LOW-LYING ZEROS OF FAMILIES OF ELLIPTIC CURVES

Transcription

1 LOW-LYING ZEROS OF FAMILIES OF ELLIPTIC CURVES MATTHEW P YOUNG Astrct Tr is groing ody of vidnc giving strong vidnc tt zros of fmilis of L-functions follo distriution ls of ignvlus of rndom mtrics Tis ilosoy is non s t rndom mtrix modl or t Ktz-Srn ilosoy T rndom mtrix modl ms rdictions for t vrg distriution of zros nr t cntrl oint for fmilis of L-functions W study t lo-lying zros for fmilis of llitic curv L-functions For ts L-functions tr is scil ritmtic intrst in ny zros t t cntrl oint y t conjctur of irc nd Sinnrton-Dyr nd t imrssiv rtil rsults tords rsolving t conjctur W clcult t dnsity of t lo-lying zros for vrious fmilis of llitic curvs Our min foci r t fmily of ll llitic curvs nd lrg fmily it ositiv rn A min cllng s n to otin rsults it tst functions tt r concntrtd clos to t origin sinc t cntrl oint is loction of grt intrst An liction is n imrovmnt on t ur ound of t vrg rn of t fmily of ll llitic curvs conditionl on t Gnrlizd Rimnn yotsis GRH T ur ound otind is lss tn 2, ic sos tt ositiv roortion of curvs in t fmily v lgric rn qul to nlytic rn nd finit Tt-Sfrvic grou W so tt tr is n xtr contriution to t dnsity of t lo-lying zros from t fmily it ositiv rn rsumly from t xtr zro t t cntrl oint 1

2 2 1 Introduction T rndom mtrix modl rdicts tt mny sttistics ssocitd to zros of fmily of L-functions cn modld or rdictd y t distriution of ignvlus of lrg rndom mtrics in on of t clssicl linr grous If t sttistics of fmily of L-functions r modld y t ignvlus of t grou G tn sy tt G is t symmtry grou or symmtry ty ssocitd to t fmily T sttistic of intrst to us in tis or is t dnsity of zros nr t cntrl oint lso non s t 1-lvl dnsity T rndom mtrix modl rdicts tt t distriution of ts zros sould modld y t ignnvlus nrst 1 for on of t symmtry tys G All of t diffrnt grous G v distinct vior in tis rgrd Trfor, comuting t 1-lvl dnsity givs torticl y to rdict t symmtry ty of fmily It is stndrd to ssum t Gnrlizd Rimnn Hyotsis GRH to study t 1-lvl dnsity nd do so trougout tis or In trut, t GRH is ncssry for only ndful of our rsults, ut it simlifis rgumnts in som non-ssntil lcs so us it frly vn n it could rmovd it xtr or It is scilly intrsting to invstigt t 1-lvl dnsity for fmilis of L-functions ttcd to llitic curvs ovr t rtionls sinc zros t t cntrl oint v imortnt ritmtic informtion y t conjctur of irc nd Sinnrton-Dyr Ts invstigtions v n t min focus of tis or 11 Acnoldgmnt Tis or constituts lrg ortion of my PD tsis I tn my dvisor, Hnry Inic, for suggsting tis rolm nd for is suort nd ncourgmnt il doing tis or 2 Prliminris W gin y collcting som fcts nd stting t nottion ill us W considr n llitic curv E/Q givn in gnrl Wirstrss form 1 E : y xy + 3 y = x x x + 6, r c i Z Undr cng of vrils E cn rougt into t simlr form 2 y 2 = x 3 + x +, r nd r intgrs T cnonicl cng of vrils cf [Si1], uss t rmtrs 2, 4, 6, c 4, nd c 6, r nd T curv 1 is tn quivlnt to 2 = , 4 = , 6 = , c 4 = , c 6 = y 2 = x 3 27c 4 x 54c 6 Wn givn y t form 2, E s discriminnt = ,

3 LOW-LYING ZEROS OF FAMILIES OF ELLIPTIC CURVES 3 ic is ncssrily non-zro for t curv E to llitic T Wirstrss qution for t llitic curv 1 is not uniqu Any to Wirstrss qutions for t sm curv r rltd y t dmissil cng of vrils 3 x = u 2 x + r y = u 3 y + su 2 x + t, r u, r, s, nd t r intgrs nd u 0 Undr tis cng of vrils t discriminnt trnsforms y u 12 = Liis, u 4 c 4 = c 4 nd u 6 c 6 = c 6 A stndrd tcniqu in studying E is to rduc t qution 1 mod for vry rim T qution 1 is miniml for t rim if t or of dividing cnnot dcrsd y n dmissil cng of vrils T qution 1 is glol miniml Wirstrss qution if it is miniml for ll rims simultnously For ny Wirstrss qution tr is n dmissil cng of vrils lcing it in glol miniml Wirstrss form cf [Si1], Corollry 83 W rcord r tt if t dmissil cng of vrils 3 lcs 1 in glol miniml Wirstrss form tn t only rims dividing u r tos for ic 1 is not miniml W rmr tt for > 3 if t qution 2 is not miniml t tn 4 nd 6 nd trfor 12 Suos E is givn y glol miniml Wirstrss qution 1 T conductor N of E is tn dfind y N = f r for > 3 f = { 1 if c 4, i E s multilictiv rduction t, 2 if c 4, i E s dditiv rduction t Wn = 2 or 3 t dfinition of f is mor comlictd cf [Si2], IV, 10, ut it ill usully noug for our uross tt N nd f 8 for ll rims If ncssry, t conductor cn comutd using Ogg s formul nd Tt s lgoritm Continuing to ssum 1 is glol miniml Wirstrss qution for E, t L-function of E is dfind y 4 Ls, E = 1 s + 1 2s 1 1 s 1, r 5 = #EF, nd #EF is t numr of ffin oints on E, n rducd mod T cntrl oint is t s = 1 Sinc 1 is miniml t cng of vrils ting 1 to 2 dos not ltr for > 3 Tus, for ny > 3, is givn y = x 3 + x + x mod W rmr tt for rims > 3 dividing t conductor v = ±1 if E s multilictiv rduction t nd = 0 if E s dditiv rduction t T infinit roduct 4 convrgs solutly nd uniformly for R s > 3 2, y Hss s stimt < 2 According

4 4 MATTHEW P YOUNG to t Simur-Tniym conjctur rovd y Wils t l [W], [TW], [CDT] tr xists igt to rimitiv cus form fz on Γ 0 N suc tt Ls, E = Ls, f Furtr, Ls, E s nlytic continution to t comlx ln nd stisfis t functionl qution s N Λs, E := ΓsLs, E = Λ2 s, E, 2π r = ±1 is t root numr of E Trougout tis or ill ssuming t Gnrlizd Rimnn Hyotsis olds, nmly tt ll t nontrivil zros of n ritmtic L-function li on its lin of symmtry Using t nottion of Inic-Luo-Srn [ILS] dfin for n L-function Ls, E t quntity DE; φ = γe φ 2π γ E r φ is n vn Scrtz clss tst function os Fourir trnsform 1 φ s comct suort, γ E runs troug t imginry rts of t nontrivil zros ρ E = 1 + iγ E of Ls, E countd it multilicity, nd X is rmtr t our disosl gnrlly of siz N E, t conductor of E; lloing X to only roximtly N E givs us mor frdom in vrging ovr fmily T scling fctor 2π 1 is insrtd to normliz t numr of zros countd y t tst function φ, so tt DE; φ sould tougt of s rrsnting t dnsity of zros of Ls, E nr t cntrl oint s = 1 W ill intrstd in vrging DE; φ ovr crtin fmilis of utomoric forms rising from llitic curvs Ec fmily study ill of t form F = {E d } r d rngs ovr st A, ic is sust of Z or Z Z, suc tt c d A nturlly dfins n llitic curv E D ovr Q T curv E d ill dfind y Wirstrss qution os cofficints r olynomils in d Our min fmily ill rmtrizd y d =, Z 2 r E d : y 2 = x 3 + x + It my n tt Ls, E d = Ls, E d for d d trivilly cus glol miniml Wirstrss qution for E d quls tt of E d or mor sutly cus E d is isognous to E d W t suc forms it multilicity W xct it sould m no sttisticl diffrnc tr on ts suc forms f it multilicity or not As gnrl rul, cn sily m rstrictions on d tt forc E d E to miniml, ut tis dos not cng ny sttistics t lst in t min trm s Sction 56 for mor toroug discussion W ill oftn surss t dndnc of F on A W study t igtd vrg DF; φ, = E F DE; φe, r E d := d is smoot, comctly suortd function cutoff function on R or R 2, icvr is rorit To void trivilitis ssum dos not v totl mss zro i ŵ0 0, nd in rticulr tt is not idnticlly zro W msur t igtd sum ginst t totl igt W F = E F E Sinc DE; φ dnds on X scl our cutoff function y X lso, in ic cs us t nottion X to dnot t scling of y X nd W X to t totl igt scld y 1 Trougout dnot fy = fx xydx, x = x2πix

