : Q◦ ,→ C. We set Σ := {ι: Q ,→ C | ι ◦ = ι◦ }.
p|Q

Research supported by ISF grant 1963/20 and BSF grant 2018250. This work was partly written while the author was inresidence at MSRI/SLMath (Berkeley, CA), supported by NSF grant DMS-1928930.

2

DANIEL DISEGNITheorem A. — Let E be a CM field with Galois group GE , and let
ρ: GE → GLn (Q p )be an irreducible, geometric Galois representation of weight −1 and even dimension n. Suppose that ρ
is conjugate-symplectic, automorphic, and has minimal regular Hodge–Tate weights.
If n ⩾ 4, assume that the maximal totally real subfield F of E is not Q, and that Hypothesis 4.1 on thecohomology of unitary Shimura varieties holds.
1. Assume Hypothesis 4.3 on the modularity of generating series of special cycles. The constructionof § 4.3 attaches to ρ a pair (Λρ , Θρ ), well-defined up to isomorphism, consisting of a Q p -line Λρtogether with a Q p -linear mapΘρ : Λρ → H f1 (E, ρ),whose image is spanned by classes of algebraic cycles.
2. Suppose that E and ρ are ‘mildly ramified’ and that ρ is crystalline at p-adic places.
(a) Assume Hypothesis 4.3, as well as Conjecture 5.3 on the injectivity of certain Abel–Jacobi
maps, and that p is unramified in E. For any ι ∈ Σ, denote by Lι (ρ, s) the complex L-functionof ρ with respect to ι. Then(1)ord s=0 Lι (ρ, s) = 1 =⇒ Θρ ̸= 0.
(b) Suppose that E/F is totally split above p, that p > n, and that for every place w| p of E,
the representation ρw is Panchishkin–ordinary. Denote by XF the Q p -scheme of continuousp-adic characters of GF that are unramified outside p, by m ⊂ O (XF ) the ideal of functions
vanishing at 1, and by L p (ρ) ∈ O (XF ) the p-adic L-function of ρ. Then
ordm L p (ρ) = 1 =⇒ Hypothesis 4.3 holds and Θρ ̸= 0.
3. Assume Hypothesis 4.3 and that ρ has ‘sufficiently large’ image. ThenΘρ ̸= 0 =⇒ dimQ H f1 (E, ρ) = 1.
p

Examples of representations ρ satisfying the general assumptions of the theorem arise fromsymmetric powers of elliptic curves: namely, if A is a modular elliptic curve over F with rational
Tate module V p A, then by [NT] one may consider the natural representation ρA,n of GE on
Symn−1V p AE (1 − n/2) (see [DL24, § 1.4] for more details); in particular, for n = 2 we obtain the
representation V p AE already studied (when F = Q) by Gross–Zagier, Perrin-Riou and Kolyvagin in
the 1980s.
Part 1 of the theorem, which builds on constructions of Kudla and Y. Liu, is the main focus of
this note; it is explained in § 4, after reviewing the representation-theoretic preliminaries in § 3. The
construction is canonical up to a representation-theoretic choice described in Remark 3.5. (However, there is a ‘standard’ choice, and part 3 of the theorem indicates that this ambiguity is quite
innocuous.)In § 5, we state a pair of formulas for the Bloch–Beı̆linson and the Nekovář heights of Theta cycles,
which are essentially reformulations of a breakthrough result of Li and Liu [LL21, LL22], and of its(1)

The order of vanishing of Lι (ρ, s ) at s = 0 is conjecturally independent of ι, cf. Conjecture 2.2.

THETA CYCLES AND THE BEILINSON–BLOCH–KATO CONJECTURES

3p-adic analogue by Liu and the author [DL24]. They imply the assertions of Part 2, and take theshape〈Θρ (λ), Θρ∗ (1) (λ′ )〉⋆ = c⋆ · L′⋆ (ρ, 0) · ζ⋆ (λ, λ′ ),where ‘⋆’ stands for the relevant decorations, c⋆ are constants, and ζ⋆ are canonical trivialisations ofΛρ ⊗ Λρ∗ (1) .
Part 3 is the subject of [Dis] (itself relying on forthcoming work of Jetchev–Nekovář–Skinner),on which we only give some brief remarks in § 5.4; in particular, we sketch the relevance of theperspective proposed here for the results obtained there.
All the constructions and results should have analogues in the odd-dimensional case, in the symplectic case, and for more general Hodge–Tate types. We hope to return to some of these topics infuture work.
Acknowledgements. — It will be clear to the reader that this note is little more than an attempt tolook from the Galois side, and the multiplicity-one side, at ideas of Kudla and Liu. I would like tothank Yifeng Liu for all I have learned from him during our collaboration, and Elad Zelingher for aremark that sparked it. I am also grateful to Yannan Qiu and Eitan Sayag for helpful conversations
or correspondence, and to Chao Li and Yifeng Liu for many useful comments on a first draft.
This text is based on a talk given at the Second JNT Biennial Conference in Cetraro, Italy, in
July 2022, and I would like to thank the organisers for the opportunity to speak there. One ofthe participants reminded me of Tate’s similarly named ‘θ-cycles’ in the theory of mod- p modular
forms [Joc82, § 7]: besides the context, the capitalisation should also dispel any risk of confusion.
Homonymous objects also occur in neuroscience, in connection with a pattern of brain activity
typical of “a drowsy state transitional from wake to sleep” [McN19, pp. 60-61]; I am grateful to the
Cetraro audience for not indulging in this confusion either.
2. The conjecture of Beı̆linson–Bloch–Kato–Perrin-Riou
Let E be a number field with Galois group GE , and let
ρ: GE → GLn (Q p )be an irreducible Galois representation that is geometric in the sense of [FM95], and pure of weight−1 at all finite places (in the sense of [DL24, Definition A.11] – where at non- p-adic places, we takethe functor of [Tat79, (4.2.1)] in place of the functor WD(·) of loc. cit.).
Example. — The Galois representations attached to modular (eigencusp)forms are geometric andpure, see [Sai97, Sai00]; the weight depends on the choice of normalisation, but if the modular form
has even weight, a suitable cyclotomic twist of its Galois representation has weight −1.
2.1. Chow and Selmer groups. — A typical source of representations as above is the cohomologyof algebraic varieties. In fact, define a motivation of ρ to be an element of(2)Motρ := lim Motρ (X , k),−→(X ,k)(2)

whereMotρ (X , k) := Hom Q [G ] (Hét2k−1 (XE , Q p (k)), ρ),pE

Throughout this paper, if R → R′ is a ring map that can be understood from the context, and X is an R-scheme or an
R-module, we write XR′ := X ⊗R R′ .

