Da mon and Ro ge Res Ma h Sci (2016) 3:27
DOI 10.1186/s40687-016-0074-9
R E S E A R C H Open Access
Ellip ic cu es o ank wo and gene alised
Ka o classes
Hen i Da mon1* and Vic o Ro ge 2
In memo y o Robe Coleman.
*Co espondence:
[email p o ec ed]
1Depa men o Ma hema ics
and S a is ics, McGill Uni e si y,
805 She b ooke S . Wes ,
Mon eal, Canada
Full lis o au ho in o ma ion is
a ailable a he end o he a icle
Abs ac
Heegne poin s play an ou s anding ole in he s udy o he Bi ch and Swinne on-Dye
conjec u e, p o iding canonical Mo dell–Weil gene a o s whose heigh s encode fi s
de i a i es o he associa ed Hasse–Weil L-se ies. Ye he ui ul connec ion be ween
Heegne poin s and L-se ies also accoun s o hei main limi a ion, namely ha hey
a e o sion in (analy ic) ank >1. This pa ly exposi o y a icle discusses he gene alised
Ka o classes in oduced in Be olini e al. (J Algeb Geom 24:569–604, 2015)andDa mon
and Ro ge (J AMS 2016), s essing hei analogy wi h Heegne poin s bu explaining
why hey a e expec ed o gi e non- i ial, canonical elemen s o he idoneous Selme
g oup in se ings whe e he classical L- unc ion (o Hasse–Weil–A in ype) ha go e ns
hei beha iou has a double ze o a he cen e. The gene alised Ka o class deno ed
κ( , g, h) is associa ed o a iple ( , g, h) consis ing o an eigen o m o weigh wo and
classical p-s abilised eigen o ms gand ho weigh one, co esponding o odd
wo-dimensional A in ep esen a ions Vgand Vho Gal (H/Q) wi h p-adic coefficien s
o a sui able numbe field H. This class is ge mane o he Bi ch and Swinne on-Dye
conjec u e o e H o he modula abelian a ie y Eo e Qa ached o . One o he
main esul s o Be olini e al. (2015) and Da mon and Ro ge (J AMS 2016)is ha
κ( , g, h)liesin hep o-pSelme g oup o Eo e Hp ecisely when L(E, Vgh,1) =0, whe e
L(E, Vgh,s)is heL- unc ion o E wis ed by Vgh :=Vg⊗Vh. In he se ing o in e es ,
pa i y conside a ions imply ha L(E, Vgh,s) anishes oe eno de a s=1, and he
Selme class κ( , g, h) is expec ed o be i ial when o ds=1L(E, Vgh,s)>2. The main new
con ibu ion o his a icle is a conjec u e exp essing κ( , g, h) as a canonical poin in
(E(H)⊗Vgh)GQwhen o ds=1L(E, Vgh,s)=2. This conjec u e s eng hens and efines he
main conjec u e o Da mon e al. (Fo um Ma h Pi 3:e8, 2015)andsuppliesa amewo k
o unde s anding he esul s o Da mon e al. (2015), Be olini e al. (2015)andDa mon
and Ro ge (J AMS 2016).
Ma hema ics Subjec Classi ica ion: 11G18, 14G35
Con en s
1 Backg ound and mo i a ion ...................................
2 Hida amilies and pe iods o weigh one o ms ........................
3 Gene alised Ka o classes .....................................
3.1 Defini ion ..........................................
3.2 Basic p ope ies .......................................
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Da mon and Ro ge Res Ma h Sci (2016) 3:27 Page 2 o 32
3.3 Enhanced egula o s ....................................
3.4 The conjec u e .......................................
4 Special cases ...........................................
4.1 Beilinson–Ka o classes ...................................
4.2 Beilinson–Flach classes ..................................
4.3 Complex mul iplica ion classes and Heegne poin s ..................
4.4 Real mul iplica ion classes and S a k–Heegne poin s .................
4.5 Adjoin classes .......................................
Re e ences ..............................................
1 Backg ound and mo i a ion
The heme o modula i y o p-adic Galois ep esen a ions has occupied cen e s age in
numbe heo y o he las se e al decades, and Robe Coleman has been a majo figu e
in many o i s key de elopmen s, no ably h ough he heo y o Coleman amilies o p-adic
modula o ms and o he Coleman–Mazu eigencu e pa ame e ising hese amilies and
hei associa ed Galois ep esen a ions. By way o backg ound and mo i a ion, his sec ion
explainshowmucho hep og essachie edon heBi chand Swinne on-Dye conjec u e,
including he esul s o [11,15]and[19], can be iewed as pa o he la ge p og amme
o unde s anding he modula i y o (non-semisimple)p-adic Galois ep esen a ions.
One o he mos celeb a ed modula i y esul s is he s a emen ha all ellip ic cu es
o e Qa ise as quo ien s o sui able modula cu es: mo e p ecisely, ha an ellip ic cu e
Eo e Qo conduc o Nis equipped wi h a su jec i e pa ame e isa ion
πE:X0(N)−→ E, (1)
whe e X0(N) is he modula cu e a ached o Hecke’s cong uence subg oup 0(N). This
was p o ed in [37,40], and [12] by showing ha he p-adic ep esen a ion
H1(E):=H1
e (E¯
Q,Qp)(1) =(lim
←,n E[pn]) ⊗ZpQp
o GQ:=Gal ( ¯
Q/Q) a ises as a quo ien o he é ale cohomology g oup1
H1(X0(N)) :=H1
e (X0(N)¯
Q,Qp(1)).
The exis ence o a Galois-equi a ian p ojec ion
πE:H1(X0(N)) −→ H1(E)(2)
is he eal con en o he b eak h ough in [40]and[37], he os ensibly s onge geome ic
e sion (1) being deduced om i by in oking he Ta e conjec u e o cu es.2
Le Ebe an open sub a ie y o E, i.e. he complemen o a ze o-dimensional sub a ie y
o Eo e Q.Thep-adic Galois ep esen a ion H1(E) si s in he middle o he sho
exac excision sequence
0−→ H1(E)−→ H1(E)−→ H0()0−→ 0
o é ale cohomology g oups, whe e he subsc ip o 0 deno es he deg ee 0 elemen s o
H0(). By analogy wi h (2), he cu e Eis(p o isionally)said obemodula i H1(E)
1The sys ema ic sho hand Hi(X):=Hi
e (X¯
Q,Qp(i)) o any a ie y Xo e Qis adop ed hence o h o ligh en he
no a ions.
2Subsequen ly, (2) has been gene alised o a hos o o he p-adic Galois ep esen a ions, while analogues o (1) emain
una ailable in all bu he simples geome ic se ings.
Da mon and Ro ge Res Ma h Sci (2016) 3:27 Page 3 o 32
a ises as a subquo ien o H1(Y), whe e Yis an open sub-Shimu a a ie y o X0(N)— he
la e being defined, in he s yle o La Palice, as he complemen o a closed sub-Shimu a
a ie y.
To comple ely desc ibe he open sub-Shimu a a ie ies o he modula cu e X0(N)
o e Q, no e ha he la e is he coa se moduli space o ellip ic cu es Awi h a ma ked
subg oup scheme o o de N, and ha i s closed sub-Shimu a a ie ies a e ob ained by
imposing addi ional endomo phism ings, which can only be equal o o de s in quad a ic
imagina y fields. Gi en such an o de O⊂K, he associa ed closed sub-Shimu a a ie y
O⊂X0(N) consis s o CM poin s o Oand is he coa se moduli space o ellip ic
cu es Awi h le el Ns uc u e equipped wi h an op imal embedding ι:O−→ End(A)
( espec ing he le el s uc u e) and ac ing in a p esc ibed way on he co angen space o
A. By he heo y o complex mul iplica ion, he 0-dimensional a ie y Ois isomo phic
o e K(a leas , when he disc iminan o Ois p ime o N) oφK(N) copies o spec(HO),
whe e φK(N) is he numbe o p imi i e ideals o Ko no m Nand HOis he ing class
field o Ka ached o O, whose Galois g oup o e Kis canonically iden ified wi h he
Pica d g oup o O ia global class field heo y.
The complemen s
YO(N):=X0(N)−O
hus p o ide an exhaus i e lis o he open sub-Shimu a a ie ies o X0(N). Gi en he
modula i y o E, he modula i y o Eamoun s o he exis ence o a Galois-equi a ian
inclusion
i:H0()0−→ H0(O)0
o sui able O, ealisingH1(E) as a subquo ien o H1(YO(N)) ia he push o wa d unde
πEand he pullback unde ιo he fi s ow in he ollowing diag am wi h exac ows:
0H1(X0(N))
πE
H1(YO(N))
?
H0(O)00
0H1(E)H1(E)H0()0
i
0.
(3)
Conside he simples non- i ial se ing whe e ={P1,P
2}⊂E(Q) consis s o wo
poin s defined o e Q, so ha H0()0=Qpwi h i ial Galois ac ion. The esul ing
ex ension
0H1(E)H1(E)Qp0(4)
encodes he image o he poin P2−P1∈E(Q) unde he connec ing homomo phism
δ:E(Q)−→ H1(Q,H1(E)) :=Ex 1
GQ(Qp,H1(E))
o Kumme heo y, whe e he Ex g oup is aken in he ca ego y o con inuous p-adic
ep esen a ions o GQ. The ollowing s a emen , which gi es a “modula i y c i e ion”
o Eand encapsula es many o he deepes heo ems on he Bi ch and Swinne on-
Dye conjec u e ob ained in he las decades, is o cou se expec ed o hold o all ellip-
ic cu es E, bu he eade is cau ioned ha he p oo o he implica ion (d) ⇒(a)
cu en ly equi es ha Ebe a semis able ellip ic cu e ha ing a leas one odd p ime
o non-spli mul iplica i e educ ion o a leas wo odd p imes o spli mul iplica i e
educ ion.
Da mon and Ro ge Res Ma h Sci (2016) 3:27 Page 4 o 32
Theo em 1.1 Assume ha he poin P2−P1is o in ini e o de in E(Q). Then he ollowing
a e equi alen :
(a) The cu e E=E {P1,P
2}is modula ;
(b) he Hasse-Weil L-se ies L(E, s)has a simple ze o a s =1;
(c) he poin P2−P1gene a es E(Q)⊗Qand LLI(E/Q)is ini e;
(d) o all p imes p, he g oup Ex 1
fin(Qp,H1(E)) o ex ensions o p-adic ep esen a ions o
he Galois g oup o Q ha a e c is alline a p is one-dimensional o e Qp.
Ske ch o p oo The modula i y o Eamoun s o he s a emen ha he e exis s an o de
Oin an imagina y quad a ic field Ksuch ha he ex ension (4) can be ob ained as he
pullback o (3) ia an inclusion i:Qp−→ H0(O)GQ, whose image con ains a deg ee 0
di iso
DK∈Di 0(O)GQ⊂Di 0(X0(N))(Q).
This means ha he poin P1−P2∈E(Q) is a nonze o mul iple o he Heegne poin
PE,K :=πE(DK).
The implica ion (a)⇒(b) he e o e ollows om he G oss–Zagie o mula [21] exp ess-
ing he heigh o PE,K as a nonze o mul iple o
L(E/K, 1) =L(E, 1) ·L(EK,1),
whe e EKis he quad a ic wis o Eby K. The exis ence o a sui able K o which
L(EK,1) = 0 ollows om a non- anishing esul o Waldspu ge o can be deduced om
analy ic numbe heo y echniques (c . [28]).
The implica ion (b) ⇒(c) was subsequen ly p o ed by Koly agin [25], who pa layed he
non- i iali y o PE,K in o a bound on he Mo dell–Weil ank and he Selme g oup o E
o e K.
The implica ion (c) ⇒(d) is a di ec consequence o he defini ions: in ac (d) is os en-
sibly weake han (c), Selme g oups being less sub le o con ol han Mo dell–Weil and
Sha a e ich–Ta e g oups.
The s iking implica ion (d) ⇒(a) ollows om Skinne ’s “con e se o he G oss–
Zagie –Koly agin Theo em” [33]. This las s ep is he mos ecen and combines se e al
new ing edien s: he powe ul echniques de eloped by Skinne and U ban o p o e he
Iwasawa–G eenbe g main conjec u e o ellip ic cu es o e Q[35], an impo an a ian
explo ed by Xin Wan in his Ph.D. hesis [39], and he p-adic analogue o [21] o mula ed
and p o ed in [8].
Mo e p ecisely, choose a p ime p≥5 o good o dina y educ ion o Esuch ha E[p]
is an i educible GQ- ep esen a ion and he image o he es ic ion map Selp(E)−→
E(Qp)/pE(Qp) does no lie in he image o E(Qp)[p]. A esul o Waldspu ge ensu es he
exis ence o an odd quad a ic cha ac e χsuch ha L(E, χ,1) = 0, which can be chosen
so ha χ(2) =χ(p)=1. Le Kdeno e he imagina y quad a ic field associa ed o χ.
The p-adic Selme g oup Ex 1
K,fin(Qp,H1(E)) o Eo e K(defined as an Ex g oup in he
ca ego y o c is alline ep esen a ions o GK) decomposes as a di ec sum o eigenspaces
Ex 1
K,fin(Qp,H1(E)) ≃Ex 1
fin(Qp,H1(E)) ⊕Ex 1
K,fin(Qp,H1(E))−
wi h espec o he ac ion o complex conjuga ion. Because L(E, χ,1) = 0, he esul s
o Koly agin (o o Ka o) imply he i iali y o Ex 1
K,fin(Qp,H1(E))−. Assump ion (d)
Da mon and Ro ge Res Ma h Sci (2016) 3:27 Page 5 o 32
he e o e implies ha Ex 1
K,fin(Qp,H1(E)) is one-dimensional o e Qp. One can hen a gue
as in [33]. Namely, he unning hypo heses ensu e ha bo h Lemma 2.3.2 and P oposi ion
2.7.3 o loc.ci . apply, and hence, ha a p-adic L- unc ion o he ype ha occu s in [39]
and [8] (which in e pola es c i ical alues o he L-se ies o he Rankin con olu ion o he
modula o m associa ed o Ewi h sui able Hecke cha ac e s o Ko highe infini y- ype)
does no anish a he i ial poin , which lies ou side i s egion o classical in e pola ion.
This in u n implies, in he ligh o [33, Co olla y 2.6.2] es ing on he a ian o he
G oss–Zagie o mula o [8], ha he Heegne poin PE,K has non- i ial p-adic o mal
g oup loga i hm and is he e o e non- o sion. As al eady explained, he non- i iali y o
PE,K is equi alen o (a), and he implica ion (d)⇒(a) ollows. 
The Bi ch and Swinne on-Dye conjec u e admi s an ex ension o ellip ic cu es
wis ed by A in ep esen a ions which a ises e y na u ally in he con ex o he modu-
la i y ques ions amed abo e. Le
:Gal(H/Q)→Au (V)≃GLn(¯
Qp)
be an n-dimensional ep esen a ion o he Galois g oup o a fini e ex ension H/Q,aso-
called A in ep esen a ion, iewed as ha ing coefficien s in ¯
Qp. The pai (E, ) gi es ise
o he Hasse–Weil–A in L-se ies
L(E, ,s):=