5 LOW-LYING ZEROS OF FAMILIES OF ELLIPTIC CURVES 5 X T rcis scling dnds on t fmily nd is usd to ic out curvs it conductors N suc tt log N is symtoticlly on vrg Oftn simly t curvs it discriminnt X Usully in rndom mtrix tory on ts fmily it conductors rstrictd y cf X nd lt X tnd to infinity; for fmilis of llitic curvs tis is not rcticl sinc it is ncssry to v concrtly-givn st ovr ic to vrg Ktz nd Srn rdict tt for nturl fmily F t vrg dnsity sould stisfy DF; φ, X lim X W X F = φtwgtdt, r WG is t 1-lvl scling dnsity of ignvlus nr 1 for symmtry grou G W ill in gnrl distriution Suc rsult is clld t dnsity torm for t fmily F W v 1 if G = U sin 2πt 1 if G = S 2πt WGt = 1 + 1δ 2 0t if G = O sin 2πt 1 + if G = SOvn 2πt 1 + δ 0 t if G = SOodd, sin 2πt 2πt r δ 0 is t Dirc distriution [KS1] In rctic it is convnint to v t Fourir trnsforms of ts distriutions, ic rcord r δ 0 t if G = U δ 0 t 1ηt if G = S ŴGt = + δ 2 0t if G = O δ 0 t + 1ηt if G = SOvn δ 0 t 1 ηt if G = SOodd, 2 r 1 if t < 1, 1 ηt = if t = 1, 2 0 if y > 1 An imortnt ftur is tt t Fourir trnsforms of ŴOt, ŴSOvnt, nd ŴSOoddt ll gr for t < 1 ut r distinguisl for t > 1 Trfor, y t Plncrl Torm, to distinguis t dnsitis of ts tr symmtry tys on nds to ly tst functions φ os Fourir trnsforms r suortd outsid [ 1, 1] Inic, Luo, nd Srn rov tt t fmily H2N of rimitiv cus forms of igt 2 nd lvl N N squr-fr s symmtry ty O for tst functions φ rstrictd y su φ 2, 2 s [ILS] Ty lso rov t forms it root numr +1 v symmtry ty SOvn nd t forms it root numr 1 v symmtry ty SOodd in t sm rng Tis is rlvnt for our fmilis cus t fmily of ll llitic curvs forms sufmily of igt to rimitiv cus forms In rticulr, do not xct to dtct sttistics of t root numr of t fmily itout otining suort st 1, 1 Tis otntil cng in vior t 1 is not surrising in ligt of t roximt functionl qution, ic stts tt L1, E = 2πn 2πn n n 1/2 g + E n n 1/2 g U V n n

6 6 MATTHEW P YOUNG r g is tst function of crtin ind nd UV = N If t U > N 1+ε i sum t Fourir cofficints of lngt grtr tn N tn t first sum imlicitly cturs t root numr sinc t scond sum is tn smll rs if U N 1 ε tn t root numr xlicitly occurs in t scond sum Aftr dvloing t xlicit formul sll s t nlogy to tis drmtic sift in vior it similr sum xct ovr rims; going st suort 1, 1 is similr to ting U lrgr tn t conductor It is mor difficult to gin lrg suort for fmilis of llitic curvs tn for ll cus forms of igt to cus t forms coming from llitic curvs comos smll sufmily of t igt to cus forms Vry loosly sing, tr r roly round X 5/6 llitic curvs it conductors N X, rs tr r out X 2 igt to cus forms of lvls N X It is n on nd vry intrsting qustion to stimt t numr of llitic curvs tt v conductor N X Our figur of X 5/6 riss y simly counting t numr of ositiv intgrs nd suc tt = X Fouvry, Nir, nd Tnnum [FNT] v son tt t numr of non-isognous smi-stl llitic curvs it conductor N X is X 5/6 In t otr dirction, Du nd Kolsi [DK] uilding on or of rumr nd Silvrmn [S] v son tt t numr of llitic curvs it conductor N X is X 1+ε for ny ε > 0 Ny imrovmnt in t xonnt in tis ur ound ould vry intrsting cus it ould so tt for lmost ll lvls N tr is no llitic curvs it conductor N Not lso tt our lc of noldg in tis rgrd illustrts y cnnot vrg DE; φ ovr ll llitic curvs it N E X 3 Summry of Rsults 31 Min rsults Our min rsults r givn in tis sction Torm 31 Lt F t fmily of llitic curvs givn y t Wirstrss qutions E, : y 2 = x 3 + x + it nd ositiv intgrs Lt C 0 R + R + 2 nd st X E, = A,, r A = X 1/3, = X 1/2 X ositiv rl numr Tn for φ it su φ 7 9, 7 9 DF; φ, X [ φ φ0]w XF s X, Not tt W X F ŵ0, 0A s X so r ting out X 5/6 curvs from our fmily In t lngug of rndom mtrix tory tis torm sos tt t fmily of ll llitic curvs s symmtry ty O, insmuc s tis cn dtctd itout ving suort outsid 1, 1 Furtr, v Torm 32 Lt F t fmily of llitic curvs givn y t Wirstrss qutions E, : y 2 = x 3 + x + 2 it nd ositiv intgrs Lt C 0 R + R + nd st X E, = A,, r A = X 1/3, = X 1/4 X ositiv rl numr Tn for φ it su φ 23 48, DF; φ, X [ φ φ0]w XF s X, Hr r ting out X 7/12 curvs from our fmily Ts llitic curvs gnrlly v ositiv lgric rn if t oint 0, is torsion t Lutz-Ngll critrion imlis so instntly s tt t numr of curvs in tis fmily suc tt 0, is torsion is OX 1/3+ε, ic xlins t rsnc of t xtr φ0 contriution 2 For us R + = 0,

7 LOW-LYING ZEROS OF FAMILIES OF ELLIPTIC CURVES 7 Otining ts to torms it t sttd suort crucilly rquirs GRH for Diriclt L-functions S J Millr, in is doctorl tsis [Mil], s indndntly rovd dnsity torms for vrious fmilis of llitic curvs long it otr tings, ut mor rstrictd y t suort of φ rumr [], Ht-ron [H-], Micl [Mic], Silvrmn [Si3], nd otrs v rovd rsults on t vrg rn of crtin fmilis of llitic curvs tt cn no intrrtd to ssntilly dnsity torms Actully, in ordr to otin dnsity torm on must symtoticlly comut t vrg of log N ovr t fmily; in ordr to otin n ur ound on t vrg rn only trivil ur ound suc s N is rquird Not tt if on cn ly φ of lrg suort tn φ cn loclizd nr t origin, so t zros ρ E = 1 + iγ E r ld closr to t cntrl oint Trfor, on cllng for us s n to otin Torm 31 it t suort of φ s lrg s ossil ring suort 1, 1 ould of intrst, sinc it is t tt oint tt t Fourir trnsforms of t dnsitis of t grous O, SOvn, nd SOodd com distinguisl It rs tt t currnt tcnology is incl of roducing suc rsult vn ssuming GRH, ut v rsons to xrss t folloing Conjctur 33 Torm 31 olds for tst functions φ it no rstrictions on t suort W ill rovid som justifiction for tis conjctur in Sction 72 ftr rov Torms 31 nd 32 Tis conjctur grs it t follor conjctur tt t root numr is quidistriutd in t fmily of ll llitic curvs ut s [H] for n xtnsiv trtmnt of t vrition in sign of t root numr Millr lso rdicts t symmtry ty is O ut from n ntirly diffrnt dirction H considrs t 2-lvl dnsity of t fmily of ll llitic curvs nd uss conjcturs imlying quidistriution of root numrs to rdict symmtry ty O t 2-lvl dnsity cn distinguis tn t vrious ortogonl symmtry tys using tst functions it ritrrily smll suort ut it is ncssry to no t rcntg of curvs it givn root numr Our mtod is quit diffrnt nd rlis on sr stimts for t tr-vril crctr sum 26 for lrg vlus of P It is mystrious o t distriution of t root numr ssntilly controlld y t Möius function of t olynomil is cturd y suc crctr sum An sy consqunc of Torm 31 is t folloing Corollry 34 T fmily of ll llitic curvs ordrd y t discriminnt s in Torm 31 s vrg nlytic rn r 25/14 Conjctur 33 ing tru imlis tt r 1/2 T roof sos tt ny fmily it symmtry ty O nd suort u to ν s vrg rn oundd y Alying dnsity torm to otin n ur ound on t vrg 2 ν rn in tis fsion rquirs t Rimnn Hyotsis for ll L-functions in t fmily Using zro dnsity stimts Kolsi nd Micl [KM1], [KM2] otind n ur ound on t vrg ordr of vnising of ll igt 2 lvl q modulr L-functions, try rmoving t ssumtion of GRH for tir fmily T ound on t vrg rn is significntly lrgr tn tt ic is otind on GRH toug It ould intrsting to otin n unconditionl yt r ur ound for t vrg rn of t fmily of ll llitic curvs y mimicing tir mtods rumr [] sod tt t vrg rn r of ll llitic curvs stisfis r 23 nd Ht-ron [H-] rovd r 2, modulo f minor diffrncs in coic of tst functions nd siving of unlsnt curvs rumr s nd Ht-ron s rsults rquirs t GRH for llitic curv L-functions It is stndrd conjctur of rndom mtrix tory tt, onc t

