Elliptic curves of rank two and generalized Kato classes

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 ....................................... ©2016 The Au ho (s). This a icle is dis ibu ed unde he e ms o he C ea i e Commons A ibu ion 4.0 In e na ional License (h p://c ea i ecommons.o g/licenses/by/4.0/), which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in any medium, p o ided you gi e app op ia e c edi o he o iginal au ho (s) and he sou ce, p o ide a link o he C ea i e Commons license, and indica e i changes we e made. 0123456789().,–: ol 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 24. Kings, G., Loeffle , D., Ze bes, S.: Rankin–Selbe g Eule sys ems and p-adic in e pola ion (submi ed) 25. Koly agin, V.: Fini eness o E(Q)andX(E, Q) o a subclass o Weil cu es, Iz . Akad. Nauk SSSR Se . Ma . 52(3) 670–671 (1988); ansla ion in Ma h. USSR-Iz . 32(3) 523–541 (1989) 26. Laude ,A.:Efficien compu a iono Rankinp-adicL- unc ions.In:Böckle,G.Wiese,G.(eds.)Compu a ionswi hModula Fo ms:P oceedings o a Summe School andCon e ence,Heidelbe g, Augus /Sep embe 2011, pp.181–200. Sp inge (2014) 27. 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