4

DANIEL DISEGNI

and the limit runs over all pairs consisting of a smooth proper variety X/E and an integer k ⩾ 1 (thisis a directed system by Künneth’s fromula). We refer to (X , k) as a source of f ∈ Motρ if f is in theimage of Motρ (X , k). We say that ρ is motivic if Motρ is nonzero. According to the conjecture of
Fontaine–Mazur, every geometric irreducible Galois representation is motivic.
To a representation ρ as above is attached its Bloch–Kato [BK90] Selmer group H f1 (E, ρ).(3) To
a variety X/E as above is attached its Chow group Chk (X ) of codimension-k algebraic cycles on
X up to rational equivalence (with coefficients in Q). A central object of arithmetic interest is its
subgroup Chk (X )0 := Ker [Chk (X ) → Hét2k (XE , Q p (k))] (where the map is the cycle class). It is
Qp

endowed with an Abel–Jacobi map
AJ: Chk (X )0 → H 1 (E, Hét2k−1 (XE , Q p (k)))
Qp(see [Nek93, § 5.1]) whose image is conjectured to land in H f1 (E, Hét2k−1 (XE , Q p (k))). We can define
an analogue of the image of AJ for the representation ρ byX
H f1 (E, ρ)mot :=f∗′ AJ(Chk (X )0 ) ∩ H f1 (E, ρ) ⊂ H f1 (E, ρ),Qpf ′ ∈Motρ

where we have denoted by (X , k) any source of the motivation f ′ . By an evocative abuse of nomenclature, we refer to elements of H f1 (E, ρ)mot as cycles.
2k −1

Remark 2.1. — If ρ = Hét 0 (X0,E , Q p (k0 )) for a variety X0 and an integer k0 , then we expect thatH f1 (E, ρ)mot = AJ(Chk0 (X0 )0 ). This equality is implied by the Tate conjecture [Tat65, ConjectureQp1] for X × X0 .
2.2. The conjecture. — We say that ρ is (Panchishkin-) ordinary (see [Nek93, § 6.7], [PR92, § 2.3.1]
for more details) if for each place w| p, there is a (necessarily unique) exact sequence of De Rham GEw -

0
0
−
+
−
representations 0 → ρ+
w → ρ|GE → ρw → 0, such that Fil DdR(ρw ) = DdR(ρw )/Fil = 0. For any

subfield F ⊂ E, letw

XF := Spec Z p JGal(F∞ /F )K ⊗Z p Q p ,where F∞ /F is the abelian extension with Gal(F∞ /F ) isomorphic (via class field theory) to thec p,× .
maximal Z -free quotient of F × \A× /Op

FF

One can conjecturally attach to ρ entire L-functionsLι (ρ, s)for ι: L ,→ C and, (at least) if ρ is ordinary, a p-adic L-function
L p (ρ) ∈ O (XF )interpolating suitable modifications of the L-values Lι (ρ⊗χ|GE , 0) for finite-order characters χ ∈ XF
(see [PR95], at least when taking F = Q).
(3)

N.B.: the subscript f has nothing to do with names of objects elsewhere in this text. Galois cohomology and Selmer
groups are usually defined for representations with coefficients in finite extensions of Q p . However, it is well-known that
we can write ρ = ρ0 ⊗L Q p for some finite extension L ⊂ Q p of Q p and some representation ρ0 : GE → GLn (L) (andsimilarly for the other representations considered in this paper). Then we define H f1 (E, ρ) := H f1 (E, ρ0 ) ⊗L Q p .

THETA CYCLES AND THE BEILINSON–BLOCH–KATO CONJECTURES

5

Denote by m = mF ⊂ O (XF ) the maximal ideal of functions vanishing at the character 1 ofGal(F∞ /F ), and by ordm the corresponding valuation. The integer ordm L p (ρ) is conjecturally independent of the choice of F .
Conjecture 2.2 (Beı̆linson, Bloch–Kato, Perrin-Riou [Beı̆84, BK90, PR95])
Let ρ: GE → GLn (Q p ) be an irreducible geometric representation of weight −1. Let r ⩾ 0 be an
integer. The following conditions are equivalent:(a)∞ for any ι: Q p ,→ C, we have(b)

ord s=0 Lι (ρ, s) = r ;dimQ H f1 (E, ρ)mot = dimQ H f1 (E, ρ) = r .
ppGEIf moreover ρ is ordinary and ρ+,∗w (1) w = 0 for every w| p, then the above conditions are equivalentto(a) p ordm L p (ρ) = r ;Remark 2.3. — The first equality in (b) generalises the conjectural finiteness of the p ∞ -torsion in
the Tate–Shafarevich group of an elliptic curve. The extra condition in (a) p serves to avoid thephenomenon of exceptional zeros, cf. [Ben14].
In the following pages, under some restrictions on ρ we will define elements in H f1 (E, ρ)mot whose

nonvanishing is conjecturally equivalent to the conditions of Conjecture 2.2 with r = 1. The construction will be automorphic; in the next section, we give the representation-theoretic background.
3. Descent and theta correspondence
Suppose for the rest of this paper that E is a CM field with totally real subfield F . We denote byc ∈ Gal(E/F ) the complex conjugation, and by η: F × \A× → {±1} the quadratic character attached
to E/F .
3.1. p-adic automorphic representations. — We denote by A the adèles of F ; if S is a finite setof places of F , we denote by AS the adèles of F away from S. If G is a group over F and v is aQplace of F , we write Gv := G(Fv ); if S a finite set of places of F , we write GS := v∈S G(FS ). (Fornotational purposes, we will identify a place of Q with the set of places of F above it.) We denote by
ψ: F \A → C× the standard additive character with ψ∞ (x) = e 2πiTrF∞ /R x , and we set ψE := ψ◦TrE/F .
We view ψ|A∞ as valued in Q◦ via the embedding ι◦ .
Unitary groups. — Fix a positive integer n. For a place v of F , we denote by Vv be the set of isomorphism classes of (nondegenerate) Ev /Fv -hermitian spaces of dimension n; this consists of one element
if v splits in E, of two elements if v is finite nonsplit, and of n +1 elements if v is real. We denote by
V + the set of isomorphism classes of E/F -hermitian spaces of dimension n that are positive definite
at all archimedean place, and by V − the set of isomorphism classes of E/F -hermitian spaces of dimension n that are positive definite at all archimedean place but one, at which the signature is (n − 1, 1).
We denote by V ◦ the set of isomorphism classes of AE /A-hermitian spaces of dimension n such thatnfor all but finitely many places v, the Hasse–Witt invariant ε(Vv ) := ηv ((−1)( 2 ) detVv ) = +1, andQ
that Vv is positive definite at all archimedean places. We put ε(V ) := v ε(Vv ), and write V ◦,ε ⊂ V ◦for the set of spaces with ε(V ) = ε ∈ {±}.

6

DANIEL DISEGNI

We have a natural identification V ◦,+ = V + . We will mostly be interested in V ◦,− , which werefer to as the set of incoherent E/F -hermitian spaces, cf. [Gro21]. If V ∈ V ◦,− , then for everyarchimedean place v of F , there exists a unique V (v) ∈ V − over F such that V (v)w ∼= Vw if w ̸= v.
◦
For V ∈ V , let HV = U(V ); if V ∈ V with ε(V ) = −1, we still use the notation HV (AS ) :=Qv ∈S
/ HV , HV := U(Vv )(Fv ), and we refer to (the symbol)v

v

HV
as an incoherent unitary group.
Suppose from now on that n = 2r is even. We define the quasisplit unitary group over F
G = U(W ),where W = E n equipped with the skew-hermitian formsize r ).