8 8 MATTHEW P YOUNG symmtry ty G for t fmily s n idntifid, t dnsity torm sould old for tst function φ it ritrrily lrg suort Rndom mtrix tory rdicts tt Conjctur 33 imlis t quidistriution of root numrs vi t 2-lvl dnsity for instnc; it ould intrsting to s dirct numr-tortic rson for tis to so Using tt 25/14 = 178 < 2 nd t fmous torm tt sys tt if t nlytic rn is 1 tn t lgric rn nd nlytic rn r qul nd X is finit du in lrg rt to Kolyvgin [Ko] nd Gross-Zgir [GZ] otin Torm 35 Assum GRH Tn ositiv roortion of llitic curvs ordrd y igt v lgric rn qul to nlytic rn nd finit Tt-Sfrvic grou 32 Furtr rsults sids t rsults rcordd in t rvious sction, invstigt similr dnsity rsults for vrity of intrsting fmilis in Sctions 8-14 In rticulr invstigt numr of fmilis it roscrid torsion S [Ku], Tl 3 for t rmtriztions of t vrious torsion structurs Ts torsion fmilis v som surrisingly nic rortis For instnc, t fmily of curvs it torsion grou Z/4Z is rmtrizd y y 2 + xy y = x 3 x 2 it = As gnrl rul cnnot rov dnsity rsult for fmily r t dgr of t discriminnt is lrgr tn 3 Altoug t discriminnt s dgr 5 for tis fmily, it is sily trtd cus t irrducil fctors nd r linr In ddition t conductor is muc smllr tn t discriminnt N 2 rs 5 so tr r mny mor curvs in t fmily it conductor X tn on ould xct sd on t dgr of t discriminnt T otr torsion fmilis v similr crctristics tt m tir study xtrmly lsnt W study som oulr fmilis of curvs it comlx multiliction in Sction 13 Ts fmilis r rtr smll yt good rsults comrd it t sizs of t fmily r still otind cus of t siml ntur of t Fourir cofficints of t L-functions T rrir to otining ttr rsults it ts fmilis is lc of noldg on t oscilltion of ind of tistd cuic or qurtic, dnding on t fmily Guss sum In Sction 14 study tin squnc of qudrtic tists it ositiv rn Tis fmily is cllnging cus t conductor is ssntilly cuic olynomil Suc fmilis of qudrtic tists r studid y Ruin nd Silvrrg [RS] for xml 33 Structur of t r In Sction 4 st u t mcinry for roving dnsity rsult for fmily of llitic curvs W rov Torms 31 nd 32 in Sctions 5 nd 6; som of t mor tcnicl dtils r dfrrd to t Andics W rovid vidnc for Conjctur 33 in Sction 7 Tis conjctur nturlly follos from n ssumtion tt crtin crctr sum in tr vrils s squr-root cnclltion in c vril on of ic is summtion ovr rims W lnd crdnc to t conjctur y studying sum similr to t formntiond on ut r t summtion is xtndd to intgrs W otin strongr rsult it tis n sum; s Torm 72 In t rmining sctions rov dnsity rsults for t intrsting fmilis discussd in Sction 32 4 Gnrl Mtod of Proof In tis sction st u som mcinry to strmlin t roofs of our dnsity torms Suos r givn Wirstrss qution 1 not ncssrily miniml ic dfins n llitic curv E it conductor N nd L-function Ls, E To nlyz DE; φ ill

9 LOW-LYING ZEROS OF FAMILIES OF ELLIPTIC CURVES 9 mloy t xlicit formul for Ls, E, ic for cus forms of igt to nd lvl N ts t form s 425 in [ILS] log N 7 DE; φ = φ0 + 1 log log φ0 P E; φ + O 2 r P E; φ = log 2 log λ E φ >3 Hr is t discriminnt of t curv dfind y 1 gin, is not ssumd miniml, X is scling rmtr ny numr 2 t our disosl, nd 8 λ E = x 3 + x + x mod if t Wirstrss qution 1 dfining E is ut into t form 2 Actully, t sum P E; φ in [ILS] is rstrictd y N nd ssums λ E is t cofficint 5 of s in t Diriclt sris xnsion of Ls, E W clim t modifictions in P E; φ r cctl cus t discrncy in t formul 7 is sord y t rror trm To rov tis first not tt if nd 2 is in glol miniml Wirstrss form tn λ E is xctly t cofficint 5 Furtr, t crctr sum dfining t quntity λ E is lft uncngd y cng of vrils lcing 2 in glol miniml Wirstrss form Trfor t trms gr for On t otr nd, t trms r r sord y t rror trm It is ortil to not tt t rror trm in 7 is drivd y using t Rimnn Hyotsis for t symmtric-squr L-function Ls, sym 2 f to ndl trms of t form λ E 2 T formul 7 olds for individul E In rctic on could limint t us of t GRH y vrging ovr fmilis; v usd t Rimnn Hyotsis for simlicity nd rvity sinc r ssuming it for otr rsons nyys Nxt sum ovr t fmily W comut DF; φ, X = φ0 E F r log N E XE φ0w X F PF; φ, X + O PF; φ, X = E F P E; φ X E WX F,, nd W X F, = E F X E log log In gnrl log log ill log so tis ill tru rror trm Wit ll of our fmilis ill t X nd X so tt log N E 9 XE W X F s X E F It is gnrlly igly nontrivil to rov suc n symtotic olds; oftn it mounts to ving control on t squr divisors of olynomil of ig dgr Tis is significnt rrir to roducing dnsity torms it fmilis of ig rn W cll 9 t conductor condition for t fmily F T id in most css is to roximt N E it numrs R E ic v t sm rim divisors s N E ut r sir to comut oftn R E =, t discriminnt of t llitic

10 10 MATTHEW P YOUNG curv Tn X nd X ill cosn so tt X is roximtly R E for E in t suort of X For n xml, it t fmily y 2 = x 3 + x + givn in Torm 31 v X scld so tt X 1/3, X 1/2, X 5/6, nd t R E = T folloing lmm ill strmlin mny of our rgumnts to so 9 olds Lmm 41 Lt F = {E d } fmily of llitic curvs it nottion s in Sction 2 Lt d t discriminnt of t curv E d Suos tt tr xists n intgr olynomil Rd dividing d suc tt c irrducil fctor of d divids Rd Furtr suos 10 X E log α 1 + X E log W X F E F E F α R E α>0 R E 2 N E >3 uniformly in X, r v dfind R E = Rd for E = E d Suos Rd X for ll f in t suort of X Tn v log N E WX F XE = W X F + O E F Rmr ot sums in 10 v n intrrttion For t first sum to smll tr must not mny lrg squr divisors of R E i R E is not too lrg For t scond sum to smll it must rr for rim to divid R E to lor ordr tn N E i R E is not too smll In most lictions N E R E so t scond sum ill void In gnrl cn ndl t first sum s long s ll of t irrducil fctors of Rd r of dgr 3 or lss Proof Suos Rd is s ov Tn W X F E F log N E XE = E F logx/r E XE + E F logr E /N E XE Sinc r ssuming R E X t first sum is trivilly W X F 1 T scond sum is 1 X E log α β, E F α R E β N E y using t dditivity of t logritm to srt t rim fctors of R E nd N E ts rims v noting to do it t xlicit formul of cours First considr t trms it β > 0 y ting t trms it α > β nd α < β srtly it s clr t contriution is 1 E F α R E α>0 X E log α E F R E 2 N E >3 WX F X E log + O t rror trm coming from = 2 or 3 nd α < β in ic cs β my lrgr tn 2 No considr tos trms it β = 0 nd nc α > 0 Sinc ut N tis imlis t qution 1 dfind y d is not miniml t nd trfor α 12, so my sor ts trms into t first sum ic consists of α R E, α > 0 For ny individul fmily ill nd to stimt PF; φ, X using d oc tcniqus licl to t fmily In its vlution ill oftn m us of t folloing idntity Poisson Summtion mod l,

11 LOW-LYING ZEROS OF FAMILIES OF ELLIPTIC CURVES 11 Proosition 42 Lt Scrtz-clss function, D ositiv rl numr, nd n intgr Tn t folloing olds d 11 = D D ŵ D l l l d Z d mod l Z 5 Proof of Torm 31 y t discussion in Sction 4, to rov Torm 31 nd to so PF; φ, X W XF A nd to so t conductor condition 9 olds 51 T Conductor Condition In tis sction so 9 olds Lmm 51 Lt F, A,, nd s in Torm 31 Tn v { } log N E 1 XE = 1 + O W X F E F Proof W ly Lmm 41 W t t olynomil Rd for d =, to givn y Rd = Not Rd = d Rcll tt A = X 1/3 nd = X 1/2 so tt Rd X sinc A nd T sum X E log E F R E 2 N E is mty sinc N T otr trm is X E log α 1 = E F α R E α>0 α α>0 A, log α 1 y dfinition First suos > 3 Intrcng t ordr of summtion nd for c rim dfin γ y γ W first considr t trms r α < 3γ For ts css α 2 so α is ncssrily vn W mloy t cng of vrils = γ, = α/2 nd otin oftn us rims to indict tt summtion is crrid out it crtin corimlity conditions in lc ic sould rnt from contxt; in t nxt formul t rstriction is, =, = 1 α X α>0 α X α>0 α/3<γ log A α X α/3<γ log A α>0 log A + γ A, α/2 log α A 1 + log α 1 γ α/2 A log A + + [α/3]+1 α/2 A log α 1 α/2+[α/3]+1