de (1 −−s(F −1
)(H1(E)⊗V)I)−1,
whe e he p oduc is aken o e he a ional p imes , he a i hme ic obenius elemen
a is deno ed by F ,andIdeno es he ine ia g oup a . The equi a ian Bi ch and
Swinne on-Dye conjec u e o Eand , deno ed BSD(E, ), asse s ha
o ds=1L(E, ,s)=dim ¯
Qp(E(H)⊗V)GQ.(5)
As a fi s s ep o unde s anding BSD(E, ), i is na u al o ask which κ∈Ex 1
fin(V,H1(E))
can be ealised as a subquo ien o a sui able H1(YO(N)). The A in ep esen a ion
H0(O)0which appea s in he uppe igh mos e m o he diag am (3) is eadily analysed
using he heo y o complex mul iplica ion. Namely, he sligh ly la ge A in ep esen a-
ion H0(O) decomposes as a di ec sum
H0(O)⊗¯
Qp=⊕
φK(N)
j=1Wj,whe e Wj=⊕
ψVj(ψ),
wi h Vj(ψ)⊂Vψ:=IndQ
Kψ.
In his equa ion, he second di ec sum is aken o e he non- i ial, ¯
Qp- alued, fini e o de
cha ac e s ψo Gal (HO/K) modulo he in olu ion ψ→ ψ−1,andVj(ψ) is a non- i ial
i educible cons i uen o he wo-dimensional ep esen a ion Vψob ained by inducing
he Galois cha ac e ψ om GK o GQ. The ep esen a ion Vψis i educible p ecisely
when ψ= ψ−1, and in his case a non- i ial class κ∈Ex 1
fin(Vψ,H1(E)⊗¯
Qp) is expec ed
o be modula i and only i (any o ) he analogues o condi ions (b)–(d) o Theo em 1.1
a e sa isfied, namely:
(b’) The Hasse–Weil–A in L-se ies L(E, Vψ,s) has a simple ze o a s=1;
(c’) he ep esen a ion Vψoccu s wi h mul iplici y one in E(H)⊗¯
Qp, and he Vψ-iso ypic
componen o he LLI(E/H) is fini e;
(d’) he g oup Ex 1
fin(Vψ,H1(E)⊗¯
Qp) is one-dimensional o e ¯
Qp, and gene a ed by κ.