−1 r

1r



(here 1 r is the identity matrix of

Definition 3.1. —1. A relevant complex automorphic representation Π of GLn (AE ) is an irreducible cuspidal automorphic representation satisfying:∼ Π∨ ;(i) Π ◦ c =(ii) for every archimedean place w of E, the representation Πw is induced from the characterargn−1 ⊗ argn−3 ⊗ . . . ⊗ arg1−n of the torus (C× )n = (Ew× )n ⊂ GLn (Ew ); here arg(z) :=z/|z|.
2. A possibly relevant complex automorphic representation π of G(A) is an irreducible cuspidal automorphic representation such that for every archimedean place v of F , the representation πv is the holomorphic discrete series representation of Harish-Chandra parametern−1 n−3
3−n 1−n{ 2 , 2 , . . . , 2 , 2 }. We say that π is relevant if it is possibly relevant and stable as definedat the beginning of § 3.2 below.
3. Let V ∈ V ◦,− and let v be an archimedean place of F . A possibly relevant complex cuspidal automorphic representation σ of HV (v) (A) is an irreducible cuspidal automorphic representationsuch that σv is one of the n discrete series representation of HV (v)v = U (n − 1, 1) of Harishn−1 n−3

3−n 1−nChandra parameter { 2 , 2 , . . . , 2 , 2 }, and for every other archimedean place v ′ ̸= v of
F , we have σv ′ = 1 (as a representation of HV (v) ′ = U (n)). We say that σ is relevant if it isv
possibly relevant and stable.
Definition 3.2. —1. A relevant p-adic automorphic representation Π of GLn (AE ) is a representation of GLn (A∞E ) on a Q p -vector space, such that for every ι: Q p ,→ C, the representationιΠ is the finite component of a (unique up to isomorphism) relevant complex automorphic
representation Πι .
2. A possibly relevant, respectively relevant p-adic automorphic representation π of G(A) is representation of G(A∞ ) on a Q p -vector space, such that for every ι: Q p ,→ C, the representation
ιπ is the finite component of a (unique up to isomorphism) possibly relevant, respectively relevant, complex automorphic representation πι of G(A).
3. Let V ∈ V ◦,− . A possibly relevant, respectively relevant, p-adic automorphic representation σof HV (A) is representation of HV (A∞ ) on a Q p -vector space, such that for every ι: Q p ,→ C
and every archimedean place v of F , the representation ισ is the finite component of a (unique

THETA CYCLES AND THE BEILINSON–BLOCH–KATO CONJECTURES

7

up to isomorphism) possibly relevant, respectively relevant, complex automorphic representation σ ι,(v) of HV (v) (A).
3.2. Automorphic descent. — For a place v of F , we denote by BCv the base-change map from
L-packets of tempered representations of Gv to tempered representations of GLn (Ev ), which is injective by [Mok15, Lemma 2.2.1]. We denote by BCG and BCHV the base-change maps from automorphic representations of the unitary groups G(A) or HV (A) to automorphic representationsof GLn (AE ), respectively; we simply write BC when there is no risk of confusion. We say that a
cuspidal automorphic representation of a unitary group is stable if its base-change is still cuspidal.
Remark 3.3. — We have the following properties of the base-change maps.
(a) By [LTX+ 22, Proposition C.3.1], if Π is a relevant representation of GLn (AE ), then: the preimage of Π under BCHV consists of relevant representations of HV (A); the preimage of Π under
BCG contains a relevant representation of G(A).
(b) If v is a finite place, the base-change maps may be defined for representations with coefficientsover Q p , compatibly with any extensions of scalars ι: Q p ,→ C.
(c) As a consequence of (a) and (b), BC extends to a map from relevant p-adic automorphic representations of G(A) and HV (A) to relevant p-adic automorphic representations of GLn (AE ).
Descent to a quasisplit unitary group. — We fix the auxiliary choice of a Borel subgroup B ⊂ Gwith torus T and unipotent radical N, and (the T-orbit of) a generic linear homomorphism
Ψ : N(F )\N(A) → C× ; we call this choice (N, Ψ) a Whittaker datum. A relevant complex or p-adic
automorphic representation π of G(A) is called Ψ-generic if it for every finite place, πv is Ψv -genericin the sense that it has a non-vanishing (Nv , Ψ|Nv )-Whittaker functional .
Proposition 3.4. — Let Π be a relevant p-adic automorphic representation of GLn (AE ). Then there
exists a relevant p-adic automorphic representation π of G(A), unique up to isomorphism, which is Ψgeneric and satisfies BC(π) = Π.
Proof. — By [GRS11] and [Mor18], for each ι there exists a relevant cuspidal automorphic representation πι of G(A) that is Ψ-generic and satisfies BC(πι ) = Πι . By [Var17, Ato17], for each finite
place v, each local L-packet of Gv contains a unique Ψ-generic representation, which (together with
the injectivity of BCv ) implies that πι is unique up to isomorphism. Then by Remark 3.3 (b), thecollection (πι ) arises from a well-defined relevant p-adic automorphic representation π of G(A).
Remark 3.5. — Our construction of Theta cycles will be based on the choice of a relevant representation π with BC(π) = Π, which is not unique. For definiteness, we may pick a Whittaker datum Ψ(for which, as explained in [KMSW, § 0.2.2, § 1.6.1], there is a standard choice), and take π to be theΨ-generic representation given by Proposition 3.4.
3.3. Theta correspondence. — Let π be a relevant p-adic representation of G with BC(π) = Π.
We will need to further transfer π to a representation of unitary groups HV for V ∈ V ◦,− .
Local correspondence and duality. — We first review the local theory. Let v be a finite place of F ,and let C be either Q p or C. For Vv ∈ Vv , let ωVv = ωVv ,ψv be the Weil representation of HVv × Gv
(with respect to the character ψv ) over C , a model of which is recalled in § 4.2 below.

8

DANIEL DISEGNIWhenever □ is some smooth admissible representation of a group G ? , we denote by □∨ the contragredient, and by ( , )□ the natural pairing on □ × □∨ .
The first part of the following result (for nonsplit finite places) is known as theta dichotomy.

Proposition 3.6. — Let πv be an tempered irreducible admissible representation of Gv over C = Q p
or C = C.
1. There exists a unique Vv ∈ Vv such thatσv∨ := (π∨v ⊗ ωVv )Gv ̸= 0.
2. The representation σv∨ is tempered and irreducible. Its contragredient σv satisfies BC(σv ) =BC(πv ), and the space
Hom HV ×Gv (σv ⊗ π∨v ⊗ ωVv , C )v

is 1-dimensional over C .
3. The representation (πv ⊗ ωV∨ )Gv is canonically identified with σv .
v
4. Denote by ϑ each of the projection maps π∨v ⊗ ωVv → σv∨ , πv ⊗ ωV∨ → σv . Then the map
v

′

′

′

′

′

ζv (ϕ, φ, f ; ϕ , φ , f ) := (ϑ(ϕ, φ), f )σv∨ · (ϑ(ϕ , φ ), f ′ )σv
defines a canonical generator
ζv ∈ Hom Gv ×HV (π∨v ⊗ ωVv ⊗ σv , C ) ⊗C Hom Gv ×HV (πv ⊗ ωV∨ ⊗ (σv∨ , C ),vvv

with the property that if πv and σv are unramified and ϕ, φ, f , ϕ ′ , φ′ , f ′ are spherical vectors, thenζv (ϕ, φ, f ; ϕ ′ , φ′ , f ′ ) = (ϕ, ϕ ′ )π∨v · (φ, φ′ )ωv∨ ( f , f ′ )σv .
Proof. — We drop all subscripts v. We start by recalling the first two statements. Consider first thecase that v is finite and E is a field. Then σV∨ = (π∨ ⊗ ωV )G is the (a priori, ‘big’) theta lift of π∨ asdefined in [Har07, (2.1.5.1)]. By the local theta dichotomy proved in Theorem 2.1.7 (iv) ibid. and[GG11, Theorem 3.10], there is exactly one V ∈ V such that σV∨ is nonzero; we fix this V and drop
it from then notation. Then the other properties of σ := (σ ∨ )∨ are consequences of [GI16, Theorem
4.1] (which collects results from [Wal90, GT16, GS12, GI14]). For the case E = F ⊕ F , see [Mín08].
We now turn to the other two statements. For a character χ : F × → C × , letbn (χ ) :=

(3.1)

n
Y

L(i, χ ηi−1 ).

i=1

If C = C, then we have a canonical elementζ˘ ∈ Hom G (π∨ ⊗ ωV , C) ⊗C Hom G (π ⊗ ωV∨ , C)
given by(3.2)

ζ˘(ϕ, φ; ϕ ′ , φ′ ) :=bn (1)
L(1/2, Π)ZG(g ϕ, ϕ ′ )π∨ · (ω(g )φ, φ′ )ω d g ,where d g is the measure of [DL24, § 2.1 (G7)], Π := BC(π). It is a generator by [HKS96, § 6], where
the regularisation of the integral is also taken care of. (For the well-known comparison between thedefinition in loc. cit. and the one given here, see [Sak17, Lemma 3.1.2].) When π (hence σ) areTHETA CYCLES AND THE BEILINSON–BLOCH–KATO CONJECTURES

9unramified and all the vectors are spherical, by [Yam14, Proposition 7.1, (7.2)] we have
ζ˘(ϕ, φ; ϕ ′ , φ′ ) = (ϕ, ϕ ′ )π∨ · (φ, φ′ )ω∨ .