12 12 MATTHEW P YOUNG Tis sum is cus X log A + A A X 5/6, α X α>0 α Z α>0 log α 1 Z, nd α 0 γ<α/3 α 3γ α>r log α 1 1 α/r No considr t trms it α 3γ In tis cs 3γ 2 so γ is ncssrily vn W mloy t cng of vrils γ, 3γ/2 nd otin γ A, 3γ/2 log α 1 W slit t summtion ovr into rogrssions mod α 3γ Sinc, =, = 1 gt tt for c t numr of solutions in mod to t congrunc mod α 3γ is oundd y 2 Trfor otin 1 + A 1 + 3γ/2 log α 1, γ α X α>0 α X α>0 0 γ<α/3 α + A + + α 3 2 [α/3] α 3γ A α 1 2 [α/3] log α 1 ic is oundd y X 5/6 y t sm ty of rsoning usd for t cs α > 3γ cc α = 2 nd α 3 srtly T rims 2 nd 3 r ndld in t sm y s ov it minor cngs For = 3 my ssum α 4 y trivilly stimting t trms it α 3 T stimtion for α < 3γ is t sm s for ftr t cng of vrils α α + 3 T stimtion for α 3γ is s for ftr t cng of vrils γ γ + 1 For = 2 my ssum α 6 Aftr cnclling 2 6 in t discriminnt congrunc r lft in vry similr cs to = 3 W omit t tdious yt lmntry dtils 52 T Cntrl Estimt W ill v rovd Torm 31 onc v rovd Lmm 52 St A = X 1/3 nd = X 1/2 Tn P E; φ rovidd su φ 7 9, 7 9 Tis lmm is t rt of t mttr Proof W clcult P E; φ A, = >3 A, 2 log φ log X5/6 λ, A, Aly Poisson summtion mod in t summtion ovr nd nd otin λ, A, = A α + β λ 2 α,β ŵ α mod β mod A,

13 LOW-LYING ZEROS OF FAMILIES OF ELLIPTIC CURVES 13 T summtion ovr α nd β is vlutd in Sction 53 t vlution is comltly strigtforrd Using Lmm 56 continu, otining 12 A 2 log log 3 2 A ε φ ŵ 3/2, >3 W rmr t tis oint tt if stimt tis sum trivilly gt ound of t ordr summing u to P, sy A log A + P 1/2 + P 3/2 1, 3/2 A P ic is O A 1 n P X 5/9 rumr ssntilly otind tis rsult [] To gt lrgr suort nd to rov tr is quit lot of cnclltion in t tr vril crctr sum 12 y xloiting som cnclltion in tis sum Ht-ron [H-] s imrovd rumr s rsult to t suort rng 2/3, 2/3 Not tt ny imrovmnt on Ht-ron s rsult sos tt t vrg rn is strictly lss tn to T first st is to limint t vrition in ε To do so sum srtly ovr t rogrssions 1 mod 4 nd 3 mod 4 Effctivly, it suffics to rlc ε y ψ 4, Diriclt crctr mod 4 No r u t summtion in 12 into dydic sgmnts using smoot rtition of unity It suffics to considr sums of t form 13 H <2H K <2K P <2P log 3/2 ψ φ log ŵ A, g,,, r g is smoot comctly suortd function rising from t rtition of unity W ssum tt t rstrictions on,, nd r rdundnt, folloing from t suort of g It suffics to so tt vry sum of ty 13 is X ε, it t imlid constnt dnding only on, g, g, tc In ddition v to ccount for t contriution to 12 of = 0, ut tis contriution is ngligil y trivil stimtions Lt SH, K, P t sum givn y 13 in t nottion surss t dndnc on t tst functions Using t ound ŵx, y M 1 + x M 1 + y M my ssum tt H ε P/A 1+ε nd K ε P/ 1+ε It ill ncssry to us diffrnt tcniqus of stimtion in diffrnt rngs As first st, v t ound H 3/4 P + HP 3/4 + H 1/4 P 5/4 HKP ε P <2P H <2H T roof is stndrd y Wyl s mtod If ly tis to 13 ftr using rtil summtion to srt t vrils gt t ound 15 SH, K, P H 3/4 KP 1/2 + HKP 3/4 + H 1/4 KP 1/4 X ε, ic is usful in som rngs To covr t rngs r 15 is insufficint continu it 13 W rfr to sum ovr rltivly rim nd so dfin d = 3, 2 nd lt d 0 to t lst ositiv intgr suc tt d d 3 0 Sinc d 0 my st = d 0 0 T condition

14 14 MATTHEW P YOUNG 3, 2 = d is quivlnt to 0, 2 /d = 1 nd d 3 0/d, 2 /d = 1 Tn 3 2 = 3 0d 3 0/d 2 /d d 2 0 0, 2 /d=1 d 3 0 /d,2 /d=1 T oint is tt vryting in t xonntil is corim it 2 /d sids ossily 2 /d No my mloy t folloing lmntry rcirocity formul ū 16 v + v u 1 mod 1, uv r u, v, ū, v r intgrs suc tt u, v = 1, uū 1 mod v, nd v v 1 mod u Tis rcirocity l s lso mloyd y [H-] in is or on tis rolm In our liction u = 2 /d nd v = No SH, K, P s trnsformd into 3 17 ψ 4 0 d 3 0/d 2 /d r K <2K P <2P d 2 H d 0 0 <2 H d 0 0, 2 d =1 d3 0 d, 2 d =1 0 d 0 A U 0, d 0,, = g 0 d 0,, ŵ, 3 0 d U 0, d 0,,, 2 log 3/2 φ log To srt t vrils us t folloing xnsion of dditiv crctrs into multilictiv crctrs vi Guss sums d 3 0/d 1 = τχχ 3 2 /d φ 2 /d 0d 3 0/d, χ mod 2 /d vlid cus of t corimlity conditions, otining t idntity 19 SH, K, P = 1 τχχd 3 φ 2 /d 0/dQd,, χ K <2K d 2 χ mod 2 /d r Qd,, χ = P <2P ψ 4 χ H d 0 <2 H 0 d 0 χ d 3 0 U 2 0, d 0,, Our gol is to gt good ound for Qd,, χ nd stimt t rst trivilly W is to ly Lmm 57 to Qd,, χ To do so must dfin numr of rmtrs nd cc tt t conditions of Lmm 57 r stisfid W st v 3/2 ud0 A 20 F u, v = gud0,, vŵ, log v φ P v v T crctr χu in Lmm 57 is rlcd y χ 3 u it modulus l 1 = 2 d crctr ψv in Lmm 57 is χvψ 4 v/v it modulus l 2 = lcm4,, 2 is t conductor of v, q = 2, U = min d 3 0 { H d 0, P d 0 A }, nd V = P W v d, il t, r

15 LOW-LYING ZEROS OF FAMILIES OF ELLIPTIC CURVES 15 Clim T tst function F u, v dfind y 20 stisfis t conditions of Lmm 57 it U = min {H/d 0, P/d 0 A} nd V = P Proof of clim T roof is strigtforrd clcultion T only sligtly torny issu is tt diffrntition of ŵ it rsct to v introducs fctor ud 0 A/v α Tis cn sord for M lrg noug y t ound of 1 + u/v/d 0 A M on ny rtil drivtiv of ŵ To ly Lmm 57 to Qd,, χ must first xtnd t summtion ovr to rim ors instd of just rims It s sy to s y trivil stimtions tt cn xtnd t summtion to rim ors itout cnging t ound for Qd,, χ Trfor v Corollry 53 If χψ 4 / nd χ 3 r nonrincil tn 1/2 H Qd,, χ P H3/2 X ε d 0 P 1/2 If χψ 4 / is rincil ut not χ 3 tn los fctor P 1/2 If χ 3 is rincil ut not χψ 4 / tn los fctor H/d 0 1/2 Hving otind t ound for Qd,, χ ly it to SH, K, P W v four css ccording to ic crctrs r rincil Lt S = S 1 +S 2 +S 3 +S 4, r S 1 corrsonds to t trms r ot crctrs r nonrincil, S 2 corrsonds to t trms r χ 3 is nonrincil ut χψ 4 / is not, S 3 corrsonds to t trms r ot crctrs r rincil, nd S 4 corrsonds to t rmining trms r χ = ψ 4 / is nonrincil Cs 1 To ound t sum S 1 ly Corollry 53 to 19, otining t ound S 1 K <2K 1 φ 2 /d d 2 χ mod 2 /d H 1/2 P H3/2 P 1/2 K P 1 H d 0 1/2 1 + H3/2 P 1/2 X ε K <2K H 1/2 P 1 K 2 + H 3/2 KP 1/2 X ε d 2 1 dd 0 1/2 X ε τχ W rquir tis ound to X ε Using H P/A 1+ε nd K P/ 1+ε sos t rquirmnt is P X 7/9 ε T xistnc of 7/9 r xiits t limit of our mtod T css r on or ot of t crctrs r rincil r tdious to crry out ut do not os significnt rrir to otining lrgr suort It rmins to ound t sum 19 n on or ot of t crctrs r rincil T loss of cnclltion in ts css ill md u for y t rrity of rincil crctrs First, χ 3 is trivil for ε crctrs χ T crctr χψ 4 / is trivil for only t crctr χ = ψ 4 / χ 0 Trfor t only y ot crctrs r trivil is if χ is trivil nd / ψ 4 is trivil ic limits to squr Cs 2 In cs χ 3 is trivil ut not χψ 4 / los H/d 0 1/2 from t loss of cnclltion ut sv φ 2 /d from t rrity of suc crctrs, ic givs t ound S 2 HP H3/2 P 1/2 K X ε 1 d 2 HP H3/2 K 1/3 X ε P 1/2 K d d 0