Da mon and Ro ge Res Ma h Sci (2016) 3:27 Page 6 o 32
Al hough such a p ecise esul does no seem o appea in he li e a u e, all he ing edien s
needed o p o e i seem o be a ailable in p inciple.
The a he na ow no ion o modula i y desc ibed abo e has a ew isible d awbacks:
(1) Ve y ew A in ep esen a ions a ise in he cohomology o he 0-dimensional
Shimu a a ie ies O, which a e no e en ich enough o cap u e all o he i e-
ducible wo-dimensional A in ep esen a ions o Q. The open Shimu a a ie ies
YO(N) hus appea o gi e no pu chase on BSD(E, ) when is no induced om a
ing class cha ac e o an imagina y quad a ic field.
(2) Theo em 1.1 sugges s ha he modula i y o elemen s o Ex 1
fin(Vψ,H1(E)) is pu ely
a “ ank one phenomenon”: i his Ex g oup has dimension >1, none o i s elemen s
a e expec ed o be ealised in subquo ien s o any H1(YO(N)).
Ino de o ela eala ge classo non-semisimple Galois ep esen a ions o modula o ms,
i becomes desi able o elax he no ion o modula i y. One way in which one migh y o
do his is by eplacing he cu es YO(N) wi h mo e gene al “open Shimu a a ie ies”. These
should include all he a ie ies whose cohomology (a leas , a e semisimplifica ion) is
di ec ly ela ed o au omo phic o ms ia a sui able gene alisa ion o he Eichle –Shimu a
cong uence, and would e en ually encompass he complemen s o sub-Shimu a a ie ies
in la ge Shimu a a ie ies, as well as Kuga–Sa o a ie ies and o he na u al a ie ies
fib ed o e Shimu a a ie ies, he complemen s o Heegne cycles in such a ie ies, and so
on. Wi h his expanded no ion o modula i y, he p og amme o cha ac e ising he non-
semisimple Galois ep esen a ions ha a e modula becomes iche and mo e sub le. See
[9] o a agmen o expe imen al ma hema ics ha migh be iewed as fi ing in o his
p og amme. The ollowing ques ion seems like i migh epay u he in es iga ion, gi en
he pauci y o e idence, bo h heo e ical and expe imen al, ha has been ga he ed a ound
i so a :
Ques ion 1.2 Le V1and V2be Galois ep esen a ions o which hom(V1,V
2) is i e-
ducible. Suppose ha he e is a non- i ial κ∈Ex 1
fin(V1,V
2) a ising as a subquo ien o
he cohomology o an open Shimu a a ie y. Is Ex 1
fin(V1,V
2) necessa ily one-dimensional?
I he answe o his ques ion we e “yes”, i would imply ha he open cu e E−{P1,P
2}
discussed in Theo em 1.1 is ne e modula when ank(E(Q)) >1. (Bu see he inspi ing
a icle [29], as well as he s iking ongoing wo k o Zhiwei Yun and Wei Zhang in he
unc ion field case, o some an alising ideas in he opposi e, mo e op imis ic di ec ion.)
A second idea o enla ging he class o p-adic Galois ep esen a ions deemed o be
modula is o allow p-adic limi s o Galois ep esen a ions a ising in he cohomology
o (open) Shimu a a ie ies. This idea is e y na u al in he ligh o he classical wo k
o Deligne–Se e on A in ep esen a ions a ached o weigh one o ms, whe eby such
A in ep esen a ions a e ob ained by piecing oge he he Galois ep esen a ions a ached
o modula o ms o highe weigh s which a e ealised in he cohomology o Kuga–
Sa o a ie ies. I is ia his b oade no ion o modula i y ha all odd, i educible wo-
dimensional A in ep esen a ions o Qcan be ela ed o modula o ms. The idea o
ealising au omo phic Galois ep esen a ions as p-adic limi s has become pe asi e in
he subjec , and led o impo an ad ances: o example, i plays a key ole in he ecen
cons uc ion [22] by Ha is, Lan, Taylo , and Tho ne o Galois ep esen a ions a ached o
non-sel -dual au omo phic o ms on GLn. E en mo e ge mane o his a icle, p-adic limi s
Da mon and Ro ge Res Ma h Sci (2016) 3:27 Page 7 o 32
o au omo phic Galois ep esen a ions appea o cap u e non- i ial ex ension classes
going beyond se ings o “mul iplici y one”, as is illus a ed by he ollowing heo em o
Skinne and U ban [34, Thm. B]:
Theo em 1.3 Le E be an ellip ic cu e o e Q.I L(E, s) anishes o e en o de ≥2a s =1,
hen he Selme g oup Ex 1
fin(Qp,H1(E)) o E con ains a leas wo linea ly independen
modula classes.
The modula classes in his heo em a e cons uc ed as p-adic limi s o geome ic Galois
ep esen a ions in he cohomology o Shimu a a ie ies associa ed o he uni a y g oup
U(2,2). Al hough hese geome ic Galois ep esen a ions a e belie ed o be semisimple,
Theo em 1.3 es s on he ac ha his ea u e need no pe sis in he limi .
The p ima y goal o his a icle is o discuss a diffe en app oach o cons uc ing canon-
ical ex ension classes o by H1(E) o a la ge class o sel -dual A in ep esen a ions 
o dimension 4 (and hei lowe -dimensional sub ep esen a ions, in case is educible)
a ising as he enso p oduc =1⊗2o a pai o odd, wo-dimensional A in ep e-
sen a ions. The cons uc ion o hese classes is one o he main esul s o [19] ( esp. [11])
when bo h 1and 2a e i educible ( esp. when exac ly one o 1and 2is i educible),
and is based on p-adic limi s o non-semisimple, bu “geome ically modula ” Galois ep-
esen a ions. These limi classes a e e e ed o as gene alised Ka o classes because hei
cons uc ion is inspi ed by he seminal wo k [23]o Ka o(c .also[6,32]) on BSD(E, χ)
o χa Di ichle cha ac e . Like Heegne poin s in he se ing o BSD(E, Vψ), gene alised
Ka o classes enjoy close ela ions o (p-adic) Hasse–Weil–A in L- unc ions a ached o E
and , bu unlike Heegne poin s, hey a e expec ed o gene a e a non- i ial subg oup o
he Selme g oup a ached o Eand p ecisely when o ds=1L(E, ,s)=2. The o mulae
o [19] (c . Co olla y 3.6 below) ela ing he linea independence o wo gene alised Ka o
classes o he non- anishing o ce ain p-adic L-se ies can hus be ega ded as a p-adic
G oss–Zagie o mula “in analy ic ank wo”.
The main new con ibu ion o his a icle is a conjec u e exp essing he same gen-
e alised Ka o classes as canonical elemen s in (E(H)⊗V)GQwhen his la e space is
wo-dimensional. This conjec u e s eng hens and efines he “ellip ic S a k conjec u e”
o [15], and p o ides a amewo k o unde s anding he esul s o [11,15]and[19]. The
se ings in which is educible o en ake on special a i hme ic in e es and a e desc ibed
in de ail in he las chap e .
2 Hida amilies and pe iods o weigh one o ms
This sec ion p o ides backg ound on ce ain canonical s uc u es associa ed o a weigh
one o m g, a ising om he Hida amilies specialising in weigh one o (a p-s abilisa ion
o ) g. These a e impo an o he conjec u es o Sec . 3.4, bu Sec . 2can be skipped on a
fi s eading by he eade wishing o ge a quick eeling o he gene alised Ka o classes
desc ibed in Sec s. 3.1 and 3.2. On he o he hand, i is also wo h no ing ha Sec . 2is
en i ely sel -con ained. Conjec u e 2.1, which can be iewed as a p-adic analogue o he