(3.3)

If C = Q p , then for any ι ∈ Σ we have a tetralinear form ζ˘ι as above, and by [DL24, Lemma 3.30],
there is a ζ˘ ∈ Hom (π∨ ⊗ ω , Q ) ⊗ Hom (π ⊗ ω ∨ , Q ) such that ζ ⊗1 = ζ ι for everyGVpι ∈ Σ.
Now, we may view ζ˘ as a map
(3.4)

Qp

G

V

p

Q p ,ι

ζ˘ : (π∨ ⊗ ωV )G ⊗ (π ⊗ ωV∨ )G → C

that is, by inspection, invariant under the diagonal action of H on both factors. It follows that ζ˘gives the duality of our third statement. The fourth statement then follows from the definitions and(3.3).
Remark 3.7. — A more symmetrically defined exalinear form would be
Z Z
′
′ ′
(ϕ, φ, f ; ϕ , φ , f ) 7→
(g ϕ, ϕ ′ )π∨ · (ω(h, g )φ, φ′ )ω · (h f , f ′ )σ d g d h,
HVGwhere the integral in d g is regularised as remarked after (3.2). If σ is a discrete series, the integral ind h converges and its value equals that of ζv , times the formal degree of σ – for which [BP21] gives
a formula in terms of adjoint gamma factors. In general, regularising the integral in d h amounts toregularising the inner product of two matrix coefficients of σ. A regularisation has been proposedby Qiu [Qiu12a, Qiu12b]; however the definition of the resulting generalised formal degree is partly
conjectural, and no precise (even conjectural) formula for it appears in the literature.
Global correspondence. — We have the following global variant of Proposition 3.6.
Proposition 3.8. — Let Π be a relevant p-adic automorphic representation of GLn (AE ), and set ε =ε(1/2, Π). Let RΠ,G be the set of isomorphism classes of relevant automorphic representations π of G(A)with BC(π) = Π, and let RΠ,H be the set of pairs (V , σ), with V ∈ V ◦,ε and σ an isomorphism class of
relevant p-adic automorphic representations of HV (A).
The relation(3.5)

Hom GV (A∞ )×HV (A∞ ) (π∞,∨ ⊗ ωV∞ ⊗ σ ∞ , Q p ) ̸= 0

defines a bijection between RΠ,G and RΠ,H .
Proof. — Take any ι: Q p ,→ C. After base-change to C via ι, given π, the existence of V with ϵ(V ) =ε(1/2, Π) and of a representation σ ∞,ι of HV (A∞ ) satisfying (3.5) follows from the explicit form of
theta dichotomy in terms of the doubling epsilon factors of [Har07], whose product over all places
coincides with the standard central epsilon factor of Π by [LR05]. Again by [LTX+ 22, Proposition
C.3.1], we have that σ ∞,ι is the finite component of relevant automorphic representation σ ι ; and as
in Remark 3.3 (c), the collection σ ι arises from a relevant p-adic automorphic representation σ.
The bijective property of the resulting map RG → RH follows from [GI16, Theorem 4.1 (iv)]
and the following archimedean fact (see [NZ01] or [PT02, Theorem 4.1 (4)]): if v|∞ and πv is then n
n−1 n−3
3−n 1−nholomorphic discrete series of U ( 2 , 2 ) with Harish–Chandra parameter { 2 , 2 , . . . , 2 , 2 },then πv has a nonzero theta lift to HVv , with Vv ∈ Vv , exactly for Vv positive-definite, in which casethe theta lift σv is the trivial representation of HVv .

10

DANIEL DISEGNI

4. Theta cycles
4.1. Assumptions on the Galois representation. — Let again ρ: GE → GLn (Q p ) be irreducible,
geometric, and of weight −1. We denote by ρc : GE → GLn (Q p ) the representation defined byρc (g ) = ρ(c g c −1 ), where c ∈ GE is any fixed lift of c. (A different choice of lift would yield an
isomorphic representation.)
We suppose from now on that the following conditions are satisfied:
1. ρ is conjugate-symplectic in the sense that there exists a perfect pairing
ρ ⊗Q ρc → Q p (1)psuch that for the induced map u : ρ → ρ (1) (where ∗ denotes the linear dual) and its conjugatedual u ∗ (1)c : ρc → ρc,∗ (1)c = ρ∗ (1), we have u = −u ∗ (1)c ;2. n = 2r is even;
3. for every place w| p of E and every embedding ȷ : Ew ,→ C p , the ȷ-Hodge–Tate weights(4) of ρ
are the n integers {−r, −r + 1, . . . , r − 1};
4. ρ is automorphic in the sense that for each ι: Q p ,→ C, there is a cuspidal automorphic representation Πι of GLn (AE ) such that Lι (ρ, s) = L(Πι , s + 1/2);c∗

Associated automorphic representations. — A collection (Πι )ι: Q ,→C as in Condition 4 is uniquelypdetermined up to isomorphism if it exists, by the multiplicity-one theorem for automorphic forms
on GLn ; it is conjectured to always exist. Moreover, every Πι is relevant in the sense of Definition
3.1.1, where Condition 1 implies property (i) in the definition, and Condition 3 implies property(ii). It is then clear that (Πι )ι arises from a unique (up to isomorphism) relevant p-adic automorphic
representationΠ = Πρ
of GLn (AE ) (Definition 3.2.1). We denote by π = πρ the relevant p-adic representation of G(A)associated with Π as in Proposition 3.4,(5) and by(V , σ) = (Vρ , σρ )the pair associated with π as in Proposition 3.8. We also put H = HV .
4.2. Models of the representations. — We now fix some concrete models of the representations
ω, π, and σ.
Weil representations. — We fix the well-known model of the representation ω = ⊗′v∤∞ ωV ,v on
S (VAr∞ , Q p ) associated with ψ, on which H(A∞ ) acts by right translations, whereas the actionof G(A∞ ) is recalled in [DL24, §4.1 (H7)].
1

Denote by † the involution on G given by conjugation by the element r −1 inside GLn (E);r

it acts on any G(R)-module for any E-algebra R. The representation ω † is a model of the Weilrepresentation attached to ψ−1 .

Siegel-hermitian modular forms and their q-expansion. — The representation π may be realised inspaces of hermitian modular forms, which we briefly review.
(4)(5)

Our convention is that the cyclotomic character has weight −1.
As noted in Remark 3.5, any other relevant π with BC(π) = Π would be equally good.