16 16 MATTHEW P YOUNG using only t ovious ound d 1 0 d 1/3 from d d 3 0 Using H P/A 1+ε nd 1 K P/ 1+ε nd rquiring tis X ε mns must rquir P X 5/6 ε Cs 3 In cs ot crctrs r trivil no is squr nd tt τχ 1 Using 19 nd ounding Qd,, χ trivilly y P H/d 0 sily gt t ound of S 3 HP 1/2 X ε = K <2K 2 d 2 d d 0 = HP 1/2 X ε K 1/2 l<2k 1/2 l 4 Notic tt if d tn / 0 d/d 0 loo t c rim srtly Trfor v S 3 HP 1/2 X ε 2 HP 1/2 K 1/2 X ε, K 1/2 l<2k 1/2 ic is X ε n K P X 2/3+ε To ndl K P X 2/3+ε simly ly Wyl s ound 14 n =, K < 2K W tus otin t ound S 3 H 3/4 P 1/2 + HP 3/4 + H 1/4 P 1/4 K 1/2 X ε, ic is X ε using H P/A 1+ε nd K P X 2/3+ε n P X 7/9 ε W xct tt us of currnt tcnology ould llo us to t P lrgr tn X 7/9 r ut sinc r rstrictd to 7/9 lsr do not ursu suc rsult Cs 4 In t lst cs r χψ 4 / is trivil ut χ 3 is not los P 1/2 ut sv φ 2 /d W gt furtr sving y noticing tt ψ 4 / s conductor qul to t squr-fr rt of u to fctor 2 or 4 Trfor 2 d 1 nd nc d 2 1 W gt t ound S 4 H 1/2 P 1/2 1 + H3/2 P 1/2 K X ε K <2K 1 l d 2 1 d l 4 1/2 d As for, d/d 0 1/2 is incrsing it rsct to divisiility so gt t ound S 4 H 1/2 P 1/2 1 + H3/2 X ε 1 P 1/2 K K <2K 2 H 1/2 P 1/2 X ε 1 + H3/2 P 1/2 K For tis ound to X ε it is ncssry nd sufficint tt 21 H 2 K 1 P X ε W us diffrnt mtod to stimt Qd,, χ ic ors ll for smll In our currnt cs v Qd,, χ = 3 0 d 3 0 χ U 0, d 0,, P <2P H/d 0 0 <2H/d 0 W xloit cnclltion in 0 c 0 3 0d 3 0/ 2 it ritrry comlx cofficints c n Prcisly, us t folloing d 0 d d 0

17 LOW-LYING ZEROS OF FAMILIES OF ELLIPTIC CURVES 17 Lmm 54 Lt c n comlx numrs stisfying c n 1 nd lt RN, P, d 0, = n 3 d 3 0 c 2 n Tn P <2P N n<2n RN, P, d 0, N 1/2 P + N 1/4 P 5/4 1/2 d 3/4 0 Tis is scil cs of mor gnrl rsult ic is sttd nd rovd in Andix ; scificlly, ly Lmm 1 it fx = x 3, gx = x 1, nd Y = N 3 d 3 0P 1 2 Alying Lmm 54 to Qd,, χ vi rtil summtion givs ctully on must srt t vrils 0 nd in U 0 d 0,, for lying Lmm 54, ic cn don in ny stndrd y it no cost Corollry 55 If χψ 4 / is rincil nd χ 3 is non-rincil tn 1/2 H Qd,, χ P 1/2 + P 1/4 H 1/4 1/2 d 1 0 X ε Alying Corollry 55 to S 4 givs t ound S 4 K <2K d 0 d 2 d 1/2 1 H 1/2 d 1/2 0 P 1/2 + H1/4 1/2 P 1/4 d 0 Using t sm tcniqus s for to stimt t sum ovr d givs t ound S 4 H1/2 K 1/2 X ε + H1/4 K 1/2 X ε P 1/2 P 1/4 Using only H P/A 1+ε nd K P/ 1+ε sos t first trm is X ε n P X 5/6 ε For t scond trm in t ov ound to sufficint must v K P 1/2 H 1/2 X ε Assuming 21 dos not old i H 2 K 1 P X ε nd using H P/A 1+ε s tt * olds n P X 5/6 ε Hving considrd ll ossil css t roof is comlt 53 A Comlt Crctr Sum In tis sction vlut crctr sum ic ros in our vrging St λ, = x 3 + x + x mod Not tt tis is λ E for t llitic curv givn y t qution y 2 = x 3 + x + Nxt, for ny intgrs nd dfin T, ; = α + β λ α,β α mod β mod Lmm 56 Lt > 2 rim nd dfind y 1 mod if, = 1 nd 0 = 0 Tn v T, ; = ε 3/2 r ε is t sign of t Guss sum 3 2, X ε

18 18 MATTHEW P YOUNG Proof y dfinition, T, ; = x mod α mod β mod x 3 + αx + β T cng of vrils β β x 3 αx givs x 3 α x T, ; = x mod α mod = ε 3/2 3 2 α + β β mod β β 54 An Estimt for n Incomlt Crctr Sum in To Vrils In tis sction stlis gnrl stimt for crctr sum in to vrils ic ill lid in our or Prcisly v Lmm 57 Lt F u, v smoot function stisfying 22 F α 1,α 2 u, vu α 1 v α 2 Cα 1, α u v 2 U V for ny α 1, α 2 0, t surscrit on F dnoting rtil diffrntition Tn u c χuψvλv F u, v ε U V U c 1/2 l 1 l 2 UV ε vq V q u v r χ nd ψ r nonrincil Diriclt crctrs to t moduli l 1 nd l 2, rsctivly, Λ is t von Mngoldt function, q is nonzro rtionl numr, c is ositiv intgr, nd ε is ny ositiv numr, t imlid constnt dnding only on ε, c, nd t numrs Cα 1, α 2 In cs χ is rincil ut not ψ t sm ound olds ut it U 1/2 lost In cs ψ is rincil ut not χ t sm ound olds ut it V 1/2 lost Rcll tt r ssuming t Rimnn Hyotsis for ll Diriclt L-functions; tis lmm, of cours, rlis vily on t GRH Proof First ssum χ nd ψ r nonrincil To gin, y Mllin invrsion, u c χuψvλv F u, v vq u v 2 1 = Ls 1, χ L 2πi L s 2, ψhs 1, s 2 ds 1 ds 2, r Hs 1, s 2 = Our gol is to otin t ound Hs 1, s 2 1/2+ε 1/2+ε 0 0 u c vq F u, vu s 1 v s 2 dudv uv U σ 1 V σ 2 s 1 /c + s 2 s 1 /c + s s 2 1+ε 1 + U c for R s 1 = σ 1, R s 2 = σ 2, σ j = 1/2 + ε or 1 + ε, nd us t GRH for t ound Ls 1, χ L s L 2, ψ s 1 s 2 l 1 l 2 ε/2 Putting ts to stimts togtr ill rov t dsird rsult T full dtils of t roof of t ound for Hs 1, s 2 r contind in Andix A V q 1/2