S a k conjec u e o he adjoin o he Galois ep esen a ion a ached o a weigh one
o m, appea s o be new and may be o independen in e es .
Le g∈S1(N, χ) be a new o m o weigh one and le el Nwi h Fou ie coefficien s in a
field L,andle
Da mon and Ro ge Res Ma h Sci (2016) 3:27 Page 8 o 32
:GQ−→ Au (V)≃GL2(L)
be he A in ep esen a ion associa ed o i by he cons uc ion o Deligne and Se e. We
iew as ac ing on a wo-dimensional L- ec o space V, whe e L⊂Ccan be chosen o
be con ained in a cyclo omic field.
Le Hbe he numbe field cu ou by , so ha  ac o s h ough Gal (H/Q). Fix a
a ional p ime pand choose a p ime po Habo e p. The la e de e mines a canonical
inclusion
H⊂Hp⊂¯
Qp
o Hin i s comple ion Hpa p. Assume ha he pai (,p) sa isfies he ollowing condi ions:
(I) The p ime pspli s comple ely in L/Q, so ha Lis equipped wi h an embedding
in o Qpwhich will be fixed om now on. This assump ion, which is made solely
o ligh en he no a ions and could easily be dispensed wi h, allows  o be iewed
as a Qp-linea ep esen a ion ia he na u al ac ion o GQon he Qp- ec o space
V⊗LQp.
(II) The ep esen a ion Vis un amified a p. The e is hen a well-defined a i hme ic
obenius elemen
F p∈Gal (H/Q)
ac ing canonically on V, and he cha ac e is ic polynomial o (F p) is equal o he
Hecke polynomial
x2−ap(g)x+χ(p)=:(x−αg)(x−βg)
a ached o g.
(III) The modula o m gis egula a p, i.e. αg= βg. A e possibly enla ging L,i may
also be assumed ha his coefficien field con ains he oo s o uni y αgand βg.
(IV) The ep esen a ion gis no induced om a cha ac e o a eal quad a ic field K
in which he p ime pspli s. The a ionale o his condi ion, which seems o be
essen ial o a numbe o he cons uc ions and conjec u es p oposed in his pape ,
is explained in [15, §1.1].
The p-s abilisa ions o ga pa e he no malised eigen o ms o weigh one wi h Fou ie
coefficien s in Ldefined by
gα:=g(z)−βgg(pz),g
β:=g(z)−αgg(pz).
They a e eigen ec o s o he Up-ope a o sa is ying
Upgα=αggα,U
pgβ=βggβ.
The A in ep esen a ion Vdecomposes na u ally as a di ec sum
V=Vα⊕Vβ
in o one-dimensional eigenspaces o F p, wi h eigen alues αgand βg, espec i ely.
By a heo em o Hida, he e exis s a fini e fla ex ension go he Iwasawa algeb a and
aHida amilyg∈g[[q]] o ame le el Nand ame cha ac e χpassing h ough he p-
s abilised weigh one eigen o m gα. When gis cuspidal, he egula i y hypo hesis imposed
on gimplies ha such a Hida amily is unique, hanks o a ecen esul o Bellaïche and
Dimi o [1].
Da mon and Ro ge Res Ma h Sci (2016) 3:27 Page 9 o 32
The Hida amily gcomes equipped wi h he ollowing canonical s uc u es:
(a) The e is a locally ee g-module Vgo ank wo, affo ding Hida’s o dina y -adic
Galois ep esen a ion
g:GQ−→ Au g(Vg)
which is ealised in he in e se limi o o dina y é ale cohomology g oups associa ed
o he owe X1(Np ) o modula cu es. This ep esen a ion in e pola es he Galois
ep esen a ions associa ed by Deligne o he classical specialisa ions o g.
(b) The es ic ion o Vg o GQpadmi s a s able fil a ion
0−→ Ug−→ Vg−→ Wg−→ 0,
whe e bo h Ugand Wga e fla g[GQp]-modules ha a e locally ee o ank one
o e g, and he quo ien Wgis un amified, wi h F pac ing on Wgas mul iplica ion
by he p- h Fou ie coefficien ap(g).
(c) Le Qn
pdeno e he maximal un amified ex ension o Qpand le 
Qn
pdeno e i s p-adic
comple ion. In [30], Oh a cons uc s a canonical g-adic pe iod
ωg∈D(Wg):=(
Qn
pˆ
⊗Wg)GQp,
co esponding o he no malised -adic eigen o m gunde he isomo phism in
Theo em (A) o he in oduc ion o [30].
(d) The e is a na u al pe ec Galois-equi a ian duali y, gi en in Theo em (B) o he
in oduc ion o [30],
Ug×Wg−→ g(de (g)),
whe e GQac s on he module go he igh -hand side ia he de e minan o g.
Le
yg:g−→ Qp
be he specialisa ion map a ached o he p-s abilised weigh one o m gα. By specialising
he s uc u es abo e a ached o g ia he map yg,weob ain
(a’) A non-canonical isomo phism o Qp[GQ]-modules
gα:Vg:=Vg⊗ygQp
∼
−→ V⊗LQp.
(b’) A non- i ial GQp-s able fil a ion
0−→ Ug−→ Vg−→ Wg−→ 0
o Vgby one-dimensional subspaces, whe e Ug:=Ug⊗ygQpand Wg:=Wg⊗ygQp.
The F obenius elemen F pac s on Wgand Ugas mul iplica ion by αgand βg,
espec i ely. Since hese eigen alues a e assumed o be dis inc , he exac sequence
abo e spli s canonically, leading o he iden ifica ions
Ug=Vβ
g,W
g=Vα
g,V
g=Ug⊕Wg=Vβ
g⊕Vα
g.
(c’) Specialising Oh a’s pe iod leads o a canonical elemen
ωgα:=yg(ωg)∈D(Vα
g):=(Qn
p⊗Vα
g)GQp=(Hp⊗Vα
g)GQp.(6)
Da mon and Ro ge Res Ma h Sci (2016) 3:27 Page 16 o 32
Le
κp( , gα,h
α)= esp(κ( , gα,h
α))
deno e he image o he global class κ( , gα,h
α) in he local cohomology g oup
H1
fin(Qp,V
gh)=(H1
fin(Hp,V
)⊗Vgh)Gal (Hp/Qp)=(E(Hp)⊗Vgh)Gal (Hp/Qp).
As we desc ibe mo e explici ly below, Theo em D o [19] asse s ha his image is con-
olled by sui able p-adic a a a s o he second de i a i e o he classical L-se ies L( , Vgh,s)
a he cen al c i ical poin s=1.
These p-adic alues we e defined and explo ed in [19]and[15] and a e deno ed
Lpgα(˘
,˘
g∗,˘
h),Lpgβ(˘
,˘
g∗,˘
h),Lphα(˘
,˘
g, ˘
h∗),Lphβ(˘
,˘
g, ˘
h∗).(22)
They depend on he choice o ce ain es ec o s
(˘
,˘
g, ˘
h)∈S2(N;L)×M1(N, χ;L)×M1(N, χ−1;L)
wi h he same sys em o Hecke eigen alues as ,g,andh, espec i ely, and wi h ou ie
coefficien s in L, and on he choice o dual es ec o s
(˘
g∗,˘
h∗)∈Hom(M1(N, χ−1;L),L)×Hom(M1(N, χ;L),L)
wi h he same sys em o Hecke eigen alues as gand h. We e e o he in oduc ion o
[19] o mo e de ails on hei defini ion, con en ing ou sel es wi h ema k ha he p-adic
L- alue Lpgα(˘
,˘
g∗,˘
h) is defined essen ially as he p-adic limi o cen al c i ical alues
Lpgα(˘
,˘
g∗,˘
h):=lim
→1
E( , g,h)×C(˘
,˘
g∗,˘
h)×L(V ⊗Vg⊗Vh,(+1)/2)
g,g
,
as g anges o e he specialisa ions o (odd) weigh ≥3o heHida amilygspecialising
o gαin weigh one. He e E( , g,h)isap-adic mul iplie a ising om a ecipe o Panciskin,
whose p esence allows he p-adic in e pola ion o he special alues abo e, and C(˘
,˘
g∗,˘
h)
is a p oduc o e he p imes di iding N·∞o local e ms which depend in a simple way
on he choice o es ec o s.
Choose a basis o Vgh (o e Qp, o now) which is compa ible wi h he decomposi ion
(21), i.e. choose nonze o ec o s
αα
gh ∈Vαα
gh ,
αβ
gh ∈Vαβ
gh ,
βα
gh ∈Vβα
gh ,
ββ
gh ∈Vββ
gh .(23)
W i e
κp( , gα,h
α)=Rαα ⊗ ββ
gh +Rαβ ⊗ βα
gh +Rβα ⊗ αβ
gh +Rββ ⊗ αα
gh .(24)
The coo dina e Rξbelongs o E(Hp)F p=ξ
Qp, whe e ξ anges o e he index se
{αα =αgαh,αβ =αgβh,βα =βgαh,ββ =βgβh}.
No e ha Rξis e en he image o a global poin in E(H)Qp, assuming he fini eness o he
Sha a e ich–Ta e g oup o Eo e H.Le
logp:E(Hp)Qp−→ Hp(25)
deno e he o mal g oup loga i hm a ached o an in a ian diffe en ial on E/Q.The
ollowing heo em is s a ed in Sec ion 6.4 o [19]:

Da mon and Ro ge Res Ma h Sci (2016) 3:27 Page 17 o 32
Theo em 3.4 When L(E, Vgh,1) =0, he e exis s a choice o πin (16)and o es ec o s
o ,g,andhsuch ha hecoo dina esin(24)sa is y
logp(Rαβ)∼Lpgα(˘
,˘
g∗,˘
h),logp(Rβα)∼Lphα(˘
,˘
g, ˘
h∗),logp(Rββ)=0,(26)
whe e ∼deno es equali y up o a nonze o p-adic pe iod in H×
p.
Rema k 3.5 This heo em says no hing abou he quan i y logp(Rαα), which does no bea
any di ec ela ionship wi h p-adic L- alues in oduced abo e. We expec ha logp(Rαα)
may a he be connec ed wi h he fi s de i a i e o a pu a i e efinemen o Lp ( , gα,h
α)
in which all h ee modula o ms would be made o a y in a Hida amily.
As explained in he in oduc ion and in Sec ion 6.3. o [19], Theo em 3.4 has he ollowing
co olla y which can be iewed as a p-adic G oss–Zagie o mula in “analy ic ank wo”:
Co olla y 3.6 I L(E, Vgh,1) =0and Lpgα(˘
,˘
g∗,˘
h)= 0 o a sui able choice (˘
,˘
g∗,˘
h)o
es ec o s, hen he wo global classes
κ( , gα,h
α),κ( , gα,h
β)
a e linea ly independen in he Selme g oup H1
fin(Q,V
gh)a ached o E and Vgh, o a
sui able choice o πin (16).
Theo em 3.4 and i s co olla y mo i a ed he expe imen al s udy unde aken in [15]o he
special alues o p-adic L- unc ions appea ing in (26). This led o a p ecise conjec u e o
hese alues up o a ac o o L× a he han Q×
p.
To o mula e his conjec u e, ecall ha he class κ( , gα,h
α) is expec ed o be i ial
when o ds=1L(E, Vgh,s)>2. Assume ha his L- unc ion has a double ze o a he cen e,
which implies, by Conjec u e BSD(E, Vgh), ha (E(H)L⊗V12)GQis a wo-dimensional
L- ec o space.
Fix ec o s αα
gh ,..., ββ
gh chosen as in (23), wi h he diffe ence ha hey belong o L-
ec o space V12 a he han he Qp- ec o space Vgh. Choose a basis (P, Q) o his
L- ec o space, and w i e
P=Pαα ⊗ ββ
gh +Pαβ ⊗ βα
gh +Pβα ⊗ αβ
gh +Pββ ⊗ αα
gh ,
Q=Qαα ⊗ ββ
gh +Qαβ ⊗ βα
gh +Qβα ⊗ αβ
gh +Qββ ⊗ αα
gh ,
whe e Pξ,Q
ξa e poin s in E(H)F p=ξ
L o e e y ξ∈{αα =αgαh,αβ =αgβh,βα =
βgαh,ββ =βgβh}.
These poin s can be used o define a egula o a ached o gα, whose en ies a e he
p-adic o mal g oup loga i hms o he coo dina es a ached o he ec o s αα
gh and αβ
gh
(and simila ly o hα):
De ini ion 3.7 The egula o s a ached o Eand V12 a e
Reggα(E, V12)=de logpPββ logpPβα
logpQββ logpQβα 
=logpPββ ·logpQβα −logpQββ ·logpPβα,
Reghα(E, V12)=de logpPββ logpPαβ
logpQββ logpQαβ 
=logpPββ ·logpQαβ −logpQββ ·logpPαβ.
Da mon and Ro ge Res Ma h Sci (2016) 3:27 Page 18 o 32
The main conjec u e o [15] is he ollowing,3assuming αg
βg=±1 ( esp. αh
βh=±1) so
ha he S a k uni ugα( esp. uhα) is well defined:
Conjec u e 3.8 Assume ha L(E, Vgh,s) anishes o o de 2a s =1. Then he e exis s a
choice o es ec o s (˘
,˘
g∗,˘
h)and (˘
,˘
g, ˘
h∗)such ha
Lpgα(˘
,˘
g∗,˘
h)=Reggα(E, V12)
logpugα
,Lphα(˘
,˘
g, ˘
h∗)=Reghα(E, V12)
logpuhα
(mod L×).
Rema k 3.9 Conjec u e 3.8 lends i sel o nume ical e ifica ion and has been ex ensi ely
es ed in [15]. This is because he p-adic L- alues Lpgα(˘
,˘
g∗,˘
h)andLphα(˘
,˘
g, ˘
h∗)can
be exp essed in e ms o he a he conc e e p-adic i e a ed in eg als o loc.ci ., which
can be compu ed efficien ly using Alan Laude ’s [26] as o dina y p ojec ion algo i hms
on he space o o e con e gen modula o ms. In con as , he gene alised Ka o classes
hemsel es (like many objec s cons uc ed in é ale cohomology) seem difficul o compu e
in p ac ice, e en hough hei heo e ical use ulness is amply illus a ed in [11]and[19].
3.3 Enhanced egula o s
The goal o his a icle is o combine he insigh s a ising om Theo em 3.4 and Conjec u e
3.8 o o mula e a conjec u e on he posi ion o he gene alised Ka o classes hemsel es
in (E(H)⊗Vgh)GQ, speci ying his posi ion up o an ambigui y o L× a he han he less
p ecise Q×
pambigui y o Theo em 3.4.
The mos impo an ing edien s in he o mula ion o his conjec u e a e he so-called
enhanced egula o s