11

THETA CYCLES AND THE BEILINSON–BLOCH–KATO CONJECTURES

In [DL24, § 2.2], we have defined the following objects.(6)[r ]– A C-vector space HC = A r,hol of holomorphic forms for the group G.
[r ]– For any Q p -algebra R, an R-module HR = H r ⊗Q p R of (classical) p-adic automorphic forms
for G, such that for each ι: Q p ,→ C, we have an isomorphism
H Q ⊗ι C → H C ,pΦ ⊗ 1 7→ Φι .

In fact, only the case where E/F is totally split above p was considered in [DL24], where[r ]H r is the direct limit, over open compact subgroups U ⊂ G(A∞ ), of subspaces of sectionsof a certain line bundle on a Siegel hermitian variety Σ(U )/Q p ; let us explain why the splitting

condition is not necessary for our purposes. Define a p-adic CM type of E to be a set Φ of[F : Q] embeddings i : E ,→ Q p such that i ∈ Φ if and only if i ◦ c ∈/ Φ; in the totally split case,the choice of a p-adic CM type is equivalent to the choice of a set PCM as in [DL24, §2.1 (F2)],
which intervenes in the construction of Σ(U ) as a moduli scheme by fixing a signature type fortest objects in the sense of [LTX+ 22, Definition 3.4.3]. However, this construction, and thecomparison with complex Siegel hermitian varieties of [DL24, Lemma 2.1], go through with[r ]any p-adic CM type Φ (with the innocuous difference that, in general, Σ(U ) and H r will
only be defined over a finite extension of Q p in Q p ).
– A space SFR = SF r (R) of formal q-expansions with coefficients in the (arbitrary) ring R, and a
Siegel–Fourier expansion map q∞ = qanr : HC → SFC . By the argument at the end of the proofof [DL24, Proposition 4.18] (based on Lemma 2.11 ibid.), we deduce a Q p -linear q-expansionmap
q p : HQ → SFQppsatisfying ιq p (Φ) = qC (Φι ) for every Φ ∈ HQ◦ and every embedding ι ∈ Σ.
p

By [DL24, Lemma 3.14] (based on [Mok15]), for a relevant p-adic automorphic representation
π, the space Hom G(A∞ ) (π, HQ ) is 1-dimensional, and π∨,† is also relevant. We identify π = πρpwith the corresponding subspace of HQ . Then πρ∗ (1) is isomorphic to π∨,† .
p

Moreover, for any ring R, let SF R be the space of those formal expansionsXcT (a) q T ,cT ∈ C ∞ (GL r (A∞E ), R)T ∈Herm r (F )+satisfying ct a c T a (y) = cT (ay) for all a ∈ GL r (E); then we have a q-expansion map
q: HQ → SF Qppcharacterised by qΦ(y) = | det y|Er q(m(y)Φ). Since M(A∞ ) acts transitively on the set of connectedcomponents of Σ(U )Q for every open compact subgroup U ⊂ G(A∞ ), the map q is injective.
p

p

Shimura varieties and their cohomology. — We assume from now on that ϵ(ρ) = −1. (The opposite
case will be trivial for our purposes in Definition 4.5 below.) Then V ∈ V ◦,− , and we have an inverse
system(XK )K⊂H(A∞ )(6)

In this discussion, most new notation will be introduced by equalities whose right-hand sides reproduce the corresponding notation in [DL24].

12

DANIEL DISEGNI

of (n−1)-dimensional smooth varieties over E, with the property that for every archimedean place w
of F , with underlying place v of F , the variety XV ,K ×E,w C is isomorphic to the complex Shimuravariety XV (v),wK associated with the unitary group HV (v) and the Shimura datum attached to w
that is the complex conjugate to the one defined in [Liu21, § C.1] (and thus coincides with the one
specified in [LTX+ 22, § 3.2] and used in [LL21, DL24]); see also [Gro21, ST].
From now on we assume that each XK is projective, which is the case if and only if either F ̸= Q,
or n = 2, F = Q and ϵ(Vv ) = −1 for some finite place v. In fact, in the remaining non-compact casefor n = 2, the curve XK (closely related to a classical modular curve) can be canonically compactified
by adding finitely many cusps; in this case the constructions make sense, and the theorems hold true,
after replacing XK by its compactification.
Let
Hét2r −1 (XE , Q p (r )) := lim Hét2r −1 (XK,E , Q p (r )),←− ∞K⊂H(A )

where the transition maps are pushforwards. For each K, we have a spherical Hecke algebra for H
acting on XK ; let mρ,K be the Hecke ideal denoted by mRπ in [LL21, Definition 6.8]. We denote by
Mρ,K := Hét2r −1 (XK,E , Q p (r ))mρ,K
the localisation, and we set
Mρ := lim Mρ,K ⊂ Hét2r −1 (XE , Q p (r )).
←−K

We will assume the following hypothesis, which is a special case of [LL21, Hypothesis 6.6] (it is
known for n = 2, and it is expected to be confirmed in general in a sequel to [KSZ]).
Hypothesis 4.1. — For each open compact K ⊂ H(A∞ ), we have a Hecke- and Galois-equivariant
decompositionM∼Mρ ⊗ σ ′∨,K ,
=
ρ,K
(4.1)
′
σ

where the direct sum runs over the isomorphism classes of relevant p-adic automorphic representation σ ′of HV (A) with BC(σ ′ ) = Π.
We thus have an H(A∞ )-equivariant map(4.2)

σ −→ Hom Q [G ] (Hét2r −1 (XE , Q p (r )), ρ),pE

and we identify σ with the image of this map. We also put Mσ,K := ρ ⊗ σ ∨,K ⊂ Hét2r −1 (XK,E , Q p (r )),
and
(4.3)

Mσ := lim Mσ,K ⊂ Mρ ⊂ Hét2r −1 (XE , Q p (r )).
←−K

Then σ = Hom Q [G ] (Mσ , ρ) := lim Hom Q [G ] (Mσ,K , ρ).
−→K
p
E
p
E
•
2r
Denote by Fil ⊂ Hét (XK , Q p (r )) the filtration induced by the Hochschild–Serre spectral se-

quence H i (E, Hét2r −i (XK , Q p (r ))) ⇒ Hét2r (XK , Q p (r )). By the argument for [DL24, Lemma 4.7],
we have a canonical Hecke-equivariant projection
Hét2r (XK , Q p (r ))/Fil2 ) → H 1 (E, Mρ,K ).

13

THETA CYCLES AND THE BEILINSON–BLOCH–KATO CONJECTURES

Lemma 4.2. — The image of the composition
AJ[−]ρ : Ch r (XK )0 −→ Hét2r (XK , Q p (r ))/Fil2 → H 1 (E, Mρ,K )Qp