19 LOW-LYING ZEROS OF FAMILIES OF ELLIPTIC CURVES 19 In t css r on of t crctrs is rincil intgrt ovr t lin Rs j = 1+ε instd of R s j = 1/2 + ε for t rorit vril T ounds on t L-functions r t sm Tis ccounts for t squr-root loss T roof is no solly dndnt on t dtils of Andix A 55 Proof of Corollry 34 W follo n sy clcultion in [ILS] T 0 nd st 1 m X = X E W X F W dfin t vrg nlytic rn r y r = lim X m=1 E F ord s=1 Ls,f=m m m X, if t limit xists Our mtod rovids ound on t limsu of t ov quntity y ting tst function φ suc tt φx 0, φ0 = 1, nd suort of φ contind in [ ν, ν] driv y Torm 31 nd t Plncrl torm tt m m X g + ox, r m=1 g = φyŵoydy Tis uss GRH for ll llitic curv L-functions so tt y ositivity cn dro ll zros not t t cntrl oint y ting t Fourir ir 2 sinπνt 1 φt =, φy = 1 y πνt ν ν otin g = Ting ν lss tn 7/9 nd ltting X sos r 25/14 + ε for ν 2 ny ε > 0 An idnticl clcultion ors for ν < 1 + δ nd comlts t roof 56 A Not on Minimlity In tis sction invstigt t fmily givn in Torm 31 it t rstriction tt tr r no rims q suc tt ot q 4 nd q 6 Tis condition nsurs tt t qution E : y 2 = x 3 + x + is miniml for ll > 3 T rsult is t sm s tt sttd in Torm 31; omit t rcis sttmnt for rvity T conductor condition i t nlogu of Lmm 51 follos dirctly from Lmm 51 T sum ovr t Fourir cofficints is rdly comlictd y t divisiility rstrictions W sily v P E; φ q 4 q 6 = >3 A, = >3 2 log φ log 2 log φ log µd d q 4 q 6 λ, A, d 4 λ d 4,d 6 A, d6

20 20 MATTHEW P YOUNG = >3 2 log φ log d d,=1 µd Comlting t sum ovr nd mod lds to t sum r A >3 d d,=1 µd α β d 4 λ d 4,d 6 A, d6 λ αd 4,βd 6 α + β Φ,, = 2 log φ log A ŵ d 4, d 6 + oa Φ,, + oa, Alying t cng of vrils α αd 4, β βd 6 givs A 1 µd T d 4, d 6 ; Φ,, + oa, 2 d 10 >3 >3 d d,=1 r T is givn y Lmm 56 Alying Lmm 56 otin t sum A 2 log log µd 3 2 ε φ ŵ 3/2 d 10 d d,=1 A d 4, d 6 No simly rmov t rstriction d, = 1 nd mov t summtion ovr d to t outsid W r c to t rolm of stimting 12 xct it sligtly smllr A nd W cn simly us t sm ounds s for; t rsnc of d contriuts t orst oundd ors of Sinc sod tt c sum of ty 13 is X ε cn us t sm roof s for Lmm 52 It is ossil to imos minimlity rstrictions on t Wirstrss qutions for otr fmilis Sinc no intrsting fturs ris v limitd our discussions to tis rif not 6 Proof of t Dnsity Torm for t Min Rn On Fmily In tis sction rov Torm 32 T roof ill rocd muc t sm s Torm 31 ut t rgumnts ill mor intrict 61 T Conductor Condition To so 9 olds ill ly Lmm 41 W t Rd = Tn Rd = d X All nd to so is Proosition 61 r = α log α 1 A, X7/12 Proof W my ssum α = 2 cus similr rgumnt to tt usd in t roof of Lmm 51 it t rols of nd sitcd ill rovid t ncssry stimtion for α 3 W my furtr ssum > 3 nd tt, =, = 1 W no m to srt rgumnts to ndl rltivly smll nd rltivly lrg For t formr, v

21 Lmm 62 For P = X 1/3 LOW-LYING ZEROS OF FAMILIES OF ELLIPTIC CURVES 21 3< P,=1,=1 0 mod 2 t imlid constnt dnding only on For t lttr v Lmm 63 For P = X 11/36+ε P 2 t imlid constnt dnding only on ε nd log A, X7/12, log A, X7/12, Sinc 11/36 < 1/3 ts to lmms ill llo us to t ε smll nougt to clos t g nd comlt t roof of Proosition 61 Proof of Lmm 62 y ring t summtion ovr into congrunc clsssmod 2 sily otin t ound 1 + A log 2 A 1 + P, A P ic is sufficint rovidd P A = X 1/3, s climd Proof of Lmm 63 y mjorizing y smoot non-ngtiv function it sligtly lrgr suort my ssum 0 T conditions on imly tt if rit s d 2 l it l squrfr tn d P Trfor v log A, A, P d P 2 =d 2 l l squrfr l XP 2 l squrfr =d 2 l A, No dfin s =, l Sinc s is squrfr must v s Dfin l = sl, = s, nd = s Tn t condition on t discriminnt is 4s s 3 4 = d/4 2 l Sinc l is squrfr v s, l = 1, ic imlis s 4 1 d Tus t condition is rducd to s 4 = d/4s 2 l T imortnt ftur is tt, l = s, l = 1 nd, l = 1 or 2 St l = 2, l 3, l l, = 3,, = 2,, nd s = 2, ss Tn t discriminnt condition coms 4 3, 3 2, l 3, l , l 2, s2, 4 s 4 = d/4s 2 l 2, l y ting ll t ossil comintions of vlus for 2, l, 3, l, 3,,, r lft it finitly mny qutions of t form c c 2 s 4 = u 2 l it t condition c 1 c 2 s, l = 1

22 22 MATTHEW P YOUNG Trfor r rducd to stimting sums of t ty s L l L/s l squrfr c 1 3 +c 2 s 4 =u 2 l s A, s r L XP 2 nd t rims indict t summtion is rstrictd y c 1 c 2 s, l = 1 For squrfr l st s Ss, l = A, s c 1 3 +c 2 s 4 =u 2 l Lt Q st of rims q of siz q Q som fixd roortion, it Q t our disosl it ill cosn to X η for η smll W dtct t condition tt n intgr m is squr m y vluting t Lgndr symol for q Q ind of mlifiction tcniqu W q otin Ss, l = s A, s = Simlifying it coms 1 Q 2 1 Q 2 c 1 3 +c 2 s 4 =cl c= + 1 Q 2 c 1 3 +c 2 s 4 =cl c= c 1 3 +c 2 s 4 =cl c= q Q q Q c 1 3 +c 2 s 4 0 mod l + 1 Q 2 c 1 3 +c 2 s 4 =cl c= c q, 2 s A, s s {q Q : q c} 2 A, s c1 3 + c 2 s 4 l q 2 s A, s s {q Q : q cl} 2 A, s, sinc in t first sum v rlxd t condition tt c is squr For t scond sum, notic tt sinc q Q nd cl X {q Q : q cl} log Q Trfor t scond sum is / log Q 2 Q 2 Ss, l If lt S s, l t first sum ov, tn 1 Ss, l 1 log2 X Q 2 log 2 S s, l Q Our coics of Q nd Q ill so Ss, l S s, l so considr S s, l W xnd t summtion ovr q nd otin S s, l = 1 c1 3 + c 2 s 4 l s Q 2 q 1 q 2 A, s q 1 Q q 2 Q c 1 3 +c 2 s 4 0 mod l

23 LOW-LYING ZEROS OF FAMILIES OF ELLIPTIC CURVES 23 Lt S 2 = S 2 s, l, q 1, q 2 t ov summtion ovr nd St r = q 1 q 2 nd ly Poisson summtion in nd mod lr W gt S 2 = A c1 u 3 + c 2 sv 4 l u + v A ŵ l 2 r 2 s 2 r rl lrs, lrs u mod lr v mod lr c 1 u 3 +c 2 sv 4 0 mod l Using t Cins rmindr torm rit u = u 1 r + u 2 l it u 1 givn mod l nd u 2 givn mod r, nd similrly for v Tn gt S 2 = A A Ul, r, sŵ l 2 r 2 s 2 lrs, u1 + v 1, lrs l r Ul, r, s = u 2 mod r v 2 mod r u 1 v1 c 1 u 3 1 +c 2rsv1 4 0 mod l c1 u c 2 lsv 4 2 Aly t cng of vrils u 1 v 1 u 1 nd otin S 2 = A A Ul, r, sŵ l 2 r 2 s 2 lrs, lrs = A l 2 r 2 s 2 A Ul, r, sŵ lrs, lrs x mod r u2 + v 2 u 1 v1 c 1 u 3 1 +c 2rsv 1 0 mod l u 1 r v1 u 1 + l c1 c 2 rsu 3 1u 1 + sinc c 1 c 2 rs, l = 1 It s sy to s tt t xonntil sum ovr l fctors into xonntil sums of t form α x 3 x +, r l = nd α, = 1 A corollry of t Rimnn Hyotsis for curvs cf [Sc], Corollry 2F, g imlis tt t summtion ovr x is O 1/2 t imlid constnt solut, unlss ot nd r zro mod, in ic cs t sum is xctly 1 Trfor t summtion ovr u 1 is l 1/2,, l 1/2 τl, unlss = = 0, in ic cs t ound is l Clrly Ul, r, s r 3/2 + r 2 δ q1 q 2 so gt t ound S 2 A l 2 r r r 2 δ q1 q 2 l 1 2 +ε,, l A l lrs lrs Trfor, 0,0 A l 2 r 2 s r r 2 δ q1 q 2 l 12 +ε l2 r 2 s 2 A = r r 2 δ q1 q 2 + l 1 +ε lrs 2 l 1 2 +ε + l 1 2 +ε A rs + A lr 2 s 2 l + l S s, l Q Q l 1/2+ε + l 1/2+ε A Q sq + A 2 ls 2 Q 4,