Reg(E, V12)∈(E(H)L⊗V12)GQ⊗(E(H)L⊗V12)GQ,

Regαα(E, V12)∈(Hp)F p=βgβh⊗(E(H)L⊗V12)GQ,

Reg(E, Vgh)∈(E(H)L⊗Vgh)GQ⊗(E(H)L⊗Vgh)GQ,

Regαα(E, Vgh)∈D(Vαα
gh )⊗(E(H)L⊗Vgh)GQ,
whose defini ion is somewha in he spi i o he egula o RSdefined in equa ion (2) o
[13], and which we now p oceed o desc ibe. As in (6), he e D(Vαα
gh ):=(Qn
p⊗Vαα
gh )GQp=
(Hp⊗Vαα
gh )GQp.
De ini ion 3.10 Choose an L-basis (P, Q) o he wo-dimensional ec o space (E(H)⊗
V12)GQ,andse

Reg(E, V12):=de PP
QQ
:=P⊗Q−Q⊗P. (27)
I does no depend on he choice o basis ha was made o define i , up o mul iplica ion
by L×.
The unc ion logαα :(E(H)L⊗V12)GQ−→ (Hp)F p=βgβhdefined by
logαα(P):=logp(Pββ)
3We wa n he eade ha he e in his no e we ha e chosen o s a e he main conjec u e o [15] in e ms o he a i hme ic
obenius F pa p,whilein[15] we a he employ he geome ic obenius σp=F −1
p. I is o his eason ha he oles
o αand βa e swapped in bo h o mula ions.
Da mon and Ro ge Res Ma h Sci (2016) 3:27 Page 19 o 32
induces a linea map
logαα ⊗1:(E(H)L⊗V12)GQ⊗(E(H)L⊗V12)GQ
−→ (Hp)F p=βgβh⊗(E(H)L⊗V12)GQ,
and we se

Regαα(E, V12):=(logαα ⊗1)(
Reg(E, V12)) =logp(Pββ)⊗Q−logp(Qββ)⊗P. (28)
Recall he embedding jgh :V12 −→ VL
gh ⊂Vgh o (12). Al hough his embedding is
comple ely non-canonical and only defined up o scaling by Q×
p, he e is a canonical way
o embedding V⊗2
12 in o V⊗2
gh . This is done by exploi ing he canonical duali ies on Vgand
Vhdesc ibed in Sec . 2, which gi es ise o pe ec pai ings
Vg×Vg−→ Qp(χ),V
h×Vh−→ Qp(χ−1),V
gh ×Vgh −→ Qp.
These pai ings allow us o define L- a ional s uc u es VL∗
g,VL∗
hand VL∗
gh which a e dual
o VL
g,VL
hand VL
gh, espec i ely, by le ing VL∗
gbe he L-dual o VL
gin Vg, and likewise o
VL∗
hand VL∗
gh . We may hen choose GQ-equi a ian embeddings
j∗
g:V1−→ VL∗
g,j
∗
h:V2−→ VL∗
h,j
∗
gh :=j∗
g⊗j∗
h:V12 −→ VL∗
gh ,
which a e well defined up o scaling by L×. Replacing jgh by μ·jgh, o any μ∈Q×
p,has
he effec o eplacing j∗
gh by μ−1·j∗
gh. Hence, he map
jgh ⊗j∗
gh :V12 ⊗V12 −→ Vgh ⊗Vgh
is well defined up o scaling by L×.
De ini ion 3.11 The enhanced egula o 
Reg(E, Vgh) associa ed o Eand Vgh is

Reg(E, Vgh):=(jgh ⊗j∗
gh)(
Reg(E, V12)) ∈(E(H)⊗Vgh)GQ⊗(E(H)⊗Vgh)GQ.(29)
Finally, le
Logp:(E(H)⊗Vgh)GQ−→ (Hp⊗Vgh)GQp=D(Vgh)
be he canonical p-adic loga i hm map induced om he p-adic loga i hm o (25) ia he
fixed embedding H⊂Hp,andle
Logαα :(E(H)⊗Vgh)GQ−→ D(Vαα
gh )
be i s composi ion wi h he unc o ial p ojec ion D(Vgh)−→ D(Vαα
gh ). This loga i hm
map is jus he mo e canonical coun e pa o he map logαα: he la e depends on he
choice o a basis ec o αα
gh o Vαα and is ela ed o Logαα by he ule
Logαα :=logαα ⊗ αα
gh .
We se

Regαα(E, Vgh):=(Logαα ⊗1)(
Reg(E, Vgh)) =Logαα(P)⊗Q−Logαα(Q)⊗P. (30)
I is wo h no ing ha he enhanced egula o 
Regαα(E, Vgh) is a canonical in a ian
associa ed o Eand Vgh, i.e. i is well defined up o mul iplica ion by L×, while he less
canonical 
Regαα(E, V12) depends on he choice o a basis αα
gh o Vαα
gh . The wo egula o s
a e ela ed by

Regαα(E, Vgh)=
Regαα(E, V12)⊗ αα
gh .(31)
Da mon and Ro ge Res Ma h Sci (2016) 3:27 Page 20 o 32
3.4 The conjec u e
Recall he pe iods
ωgα∈D(Vα
g),ωhα∈D(Vα
h)
cons uc ed in (6). The main conjec u e o his no e is:
Conjec u e 3.12 Assume ha (E, Vgh)=2. The gene alised Ka o class κ( , gα,h
α)belongs
o (E(H)⊗Vgh)GQand sa is ies he ela ion
ωgαωhα⊗κ( , gα,h
α)∼L
Regαα(E, Vgh)
in D(Vαα
gh )⊗(E(H)⊗Vgh)GQ,whe e∼Ldeno es an equali y up o scaling by a ac o in L
which is nonze o o a sui able choice o πin (16).
The ollowing p oposi ion shows ha , unde Conjec u e 2.1 ( ela ing he canonical
pe iod a ached o g o he S a k uni ugα) and Conjec u e 3.2 (a mild s eng hening o
BSD(E, gh)), Conjec u e 3.12 implies he main conjec u e o [15]. Be o e dismissing his
p oposi ion as me e conjec u al ela ions be ween conjec u es, he eade is eminded
ha Conjec u e 3.8 lends i sel o expe imen and has been ex ensi ely es ed nume ically
in [15], while he s eng hening desc ibed in Conjec u e 3.12 lies o he momen beyond
he ange o explici calcula ions (c . Rema k 3.9).
P oposi ion 3.13 Assume Conjec u es 2.1 and 3.2. Then Conjec u e 3.12 implies Conjec-
u e 3.8.
P oo Conside he p oduc o pe iods
ηgαωhα=(gα⊗ β
g)·(hα⊗ α
h)=gα·hα⊗ βα
gh ∈D(Vβα
gh )
defined in Sec . 2.
The pai ing in oduced in (7) gi es ise o a pai ing
,:D(Vαβ
gh )×D(Vβα
gh )−→ D(Qp)=Qp.
As shown in he p oo o [19, Theo em 6.10 (ii)],
Logαβ κ( , gα,h
α),ηgαωhα=Lpgα( , g, h) (mod L×).(32)
On he o he hand, by he defini ion o he enhanced egula o ,
Logαβ 
Regαα(E, Vgh)=(logpPββ logpQβα −logpQββ logpPβα)⊗ αα
gh ⊗ ∗αβ
gh
=Reggα(E, V12)⊗ αα
gh ⊗ ∗αβ
gh (mod L×).
Hence, he ollowing equali y holds in D(Vαα
gh ):
Logαβ 
Regαα(E, Vgh),ηgαωhα=gα·hα·Reggα(E, Vgh)⊗ αα
gh (mod L×).(33)
By pai ing he alue o Logαβ a bo h sides o he displayed iden i y in Conjec u e 3.12
wi h he class ηgαωhαand in oking (32)and(33), we ob ain
ωgαωhα⊗Lpgα( , g, h)=gα·hα·Reggα(E, V12)⊗ αα
gh ∈D(Vαα
gh ) (mod L×).
Since
ωgαωhα=gα·hα· αα
gh (mod L×),
Da mon and Ro ge Res Ma h Sci (2016) 3:27 Page 21 o 32
i ollows ha
gαLpgα( , g, h)=gαReggα(E, V12) (mod L×),
and he e o e ha
Lpgα( , g, h)=Reggα(E, V12)
Lgα
(mod L×).
Conjec u e 3.8 now ollows di ec ly om his equali y a e in oking Conjec u e 2.1.
Rema k 3.14 As explained in a numbe o he examples co e ed in Sec . 4below, i may
happen ha all ou o he p-adic i e a ed in eg als in (22) a e equal o ze o e en when
some o he gene alised Ka o classes a e non- i ial. This sugges s ha Conjec u e 3.12 is
a genuine s eng hening o Conjec u e 3.8.
4 Special cases
This sec ion examines Conjec u e 3.12, and he special o ms aken by he enhanced
egula o s