is contained in H f1 (E, Mρ,K )Proof. — As in [DL24, Lemma 4.24], using [NN16, Theorem B] in place of [Nek00] for p-adic
places.
4.3. Construction. — We proceed in four steps. The first three steps follow works of Kudla andcollaborators [Kud97, Kud03, KRY06], and of Liu and collaborators [Liu11a, DL24].
0. Special cycles in X . — For each x ∈ VAr∞ and each open compact K ⊂ H(A∞ ), we have acodimension-r special cycle
Z(x)K ∈ Ch r (XK )defined in [Liu11a, § 3A]. Putting
T (x) := ((xi , x j )V )i j ,where ( , )v is the hermitian form on V , we recall the definition in two basic cases. Denote byHerm r (F )+ the set of r × r matrices over E that satisfy T c = T t and that are totally positive semidefinite. First, Z(x)K = 0 if T (x) ∈/ Herm r (F )+ . Second, assume that T (x) ∈ Herm r (F )+ is positive
definite. Let V x ⊂ V be the incoherent hermitian space that is (place by place) the orthogonal
complement of the span of (x1 , . . . , x r ). The corresponding embedding U(V x ) ,→ U(V ) of incoherent unitary groups. induces a map of towers of Shimura varieties α x : XVx → XV ; then we define
Z(x)K ∈ Ch r (XV ,K ) to be the class of the image cycle.
1. Theta kernel. — The special cycles just defined may be assembled into a generating series. Letφ ∈ ω. For every K ⊂ HV (A∞ ) fixing φ, we defineXqΘ(φ)ρ,K (a) := vol(K)φ(xa)[Z(x)K ]ρ q T (x) ,x∈K\VAr ∞where vol(K) is as in [LL21, Definition 3.8]. Then q Θ(φ)ρ,K is an element of H f1 (E, Mρ,K )⊗Q SFQ ,ppand the construction is compatible under pushforward in the tower XK . (The reason why we preferour Θ(φ)−,ρ to be compatible with pushforwards rather than pullbacks is that this allows to pairthem, in Step 3, with elements of the automorphic representation σ under the identification (4.2).)
The following conjecture, which is a variant of [DL24, Hypothesis 4.16], asserts the modularityof the generating series, and from now on we will assume it holds.
Hypothesis 4.3. — For every φ ∈ ω and any K ⊂ HV (A∞ ) fixing φ, there exists a uniqueΘ(φ)ρ,K ∈ H f1 (E, Mρ,K ) ⊗Q HQppsuch thatq (Θ(φ)K,ρ ) = q Θ(φ)ρ,K .
pRemark 4.4. — A recent piece of evidence for this modularity conjecture is provided in [DL24,
Theorem 4.20], which is recalled as part of Theorem 5.5.2; moreover,(7) an analogous conjecture(7)

I am grateful to Yifeng Liu for bringing this to my attention.

14

DANIEL DISEGNI

for orthogonal Shimura varieties can be deduced from [Kud21]. Hypothesis 4.3 is implied by thevariant for Chow groups of [LL21, Hypothesis 4.5]. See Remark 4.6 ibid. for comments on the
supporting evidence for that conjecture until then, to which we should add the recent [Xia22]. For
the history, which traces back to the work of Gross–Kohnen–Zagier on generating series of Heegner
points [GKZ87], see [Li23, Remark 3.5.5], cf. also ibid. § 6.4.
2. Arithmetic theta lifts. — Denote by Φ 7→ Φπ the Hecke-eigenprojection HQ → π, and byp〈 , 〉π∨ : π∨ ⊗ π → Q p the canonical duality. (We also use the same names for any base-change.)
Then for every ϕ ∈ π∨ , we may define(4.4)

Θ(ϕ, φ)K := 〈ϕ, Θ(φ)K,ρ,π 〉π∨∈ H f1 (E, Mρ,K ).

Since the map (ϕ, φ) 7→ Θ(ϕ, φ)K is equivariant under the action of Q p [K\HV (A∞ )/K], Proposition 3.8 implies that Θ(ϕ, φ)K belongs to the subspace H f1 (E, Mσ,K ) ⊂ H f1 (E, Mρ,K ).
3. Theta cycles. — For every f ∈ σ,ϕ ∈ π∨ , and any K ⊂ HV (A∞ ) fixing f and φ, we defineΘρ (ϕ, φ, f ) := f∗ Θ(ϕ, φ)K ∈ H f1 (E, ρ).
The following definition then satisfies the first property asserted in Theorem A.
Definition 4.5. — Let ρ be a Galois representation satisfying the assumptions of § 4.1.
If ϵ(ρ) = +1, we may put Λρ = Q p and Θρ := 0.
If ϵ(ρ) = −1, assume that F ̸= Q and that Hypotheses 4.1 and 4.3 hold, and let π, V , σ be as
above. Then we defineΛρ := (π∨ ⊗ ω ⊗ σ)G(A∞ )×H(A∞ ) ,andΘρ : Λρ → H f1 (E, ρ),[(ϕ, φ, f )] 7→ Θρ (ϕ, φ, f ).

Remark 4.6. — Suppose that n = 2 and that ρ = V p AE for a modular abelian variety A of GL2 -typeover F . Then the image of Θρ consists of classes of Heegner points. This follows by comparing theheight formulas for the two objects in [YZZ12] and [Liu11b], against the backdrop of [Nek07]. A
direct comparison is also possible: for n = 2, all the Z(x) are CM points on unitary Shimura curves,which can be related along the lines of [Car86, § 4] to the modular curves and the quaternionic
Shimura curves used to construct Heegner points in [GZ86, YZZ12].

5. Relation to L-functions and Selmer groups
We continue to denote by ρ a Galois representation satisfying the assumptions of § 4.1.
5.1. Complex and p-adic L-functions. — For every ι: Q p ,→ C, and every finite-order characterχ ′ : GE → Q×p , we have the L-functionLι (ρ ⊗ χ ′ , s) = L(s + 1/2, Πι ⊗ ιχ ′ ),which is holomorphic and has a functional equation with center at s = 0 and sign ϵ(ρ).
At least under the following assumption, we also have a p-adic L-function.

THETA CYCLES AND THE BEILINSON–BLOCH–KATO CONJECTURES

15

Assumption 5.1. — The extension E/F is totally split above p, and for every place w| p of E, the representation ρw is crystalline and Panchishkin-ordinary.
We need to make the auxiliary choice of an isomorphism α: π∨,† → πρ∗ (1) (where π = πρ , πρ∗ (1) ⊂HQ ), which yields for each ι: Q p ,→ C, an element Pρ,ι = Pρ,α,ι (ρ) ∈ C× such thatp((αϕ1 )ι,† , ϕ2ι )Pet†
ι(ϕ1 , ϕ2 )π∨ =Pρ,ι
for every ϕ1 ∈ π∨,† , ϕ2 ∈ π; here′

(ϕ, ϕ )Pet :=Z
G(F )\G(A)ϕ(g )ϕ ′ (g ) d g

where d g is the measure of [DL24, § 2.1 (G7)].
Q
For a character χ of GF , we put χE := χ|GE , and bn (χ ) := v∤∞ bn (χv ), where the factors are asin (3.1); we also define a constant[F :Q]Γ (1) · · · Γ (r )r −r 2 −r r 2
c∞ = (−1) 2π.
Γ (r + 1) · · · Γ (2r )Finally, we denote by K (XF ) the fraction field of O (XF ).
Proposition 5.2. — Suppose that ρ satisfies Assumption 5.1. There is a meromorphic function
L p (ρ) = L p,α (ρ)∈ K (XF )characterised by the following property: for every finite-order character χ ∈ XF (Q p ) and every embedding ι: Q p ,→ C, we havec L (ρ ⊗ χE , 0)
ιL p (ρ)(χ ) = ιe p (ρ, χ ) · ∞ ι.
bn (χ )Pρ,ι
Here, ιe p (ρ, χ ) =of F , andQ

w|v| p ιew,ι (ρ, χ ) ∈ ιQ p , in which the product ranges over the p-adic places of E and−1ιew (ρ, χ ) := γ (ιWD(ρ+w ⊗ χE,w ), ψE,w )bn,v (χ )Lι (ρw ⊗ χE,w ).

where the Deligne–Langlands γ -factor and Fontaine’s functor ιWD are as recalled in [Dis23, (1.1.4)].
Proof. — This follows by multiplying the incomplete p-adic L-function of [DL24, Theorem 1.4] by
local L-factors at ramified and p-adic places, as in the proof of Proposition 3.39 ibid.(8)5.2. Pairings. — Let ρ be a representation satisfying the assumptions of Definition 4.5, and let πρ ,V , σρ , Λρ , and Θρ be the associated objects. We denote by πv and σv the local components of πρand σρ at the place v (which are well-defined up to isomorphism).
Dual Theta cycles. — The representation ρ∗ (1) also satisfies those assumptions, and we have the corresponding mapΘρ∗ (1) : Λρ∗ (1) → H f1 (E, ρ∗ (1)).
(8)

Before [DL24], a p-adic L-function that extends L p (ρ) to a larger space was constructed in [EHLS20]; the rationality
property proved there is weaker than stated here.