24 24 MATTHEW P YOUNG On summtion ovr l Ls 1 nd s L gt S s, l 1 + Q Q s L l Ls 1 Q 3 L 3/2+ε + AQL 1/2+ε + A log L Q W t Q = X ε nd Q Q ic imlis Ss, l S s, l T ncssry ound on tis sum is X 7/12 ε, ic mns t rquirmnt on L is L X 7/18 ε Sinc L X/P 2 t rquirmnt on P is P X 11/36+ε No t roof of Lmm 63 is comlt 62 Estimting t Sum of t Fourir Cofficints In tis sction vlut PF; φ, X for t fmily givn in Torm 32 Tis is ccomlisd it Lmm 64 St A = X 1/3 nd = X 1/4 Tn P E; φ A, X 7/12 = φ0w X F + O rovidd su φ 23 48, W mimic t roof of Lmm 52 rgumnts Proof On Poisson summtion, r T dtils r similr so oftn condns our P E; φ A, = A 2 log 3 φ log T, ; ŵ >3 T, ; = α mod β mod α + β λ α,β 2 A, T comlt sum T cn vlutd xlicitly; t clcultion is md it Lmm 66 T δδ 2 trm in T givs t xtr φ0w X F y t Prim Numr Torm T trm is ngligil vi trivil stimtions W r lft it stimting t sum A 2 log ε 3/2 φ log A ŵ, >3 Estimting tis sum trivilly otins our rsult for suort u to 7/18 To gt lrgr suort nd to so tr is cnclltion in t sum W stimt it in xctly t sm y did in t roof of Lmm 52 First rlc ε y crctr ψ 4 mod 4 Tn r u t sum into dydic sgmnts using rtitions of unity It suffics to considr sums of t ty 23 ψ 4 H <2H K <2K P <2P log φ 3/2 log A ŵ,, g,,, r g is function rising from t rtitions of unity T contriution from = 0 is ngligil y trivil stimtions Lt SH, K, P t sum givn y 23 It suffics to so SH, K, P X ε

25 LOW-LYING ZEROS OF FAMILIES OF ELLIPTIC CURVES 25 Using t ound ŵx, y 1 + x M 1 + y M my ssum tt H P/A 1+ε nd K P/ 1+ε In ordr to sum ovr corim intgrs st d = 4, nd dfin d 0 to t lst ositiv intgr suc tt d d 4 0 Sinc d 0 my st = d 0 0 T condition 4, = d is quivlnt to t to conditions d 4 0/d, /d = 1 nd 0, /d = 1 Trfor SH, K, P is r H <2H d K P <2P d 0 <2 K 0 d d 4 0 d, =1 d 0,2 6 3 =1 d ψ 4 d 4 0/d /d U 0, d,,, A U 0, d,, = g, 0 d 0, ŵ, d 0 0 log log φ 3/2 W ly t lmntry rcirocity formul nd t xnsion of t xonntil into multilictiv crctrs s in t roof of Lmm 52 nd otin 24 SH, K, P = r Qd,, χ = P <2P H <2H 1 φ2 6 3 /d d χ mod /d ψ 4 χ K/d 0 0 <2K/d 0 τχ χd 4 0/dQd,, χ d χ U , d,, Sinc t rgumnts r no xtrmly similr to tos usd in t roof of Lmm 52 ill rif W ly Lmm 57 to Qd,, χ nd otin Lmm 65 If χψ 4 / nd χ 4 r nonrincil tn 1/2 Qd,, χ P 1 K 1/2 d 1/ K4 X ε P H 3 If χ 4 is rincil ut χψ 4 / is not rincil los fctor K 1/2 d 1/2 0 Lt S = S 1 + S 2 + S 3, r S 1 corrsonds to t trms r ot crctrs r nonrincil, S 2 corrsonds to t trms r χ 4 is rincil ut χψ 4 / is not rincil, nd S 3 corrsonds to t rmining trms r χψ 4 / is rincil nd ncssrily χ 4 is rincil Cs 1 W ly Lmm 65 to S 1 nd otin t ound 1/2 S 1 P 1 K 1/2 1 + K4 X ε 3/2 dd P H 3 0 1/2 H <2H d P 1 H 5/2 K 1 1/2 K 2 + X ε, P 1/2 H 3/2 ic is X ε n P X 23/48 ε

26 26 MATTHEW P YOUNG Cs 2 Alying Lmm 65 to S 2 givs S 2 1/2 P 1 K 1 + K4 X ε P H 3 P 1 K 2 K 1 + X ε P 1/2 H 3/2 H <2H 3/2 d 1/2 d 1 0 d Using only K P/ 1+ε nd H 1 sos tis is X ε n P X 1/2 ε Cs 3 No considr t cs r χψ 4 / nd nc χ 4 r rincil Sinc ψ 4 / s conductor qul to t squr fr rt of u to fctor of 2 or 4 gt xtr sving from t rstriction d Tus χ s conductor of siz so τχ 1/2 W trfor v t ound S 3 P 1/2 H 5/2 KX ε d d 0 H <2H d / P 1/2 KX ε 1 + 3/2 + ic is X ε n P X 1/2 ε sinc t infinit roduct convrgs Hving considrd ll ossil css t roof is comlt 63 A Comlt Crctr Sum As in t roof of Torm 31 nd to vlut comlt crctr sum W do tis no St T, ; = α + β λ α,β 2 W v α mod β mod Lmm 66 Lt > 2 rim nd dfind y 1 mod if, = 1 nd 0 = 0 Tn v T, ; = 2 δδ ε 3/2 6 +, r δ is t Kroncr dlt function mod Proof y dfinition, = α T, ; = β 2 sy W sily v β x mod α mod β mod x 3 + αx + β 2 α + β x 3 + αx + β 2 x 0 α β = T 1 + T 2, 1 if 0 mod, T 1 = if 0, 0 mod, 0 if 0 mod α + β α + β

27 LOW-LYING ZEROS OF FAMILIES OF ELLIPTIC CURVES 27 T sum T 2 is, ftr t linr cng of vrils α α x 2 β 2 x, givn y T 2 = x β2 x + β x 2 α α x 0 β α x x = ε 1/2 2 β2 x + β x 0 β x x = ε 1/2 2 xβ 2 + xβ from β xβ x 0 To vlut t summtion ovr β ly t formul x 2 + x ε ā 2 4 if, = 1, 25 = if 0 mod, x mod 0 otris W otin T 2 = ε x x x 0 Alying 25 gin otin T = ε Gtring trms nd simlifying finis t roof of t lmm β 1 7 Conjcturlly Enlrging t Suort for t Fmily of ll Ellitic Curvs 71 A Conjctur on t Siz of Crctr Sum In tis sction invstigt uristiclly t vior of DF; φ, X for t fmily givn in Torm 31 n xtnd t suort of φ outsid t rng 1, 1 Rcll tt tis is t slitting oint for t symmtry tys O, SOvn, nd SOodd W rdict tt t symmtry ty is O To rovid vidnc for tis conjctur, nd to rgu tt P E; φ A, = oa for φ it lrg suort From 12 nd 17 t rolm is siclly rducd to stimting t folloing crctr sum , 2 2 K H P it crtin rltions on t sizs of H, K, nd P On of t rltions is H 3 /P K 2 1 so for uross of tsting ignor t 3 / 2 trm Conjctur 71 Tr xists δ > 0 nd ε > 0 suc tt if P, is not squr i t crctr / is nonrincil, nd H = P 2/3+δ, tn 3 P 1 3δ/2 ε 2 H P

28 28 MATTHEW P YOUNG On summtion ovr P 1/2+3δ/2 tis conjctur ould indict tt Torm 31 rmins tru it tst functions os Fourir trnsforms φ v suort outsid of 1, 1 T xtnt to ic t suort could xcd 1, 1 ould dnd on t vlu of δ Prcisly, ould otin suort u to 1 3δ 1 T uristic usd in Sction 72 lnds suort to t vlu δ = 1/48, ic ould giv suort u to 16/15 72 Evidnc for Conjctur 71 To lnd suort for Conjctur 71 invstigt t sm sum ut it rnging ovr ositiv intgrs corim it instd of rims W v Torm 72 Lt H = P 2/3+δ, K = H 3/2 P 1/2 Tn for ny ε > 0 v m 3 m 2 H, m ε P 5/6+cδ+ε, P m r t rim indicts t summtion is rstrictd to, = 1 nd c is ositiv constnt t roof givs c = 13/2 Ting δ < 1 6 c nd ε smll noug ill so tt t sum is P 1 3δ/2 ε for ositiv δ nd ε T vlu c = 13/2 llos us to t ny δ < 1/48 Proof Lt S = S t sum to stimtd y Poisson summtion in m mod 2, S = m 3 m 2 H, m P m = P x 3 x + lx 2 2 H, lp, 2 l x mod 2 r t t ovr t scond vril indicts v tn t Fourir trnsform in tt vril only Rcll K 2 = H 3 /P = P 1+3δ so 2 /P P 3δ No rit x = y1 + z r y nd z rng ovr rrsnttivs mod Tn x = ȳ1 z r ȳ is t multilictiv invrs of ymod 2 W otin S = P 2 = P 2 = P l l H, lp 2 H, lp 2 l y mod ly 2 3 mod y y mod z mod y mod y 3ȳ + ly 2 y 3ȳ1 z + ly1 + z 3ȳ + ly 2 H, lp 2 z mod 2 z 3ȳ + ly Writ = r 0 H, 0, = 1, 1 H/, 0 ts vlus in n intrvl of lngt, nd 1 ts vlus in n intrvl of lngt H/ No xtnd t summtion ovr 0 to n intrvl of lngt H, so t sum is rtd H/ tims W otin ly S P H 0 l y mod ly mod 1 y 3 0 ȳ ȳ H, lp 2