Regαα(E;V12),
Regαβ(E;V12),
Regβα(E;V12),
Regββ(E;V12),
in he a i hme ically in e es ing cases whe e Vgh is educible. Acco ding o Da mon e al.
[16, §2], he ollowing is a comple e lis o scena ios whe e his occu s:
(1) The o iginal Beilinson–Ka o se ing whe e Vgand Vha e bo h educible, i.e. whe e
gand ha e bo h Eisens ein se ies o weigh one;
(2) he Beilinson–Flach se ing whe e exac ly one o Vgo Vhis educible, i.e. whe e
exac ly one o go his cuspidal;
(3) he complex mul iplica ion case whe e Vgand Vha e bo h induced om cha ac e s
o a common imagina y quad a ic field;
(4) he eal mul iplica ion case whe e Vgand Vha e induced om cha ac e s o mixed
signa u e o a common eal quad a ic field;
(5) he adjoin case whe e his (a wis o ) he dual o g, so ha Vgh is he di ec sum o a
one-dimensional ep esen a ion and a wis o he adjoin o Vg.
The eade will no ice ha some o he abo e se ings a ise when gand/o ha e
educible, while in Sec s. 2and 3 hese ep esen a ions we e assumed o be i educible.
This assump ion was imposed o a la ge ex en o he sake o simplici y o he exposi-
ion, and he s a emen (and p esumed alidi y) o Conjec u e 3.12 does no ely on i .
Fo comple eness, we ha e he e o e desc ibed he enhanced egula o s ha appea in
Conjec u e 3.12 in all o he abo e cases.
4.1 Beilinson–Ka o classes
Assume ha gand ha e bo h Eisens ein se ies. A e possibly wis ing go h, he e is no
eal loss o gene ali y in assuming ha he e exis Di ichle cha ac e s χ1,χ2such ha g
and ha e gi en by
g=E1(χ1,χ2),h=E1(1,χ−1
12 ),whe e χ12 =χ1χ2.
We e e o e.g. [10, §2.1.2] o he defini ion o hese weigh one Eisens ein se ies in
e ms o hei q-expansions. The Galois ep esen a ions a ached o gand ha e educible,

Da mon and Ro ge Res Ma h Sci (2016) 3:27 Page 22 o 32
namely
V1=L(χ1)⊕L(χ2),V
2=L⊕L(χ−1
12 ),
V12 =L(χ1)⊕L(χ−1
1)⊕L(χ2)⊕L(χ−1
2),(34)
whe e he coefficien field Lis he cyclo omic field gene a ed by he images o χ1and χ2.
These ep esen a ions ac o h ough he Galois g oup Gal (H/Q) o an abelian ex ension
Ho Q.Wemayse
αg=χ1(p),βg=χ2(p),αh=1,βh=χ−1
12 (p).
The egula i y assump ion implies ha V1and V2decompose uniquely as a di ec sum o
wo GQp-s able lines, which a e also s able unde GQ. Mo e p ecisely,
Vαα
12 =L· χ1,V
ββ
12 =L· ¯χ1,V
αβ
12 =L· ¯χ2,V
βα
12 =L· χ2,
whe e ( χ1, ¯χ1, ¯χ2,
χ2)isabasis o V12 on which GQac s ia he cha ac e s χ1,¯χ1,¯χ2,
and χ2, espec i ely.
The class κ( , gα,h
α)=κBK( , gα,h
α) was cons uc ed by Ka o as a p-adic limi o Beilin-
son elemen s a ached o pai s o modula uni s whose loga i hmic de i a i es a e weigh
wo Eisens ein se ies. Theo em 3.1 in his case boils down o Ka o’s ecip oci y law, which
asse s ha κ( , gα,h
α) belongs o he Selme g oup o Eo e Hi and only i he L- unc ion
L(E, Vgh,s)=L(E, χ1,s)L(E, ¯χ1,s)L(E, χ2,s)L(E, ¯χ2,s)
anishes a s=1. In his case, i clea ly anishes o e en o de and anishes o o de wo
i and only i (a e e en ually in e changing he cha ac e s χ1and χ2)
o ds=1L(E, χ1,s)=o ds=1L(E, ¯χ1,s)=1,L(E, χ2,1),L(E, ¯χ2,1) = 0.
Assuming ha his is he case, Conjec u es BSD(E, χ1) and BSD(E, χ2)p edic ha
(E(H)L⊗V12)GQis wo-dimensional o e Land ha a basis o i can be chosen o
be
P:=P¯χ1⊗ χ1,Q:=Qχ1⊗ ¯χ1,
whe e P¯χ1and Qχ1a e global poin s in E(H)Lgene a ing he ¯χ1and χ1eigenspaces,
espec i ely, o he na u al ac ion o GQ. Wi h hese no a ions, we ha e
Pαα =Pαβ =Pβα =0,P
ββ =P¯χ1,
Qαβ =Qβα =Qββ =0,Q
αα =Qχ1.
This immedia ely implies ha

Regαα(E, V12)=logp(P¯χ1)·Q, 
Regαβ(E, V12)=0,

Regβα(E, V12)=0,
Regββ(E;V12)=logp(Qχ1)·P.
I ollows ha
Reggα(E;V12)=Reggβ(E;V12)=Reghα(E;V12)=Reghβ(E;V12)=0.
This accoun s o he ac ha he p-adic i e a ed in eg als
Lpgα( , g, h),Lpgβ( , g, h),Lphα( , g, h),Lphβ( , g, h)
Da mon and Ro ge Res Ma h Sci (2016) 3:27 Page 23 o 32
sys ema ically anish4when gand ha e Eisens ein se ies ha a e egula a p. Conjec u e
3.12 makes he s onge p edic ion ha he gene alised Ka o classes hemsel es a e non-
i ial, and is consis en wi h a Conjec u e o Pe in-Riou, since i p edic s ha
logββ(κ( , gα,h
α)) =logαα(κ( , gβ,h
β)) =logp(P¯χ1)logp(Qχ1) (mod L×).
4.2 Beilinson–Flach classes
In heBeilinson–Flachse ing,i canbeassumedwi hou losso gene ali y ha gisaweigh
one cusp o m wi h neben ypus cha ac e χand Galois ep esen a ion Vg=V1⊗LQp,
and ha h:=E1(1,χ−1) is he weigh one Eisens ein se ies a ached o he pai (1,χ−1)
o Di ichle cha ac e s. The ele an ou -dimensional ep esen a ions a e hen equal o
Vgh =Vg⊕V¯
g;V12 =V1⊕¯
V1,
and he Hasse–Weil–A in L-se ies
L(E, Vgh,s)=L(E, Vg,s)L(E, ¯
Vg,s)
has a double ze o a s=1 p ecisely when each o he p imi i e L-se ies L(E, Vg,s)and
L(E, V¯
g,s) ha e a simple ze o a s=1. Conjec u e BSD(E, Vg) hen implies ha each o
he L- ec o spaces on he igh -hand side o
(E(H)L⊗V12)GQ)=(E(H)L⊗V1)GQ⊕(E(H)L⊗¯
V1)GQ,
is one-dimensional. Le Pbe an L-basis o (E(H)L⊗V1)GQand le ¯
Pbe he associa ed
L-basis o (E(H)L⊗¯
V1)GQ, ob ained by applying complex conjuga ion o he coefficien s
in L.
A e fixing an o de ing αg,βg∈L o he eigen alues o F pon V1, and se ing
αh=1,βh=χ−1(p)=(αgβg)−1,
we ha e
Vαα
12 =Vαg
1,V
αβ
12 =¯
Vβ−1
g
1,V
βα
12 =Vβg
1,V
ββ
12 =¯
Vα−1
g
1,
and hence
Pαα =Pαg,P
βα =Pβg,P
αβ =0Pββ =0,
¯
Pαα =0¯
Pβα =0,¯
Pαβ =¯
Pβ−1
g,¯
Pββ =¯
Pα−1
g.
A di ec calcula ion e eals ha , up o mul iplica ion by L×,