16

DANIEL DISEGNI

Pairings. — Let 〈 , 〉: Mρ ⊗ Mρ∗ (1) → Q p (1) be the pairing induced by Poincaré duality. Then we
define a pairing( , )σ : σρ ⊗ σρ∗ (1) → Q p(5.1)

by ( f , f ′ )σ := f ◦ u( f ′∗ (1)), where f ′∗ (1): ρ∗σ ∗ (1) (1) → Mρ∗∗ (1) (1) is the transpose, and u : Mρ∗∗ (1) (1) →Mρ is the isomorphism induced by 〈 , 〉. Thus σρ∗ (1) is identified with σρ∨ = σ ∨ .
We also have a canonical pairing on ω ⊗ ω † defined byZ′(φ, φ )σ =φ(x)φ′ (x) d x
(5.2)
VAr ∞

for the product of ψ-selfdual measures. Thus ω † is identified with ω ∨ . Similarly, if we denoteι□ := □ ⊗Q ,ι C, and complex conjugation in C by a bar, we have ιω = ω ∨ . Let vol(H∞ ) be thep

1♮volume of H(F∞ ) for the measure denoted b (0) d hv in [LL21, Definition 3.8], which is a rational
2r
number by [DZ, Lemma 2.2.1].
Then:
∨,†

– for every isomorphism α: πρ → πρ∗ (1) , we have a pairing(5.3)

ζα := vol(H∞ ) · ⊗v∤∞ ζv ◦ ( )† ◦ jα : Λρ ⊗ Λρ∗ (1) → Q p ,††where jα identifies the factor πρ∨∗ (1) of Λρ ⊗Λρ∗ (1) with πρ via the dual of α, and ( )† maps πρ ⊗ωto πρ ⊗ ω † = πρ ⊗ ω ∨ ;∼=– for every ι ∈ Σ we have an identification jι : ιπρ∨∗ (1) −→ ιπρ via the restriction of ( , )Pet to πρι ⊗πρ∗ (1) . Then we obtain a pairingζι := vol(H∞ ) · ⊗v∤∞ ζv ◦ ( ) ◦ jι : ιΛρ ⊗ ιΛρ∗ (1) → C
where ( ) maps ιπρ ⊗ ιω to ιπρ ⊗ ιω = ιπρ ⊗ ιω ∨ .
p-adic height pairing. — Assume that ρ is Panchishkin-ordinary. Then the construction of Nekovář
[Nek93] (see [DL24, § 4.2] for a verification of the assumptions) yields a p-adic height pairing
ˆ p.
〈 , 〉: H f1 (E, ρ) ⊗ H f1 (E, ρ∗ (1)) → ΓF ⊗Q
Complex height pairings. — On the other hand, assume that p is unramified in E, and let K ◦p =Q
pv| p Hv ⊂ H p be a product ot maximal hyperspecial subgroups. Then for open compact K ⊂H(A p∞ ), setting K := K p K ◦p ⊂ H(A∞ ), the variety XK has good reduction at all p-adic places.
DefineCh r (XK )〈 p〉 ⊂ Ch r (XK )0
to be the Q-subspace of algebraic cycles whose class in H 2r (XK,Ew , Q p (r )) is trivial for every finite
place w ∤ p of E. Li and Liu [LL21] observed that the construction of Beı̆linson [Beı̆87] unconditionally defines a height pairing
(5.4)

〈 p〉

〈 p〉

〈 , 〉BB : Ch r (XK )C ⊗C Ch r (XK )C → C ⊗Q Q pthat is C-linear in the first factor and C-antilinear in the second factor. (It is conjectured that thepairing takes values in C ⊂ C⊗Q Q p ; this turns out to be the case in the application to Theta cycles.)17

THETA CYCLES AND THE BEILINSON–BLOCH–KATO CONJECTURES

In order to descend this pairing to Selmer groups, we need to assume a case of a standard conjectureon the injectivity of Abel–Jacobi maps. Whenever K ⊂ H(A∞ ) is an open compact subgroup thatis understood from the context, denote mρ = mρ,K , mρ∗ (1) = mρ∗ (1),K
Conjecture 5.3. — For ρ? ∈ {ρ, ρ∗ (1)} and for each open compact subgroup K p ⊂ H(A p∞ ), the Abel–
Jacobi map


〈 p〉
r
AJ
:
Ch
(X
)
→ H f1 (E, Mρ? ,K p K ◦p )p,K p K ◦p
K p K ◦p
(5.5)
Qp

mρ?

is injective.
Assume that ρ is crystalline at all p-adic places. Fix a maximal hyperspecial subgroup K ◦p ⊂H(A p∞ ), and assume that Conjecture 5.3 holds. Denote by H f1 (E, Mρ? ,K p K ◦p )X the image of (5.5),
and letX ◦H f1 (E, ρ? ) K p :=XX

σ ′ ,K p f ′ ∈(σ ′ )K p K pf∗′ H f1 (E, Mρ? ,K p K ◦p )X

where the first sum is as in (4.1) for ρ? . Then for every ι: Q p ,→ C and every K = K p K ◦p , we have apairing(5.6)〈 , 〉ιK : H f1 (E, Mρ,K )X ⊗Q H f1 (E, Mρ∗ (1),K )X ⊗Q ,ι C → C ⊗Q Q ppptransported from (5.4) via the maps AJ p,K p K ◦p ⊗ι 1. We may deduce from it a pairing
X ◦

X ◦

〈 , 〉ι : H f1 (E, ρ) K p ⊗Q H f1 (E, ρ∗ (1)) K p ⊗Q ,ι C → C ⊗Q Q p

(5.7)

p

p

defined as follows.
For i = 1, 2 let
ci = fi,∗ AJ p,K ci′
for some K = K p K ◦p , some fi ∈ σiK , and some〈 p〉c1′ ∈ Ch r (XK p K ◦p )
,
Qp

mρ



〈 p〉
c2′ ∈ Ch r (XK p K ◦p )Qp.

mρ∗ (1)

If σ1 ≁ σ2∨ , we put=(5.8)〈c1 , c2 〉ι := 0.

If σ1 ∼= σ2∨ , we have the pairing ( , )σ1 of (5.1) on σ1 ⊗ σ2 , through which we identify σ2 = σ1∨ . LettK ( f1 ⊗ f2 ) ∈ Hom (σ1∨,K , σ2K ) = End (σ1∨,K ) = End Q [G ] (Mσ1 ,K )pE

be given bytK ( f1 ⊗ f2 )(v1 ) = vol(K) · (v1 , f1 )σ1 · f2 ,and let(5.9)

t( f1 ⊗ f2 )(v1 ) = vol(K) · tK ( f1 ⊗ f2 );the normalising volume factor makes t into a well-defined map σ1 ⊗ σ1∨ → End Q [G ] (Mσ1 ). Thep

E

existence of a Hecke correspondence acting as t( f1 ⊗ f2 ) implies that the action of t( f1 ⊗ f2 ) on Selmer

18

DANIEL DISEGNI
X ◦groups preserves the subspace H f1 (E, Mσ1 ,K ) K p . Then we define(5.10)〈c1 , c2 〉ι := 〈t( f1 ⊗ f2 )c1′ , c2′ 〉ιK .