29 LOW-LYING ZEROS OF FAMILIES OF ELLIPTIC CURVES 29 Alying t cng of vrils y 2 0ȳ givs S P y 2 0 lȳ H 2 l y mod y 2 l 0 mod y H, lp 2 Tr ill virtully no oscilltion in y/ 2 if y P ε 2 /H P 1/3+2δ+ε On t otr nd, if y P 1/3+2δ+ε tn it is sily son tt t summtion ovr 1 is ε,m P M Trfor v fixd 0 < y P ε K 2 /H Tus t sum is rducd to S P H 1 H/ 0 l 0<y P ε K 2 /H y 2 l 0 mod y 2 0 lȳ 2 1 0, 1,, l, y, r 1 is t n tst function otind y soring t non-oscilltory xonntil fctor into ny rocss of diffrntition of 1 it rsct to y introducs only fctors of siz P ε No ly t lmntry rcirocity l 16, otining S P H 1 H/ 0 l 0<y P ε K 2 /H y 2 l 0 mod y l y 2 0 l y 2 1 0, 1,, l, y T qulity y 2 = l 0 + s mns s P 2/3+4δ / 1+ε P 1/6+5δ/2+ε Tn t sum is rducd to S P y s2 l y 2 s 2 H y ly 2 1 0, 1,, l, y = P H 1 H 1 H/ 0 l 0<y P ε K2 H y 2 =l 0 +s s l 0<y P ε K 2 /H y 2 s mod l y s2ȳ l s2 ly + y2 s 2 ly 2 2 r 2 = 2 s, 1,, l, y, is t rlcmnt of 1 ftr t cng of vrils ic limints 0 nd introducs s No r t summtion ovr y into rogrssions mod l nd ly t Póly-Vinogrdov ound of y ly 1/2 log ly y Y y λ mod l to S vi rtil summtion Tis givs t finl ound of P 5/6+cδ+ε r c = 13/2 8 Curvs it Torsion Z/2Z Z/2Z In t folloing sctions study som intrsting fmilis of llitic curvs tt v torsion oints Ec fmily s its on intrsting fturs On common ftur is tt t squr divisors of t conductor r gnrlly rtr sy to control T sourc of tis s is tt t discriminnts fctor into olynomils of smllr dgr In ddition, t conductors r oftn muc smllr tn t discriminnt cus of ig multilicity in on or mor of ts olynomil fctors Tis fct cuss ts torsion fmilis to v rtr lrg numr of curvs it conductor N X

30 30 MATTHEW P YOUNG W rfr to t r of Kurt [Ku] s rfrnc for ts torsion fmilis In rticulr Tl 3 contins ssntilly ll t informtion us In tis sction invstigt rticulrly intrsting fmily of llitic curvs, givn in Wirstrss form y E s discriminnt E : y 2 = xx x + = T torsion grou is gnrtd y t oints 0, 0 nd, 0 nd λ = xx x + x mod Hlfgott s son tt t root numr in tis fmily is quidistriutd [H] W v t folloing Torm 81 Lt F t fmily of llitic curvs givn y t Wirstrss qutions E, : y 2 = xx x + it nd ositiv intgrs St A = = X 1/3, lt smoot comctly suortd function on R + R +, nd st X E, = A, Tn for φ it su φ 2 3, 2 3 DF; φ, X [ φ φ0]w XF s X, Tis fmily s rticulr intrst cus cn sum rims u to t siz of t fmily r ting A = X 2/3 curvs, nturl rrir for ny fmily sinc squr-root cnclltion coming solly from vrging ovr t fmily givs us tis suort; to go furtr rquirs dditionl cnclltion in t λ s s vris, t lst on vrg To rov Torm 81 nd to lmms For t conductor condition v Lmm 82 Lt F t fmily givn in Torm 81 Tn log N A, = W X F + O + 2 N, >3 A Proof W ill ly Lmm 41 W t Rd = + Tn it s clr tt Rd X nd tt t irrducil fctors of Rd ll divid d W first considr A, log Tis sum is mty cus 2 N imlis, sinc xx x + s tril root mod To stimt t sum A, log α 1 α + suos γ nd δ Tus α γ δ + W first suos γ δ y symmtry my ssum γ < δ ic imlis γ + nd γ < α/3 Sinc A = = X 1/3

31 LOW-LYING ZEROS OF FAMILIES OF ELLIPTIC CURVES 31 lys v X 1/3 Tn gt t ound α X α>0 X 1/3 γ α/3 α X α>0 X 1/3 1 + A γ 1 + α 2γ γ α/3 α 3γ + log α 1 X 2/3 In cs γ = δ ly t cng of vrils γ, γ nd gt t ound γ A, γ log α 1 For fixd, is dtrmind mod α 3γ so gt t ound 1 + A 1 + γ α 2γ α X α>0 X 1/3 γ α/3 log α 1, ic is t sm ound v for γ δ, so t roof is comlt T sum ovr t Fourir cofficints is ndld it Lmm 83 Lt F t fmily givn in Torm 81 Tn P E; φ A, X2/3 rovidd su φ 2 3, 2 3 Proof St so tt S = λ A, P E; φ A, = S 2 log φ log >3 Tn S = xx x + A, x mod = A xx ρx + σ ρ + σ 2 x,ρ,σ mod = A xρσ ρ + σ + x 2 x,ρ,σ mod = A ε A 1/2 ŵ, A ŵ ŵ, A, To gt furtr cnclltion sum ovr To ndl t vrition of ε introduc crctr ψ 4 mod 4 vlutd t For tos trms it / ψ 4 nontrivil

32 32 MATTHEW P YOUNG l to t Rimnn Hyotsis for Diriclt L-functions to otin t ound summing P A P P ε P 2 A = P 1+ε, ic is X 2/3 ε n P X 2/3 ε, i t rstriction on t suort of φ is 2/3 Wn / ψ 4 is trivil do not otin ny sving in t summtion ovr T crctr is trivil only if = ± so t rolm mounts to stimting t numr of suc solutions Tis is crrid out y t folloing Lmm 84 Lt Tn CY ε Y 1+ε CY = {, Z 2 : + Y, = ± } for roving t lmm ly it to our sum y t rid dcy of t Fourir trnsform my ssum, P A 1 1+ε Using t vlu Y = P A 1 1+ε in t lmm otin t ound A CY P 1/2+ε P 1/2 P on t contriution of t trms it trivil crctr / ψ 4 T contriution is X 2/3 ε n P X 2/3 ε, s dsird Tis ill comlt t roof of Lmm 83 nd nc Torm 81 onc rov Lmm 84 Proof of Lmm 84 Suos v solution = ± r + Y y ossily cnging t signs of nd nd sitcing t vlus of nd my ssum > 0, > 0, nd > t css r = 0 r trivil St g =, nd lt g = d 2 l r l is squrfr St = g nd = g Tn v l = Sinc, = 1 nd l is squrfr must v l = l 1 l 2 l 3 r l 1, l 2, l 3 nd nc c of l 1, l 2, nd l 3 must squr Tus = l 1, = l 2, nd = l 3 Tus vry solution to = is givn y r = d 2 l 2 1l 2 l 3 x 2, = d 2 l 1 l 2 2l 3 y 2, 27 l 1 x 2 = l 2 y 2 + l 3 z 2, it t rstrictions x, y = x, z = y, z = 1, l = l 1 l 2 l 3 is squrfr, nd c l i is ositiv ut no notic tt t solution 27 givs ris to t fctoriztion l 1 l 2 x 2 = l 2 y + l 2 l 3 zl 2 y l 2 l 3 z T numr of suc fctoriztions is clrly oundd y d Q l1 l 2 lx 2, t divisor function in t ring of intgrs of t fild Q l 2 l 3 It s ll-non tt d K n cε N K n ε r N K n is t norm in t numr fild K nd cε dos not dnd on t fild K Using = d 2 l 1 lx 2 d 2 lx 2 sily gt t ound CY Y 1+ε