Regαα(E, V12)=logp(¯
Pα−1
g)·P, 
Regαβ(E, V12)=logp(Pβg)·¯
P,

Regβα(E, V12)=logp(¯
Pβ−1
g)·P, 
Regββ(E;V12)=logp(Pαg)·¯
P.
I ollows ha
Reggα(E, V12)=logp(¯
Pα−1
g)·logp(Pβg),Reggβ(E, V12)=logp(¯
Pβ−1
g)·logp(Pαg),
Reghα(E, V12)=0,Reghβ(E, V12)=0.
as desc ibed in [15,§6].
4Bu see he expe imen s desc ibed in [15, §7] in he case whe e gis i egula a p, which sugges ha he i egula
se ing o Conjec u e 3.12 would me i u he in es iga ion.
Da mon and Ro ge Res Ma h Sci (2016) 3:27 Page 24 o 32
4.3 Complex mul iplica ion classes and Heegne poin s
In his chap e we conside he se ing whe e gand ha e he a se ies a ached o cha ac e s
ψgand ψho he same imagina y quad a ic field K, and wi h in e se neben ypus cha ac e .
Gi en any cha ac e ψo GK,le ψdeno e he cha ac e ob ained by conjuga ing i wi h
he in olu ion in Gal (K/Q). Then
Vg=IndQ
Kψg=IndQ
Kψ
g,V
h=IndQ
Kψh=IndQ
Kψ
h,
and he e o e
Vgh =IndQ
Kψ•⊕IndQ
Kψ◦,whe e ψ•=ψgψh,ψ◦=ψgψ
h.
The sel -duali y assump ion implies ha ψ•and ψ◦a e a e ing class cha ac e s, i.e. hey
sa is y
ψ
•=ψ−1
•,ψ
◦=ψ−1
◦.
Assume ha he induced ep esen a ions
V•:=IndQ
Kψ•,V
◦:=IndQ
Kψ◦
appea ing in he decomposi ion
V12 =V•⊕V◦(35)
( iewed as ep esen a ions wi h coefficien s in he numbe field L)a ei educible, which is
always he case unless ψ•o ψ◦is a quad a ic, i.e. a genus cha ac e . (The mo e degene a e
case whe e his a ises can be subsumed unde he “adjoin se ing” conside ed in Sec . 4.5.)
The Hasse–Weil–A in L-se ies
L(E, Vgh,s)=L(E, V•,s)L(E, V◦,s)=L(E/K, ψ•,s)L(E/K, ψ◦,s)
has a double ze o a s=1 in one o he ollowing wo cases:
(1) The p imi i e L-se ies L(E, V•,s)andL(E, V◦,s) each ha e a simple ze o a s=1. This
se ing, which esembles mo e closely he phenomena desc ibed in he p e ious wo
sec ions on Beilinson–Ka o and Beilinson–Flach elemen s, will be e e ed o as he
ank (1,1) se ing o Conjec u e 3.12.
(2) Exac ly one o he p imi i e L-se ies L(E, V•,s)o L(E, V◦,s)hasadoubleze oa s=1,
and he o he is non- anishing a he cen e. This case shall be e e ed o as he ank
(2,0) se ing o Conjec u e 3.12. The possible non- i iali y o he gene alised Ka o
classes in he p esence o a “genuine” double ze o o a p imi i e Hasse–Weil–A in L-
unc ion ep esen s a no el ea u e ha did no a ise in he se ing o Beilinson–Ka o
o Beilinson–Flach elemen s.
4.3.1 The ank (1,1) se ing
In his case, Conjec u es BSD(E, V•) and BSD(E, V◦) p edic ha he Mo dell–Weil g oups
(E(H)L⊗V•)GQand (E(H)L⊗V◦)GQa e bo h one-dimensional L- ec o spaces, wi h
gene a o s P•and P◦, espec i ely. I is na u al o w i e
P•=Pψ•⊗ ψ
•+Pψ
•⊗ ψ•,P
◦=Pψ◦⊗ ψ
◦+Pψ
◦⊗ ψ◦,(36)
whe e Pψ•,Pψ
•,Pψ◦,andPψ
◦a e gene a o s o he one-dimensional subspaces o E(H)L
on which GKac s ia he cha ac e s ψ•,ψ
•,ψ◦,andψ
◦, espec i ely.
Da mon and Ro ge Res Ma h Sci (2016) 3:27 Page 25 o 32
The desc ip ion o he enhanced egula o s a ached o V12 and o (P•,P
◦) can be u he
subdi ided in o wo cases, wi h ma kedly diffe en ea u es: he case whe e he p ime pis
spli in K, and he case whe e i is ine in K.
a) The case whe e p is spli in K. In his case, le p=pp
be he ac o isa ion o pin o
dis inc p imes o K.Wecan hense
αg=ψg(p),βg=ψg(p),αh=ψh(p),βh=ψh(p),
so ha
αgαh=ψ•(p),αgβh=ψ◦(p),βgαh=ψ◦(p),βgβh=ψ•(p).
The decomposi ion o he GKp=GQp ep esen a ions a ached o (35) in o F p-
eigenspaces is also s able unde he ac ion o he global Galois g oup GK, and is desc ibed
by:
Vαα
12 =Vψ•
•,V
αβ
12 =Vψ◦
◦,V
βα
12 =Vψ
◦
◦,V
ββ
12 =Vψ
•
•.
I ollows ha , up o mul iplica ion by L×,

Regαα(E, V12)=logp(Pψ
•)·P◦,
Regαβ(E, V12)=logp(Pψ
◦)·P•,

Regβα(E, V12)=logp(Pψ◦)·P•,
Regββ(E, V12)=logp(Pψ•)·P◦,
and he e o e ha
Reggα(E, V12)=logp(Pψ
•)·logp(Pψ
◦),Reggβ(E, V12)=logp(Pψ•)·logp(Pψ◦),
Reghα(E, V12)=logp(Pψ
•)·logp(Pψ◦),Reghβ(E, V12)=logp(Pψ•)·logp(Pψ
◦).
The co esponding o mulae o he p-adic i e a ed in eg als Lpgα( , g, h), Lpgβ( , g, h),
Lphα( , g, h), and Lphβ( , g, h) we e p o ed in [15,§3],byusing hep-adic G oss–Zagie
o mula o [8] o exp ess hese L- alues in e ms o p oduc s o p-adic loga i hms o
Heegne poin s. Theo em 3.3 o loc.ci . is one o he ew pieces o heo e ical e idence in
suppo o Conjec u e 3.12.
b) The case whe e p is ine in K. In his case, he eigen alues o he F obenius au omo -
phism F pac ing on Vgand Vha e o he o m
αg,βg=−αg,αh=α−1
g,βh=−α−1
g.
Le ( ψg,
ψ
g) be a eigenbasis o Vg o he ac ion o GK ela i e o he dis inc cha ac e s
ψgand ψ
g,andle ( ψh,
ψ
h) be a simila basis o Vh. These ec o s can be scaled so ha
F pac s on hem as
F p( ψg)=αg· ψ
g,F p( ψ
g)=αg· ψg,F p( ψh)=α−1
g· ψ
h,
F p( ψ
h)=α−1
g· ψh,
and he e o e we may se
Vα
g=L·( ψg+ ψ
g),V
β
g=L·( ψg− ψ
g),V
α
h=L·( ψh+ ψ
h),
Vβ
h=L·( ψh− ψ
h).
A e se ing
ψ•:= ψg⊗ ψh,
ψ
•:= ψ
g⊗ ψ
h,
ψ◦:= ψg⊗ ψ
h,
ψ
◦:= ψ
g⊗ ψh,
and le ing
+
•:= ψ•+ ψ
•,
−
•:= ψ•− ψ
•,
+
◦:= ψ◦+ ψ
◦,
−
•:= ψ◦− ψ
◦,
Da mon and Ro ge Res Ma h Sci (2016) 3:27 Page 32 o 32
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