The definition of (5.7) in the general case follows from (5.8), (5.10) by bilinearity.
ˆ p -valued Nekovář pairings 〈 , 〉K analogous
Remark 5.4. — In the p-adic case, we also have ΓF ⊗Q
to (5.6) (whose construction takes as input the pairing on Mρ,K ⊗ Mρ∗ (1),K deduced from Poincaré
duality). The analogous formula to (5.10) holds true as a consequence of the definitions and the
projection formula [DZ, Lemma A.2.5].
5.3. The height formulas. — We may now state the main known results on Theta cycles. They
parallel those of [GZ86, PR87, Kol88] on Heegner points.
We will say that E and ρ are mildly ramified if E and πρ satisfy the hypotheses of [DL24, Assumption 1.6], except possibly for the ones about p-adic places.
Theorem 5.5. — Suppose that F ̸= Q or n = 2, that E and ρ are mildly ramified, and that ρ is crystalline
at all p-adic places. Assume Hypotheses 4.1.
1. Assume the Modularity Hypothesis 4.3, Conjecture 5.3, and that p is unramified in E. Then forevery λ ∈ Λρ , λ′ ∈ Λρ∗ (1) and for every ι ∈ Σ, we have〈Θρ (λ), Θρ∗ (1) (λ′ )〉ι =c∞ L′ι (ρ, 0)· ζι (λ, λ′ )bn (1)in C.
∨,†
2. Suppose that Assumption 5.1 holds and that p > n. Let α: πρ ∼= πρ∗ (1) . Then:– if the order of vanishing of L p,α (ρ) at 1 is one, then the Modularity Hypothesis 4.3 holds, andfor every λ ∈ Λρ , λ′ ∈ Λρ∗ (1) , we have〈Θρ (λ), Θρ∗ (1) (λ′ )〉 = e p (ρ, 1)−1 · dL p,α (ρ)(1) · ζα (λ, λ′ ).
ˆ p = T ∗ XF .
in ΓF ⊗Q1
– if the order of vanishing of L p,α (ρ) at 1 is not one and the Modularity Hypothesis 4.3 holds,
then for every λ ∈ Λρ , λ′ ∈ Λρ∗ (1) , we have〈Θρ (λ), Θρ∗ (1) (λ′ )〉 = 0.
Proof. — Write λ = [(ϕ, φ, f )], λ′ = [(ϕ ′ , φ′ , f ′ )]. Consider the p-adic case. The modularity result is[DL24, Theorem 4.20], after projection H f1 (E, Mρ ) → H f1 (E, Mσ ′ ) for any relevant σ ′ with BC(σ ′ ) =Π; but this is equivalent to the modularity in H f1 (E, ρ) by Hypothesis 4.3.
For the first height formula, by the definitions and Remark 5.4, it is equivalent to prove(5.11) 〈t( f ⊗ f ′∨ ) Θ(ϕ, φ), Θ(ϕ ′ , φ′ )〉 = e p (ρ, 1)−1 · dL p,α (ρ)(1) · ζ˘α (t( f ⊗ f ′∨ )ϑ(ϕ, φ′ ); ϑ(ϕ ′ , φ′ ))where the Θ’s are the arithmetic theta liftings for ρ and ρ∗ (1) as in (4.4), andζ˘α = vol(H∞ ) · ⊗v∤∞ ζ˘v ◦ ( )† ◦ jα
is defined analogously to (5.3) based on the pairings (3.4). Pick a K ⊂ HV (A∞ ) fixing f , f ′ , φ, φ′ ,and let T ∈ H (H(A∞ )) be a Hecke operator acting as vol(K)−1 t( f ⊗ f ′∨ ) on σ ∨ . Then (5.11) isTHETA CYCLES AND THE BEILINSON–BLOCH–KATO CONJECTURES

19

equivalent to [DL24, Theorem 1.8 (1)] in level K for(ϕ, T φ; ϕ ′ , φ′ ).
(Note that our definitions of the arithmetic theta lifts Θ(−, −) differ from those of [DL24] by afactor vol(K); in the height formula, one factor is accounted for by (5.9), and another one by thenormalisation of height pairings in loc. cit.. The term vol♮ (K) in [DL24] equals our vol(H∞ )vol(K):this difference is accounted for by the factor vol(H∞ ) in the pairing ζα .)The p-adic height vanishing formula is likewise equivalent to [DL24, Theorem 1.8 (2)].
The complex case is similarly reduced to [LL22, Theorem 1.8]. As ρ is crystalline at all w| p, therepresentations Πw , σw , πw are unramified, so that we can take representatives ( f , ϕ, φ; f ′ , ϕ ′ , φ′ )of λ, λ′ ̸= 0 that are fixed by a maximal hyperspecial K ◦p . Then the fact that 〈 , 〉ι is well-defined on
Theta cycles follows from the definitions and [LL21, Proposition 6.10 (3)].
Part 2 of Theorem A is then an immediate consequence of Theorem 5.5. For a beautiful expositionof some key aspects of the proofs of the formulas in [LL21, LL22, DL24], see [Li23].
The proof of Theorem 5.5 suggests that from the point of view of height formulas, Theta cycles
offer no material advantage over previous constructions. This is not so from the point of view ofEuler systems, as we explain next.
5.4. An Euler system. — The main technique for bounding Selmer groups is that of Euler systems,
originally introduced by Kolyvagin to study Heegner points [Kol88, Kol90]. Roughly speaking, an
Euler system for a representation ρ of GE is a collection of integral Selmer classes defined over certain
abelian extensions of ρ and satisfying certain compatibility relations; the (one) class defined over E
itself is called the base class of the Euler system.
In a forthcoming work, Jetchev–Nekovář–Skinner theorise a variant of this notion, that we shallcall a JNS Euler system. It is adapted to conjugate-symplectic representations over CM fields, where
the abelian extensions are ring class fields ramified at the primes of E split over the totally real subfield F (see [Ski]). Their main result is that if ρ has ‘sufficiently large’ image, then the existence ofa JNS Euler system with nontrivial base class z implies that z generates the Selmer group of ρ: fora precise statement (when F = Q), see [ACR23, Theorem 8.3 and Remark 8.4], where JNS Euler
systems are called ‘split anticyclotomic Euler systems’ (ibid., Definition 8.1).
The following is the main result of [Dis]. Granted the results of Jetchev–Nekovář–Skinner, it
implies part 3 of Theorem A.
Theorem 5.6. — Let ρ: GE → GLn (Q p ) be a representation satisfying the assumptions of § 4.1. Thenfor any λ ∈ Λρ , there exists a JNS Euler system based on Θρ (λ).
Multiplicity-one principles are remarkably useful to prove relations between special cycles and,in particular, compatibility relations in Selmer groups – as first observed in [YZZ12] and [LSZ22].
The proof of Theorem 5.6 is no exception: this is the main technical advantage of having constructeda cycle depending on one parameter only.
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DANIEL DISEGNI, Department of Mathematics, Ben-Gurion University of the Negev, Be’er Sheva 84105, Israel • AixMarseille University, CNRS, I2M - Institut de Mathématiques de Marseille, campus de Luminy, 13288 Marseille,
France • E-mail : daniel.disegni@univ-amu.fr