THE F -PURE THRESHOLD OF AN ELLIPTIC CURVE BHARGAV BHATT ABSTRACT. W calculat th F -pur thrshold of th affin con on an lliptic curv in a fixd positiv charactristic p. Th mthod mployd is dformation-thortic, and th answr dpnds on th arithmtic proprtis of th lliptic curv. 1. MOTIVATION Fix a bas fild k of charactristic p > 0. Our goal is to xplain how to comput th F -pur thrshold of th con on an lliptic curv E ovr a fild k in trms of th arithmtic of E. W first rcall th ncssary dfinitions, and provid som contxt for our calculation. Dfinition 1.1. Lt (R, m) b a rgular local k-algbra, and lt a m b an idal. For a fixd intgr 1, w dfin ν a (p ) = max{r a r m [p] }. Th F -pur thrshold fpt(a) is thn dfind as th limit which on chcks xists and is finit. fpt(a) = lim ν a (p ) p F -pur thrsholds ar bst thought of as charactristic p analogus of th log canonical thrsholds of charactristic 0. Indd, thr xist thorms and conjcturs rlating th log canonical thrshold of a complx singularity with th F -pur thrshold of its rduction to charactristic p, at last for infinitly many choics of p. Such rsults oftn allow on to comput/guss th F -pur thrsholds for infinitly many mod p rductions of a givn complx singularity. Th primary drawback of this approach is that on cannot say much for a fixd prim p. Our goal in th squl is to xplain how to comput this invariant in a fixd charactristic p for crtain singularitis. Th singularitis w tackl hr ar cons on lliptic curvs, though it sms rasonabl to hop that th mthod can b pushd a littl furthr. Acknowldgmnts. W thank th following individuals and groups at AMS-MRC workshop on Commutativ Algbra: Mirca Mustaţǎ for suggsting th problm addrssd hr, th positiv charactristic group for calculating xampls which ld to th statmnt of Thorm 2.1, and th NSF and th AMS for providing th wondrful working conditions whr th solution prsntd hr was workd out. In addition, w would lik to thank Johan d Jong for a usful convrsation. 2. THE CALCULATION Fix an lliptic curv E ovr k and a closd immrsion E P 2 ; w st R = kx, y, z to b th compltd homognous co-ordinat ring of P 2, and a = (f) R to b th idal dfind by a homognous cubic form f dfining E. Our goal is to xplain th following thorm: Thorm 2.1. If E is ordinary, thn fpt(a) = 1. If not, thn fpt(a) = 1 1 p. 1
Sktch of proof. Th main ida of th proof is th following. W first intrprt th dfinition of F -pur thrsholds in trms of th Frobnius action on local cohomology, and thn translat th lattr to to a qustion about th bhaviour of Frobnius on th cohomology of E and crtain thicknings of it. This lattr qustion is wll-suitd for a dformation thortic study, and w succssfully implmnt this approach using an old thorm on Igusa on th rducdnss of th suprsingular locus in a vrsal family of lliptic curvs. W now sktch th proof. As th argumnt is slightly long, it is brokn into a numbr of stps. Th first fw stps xplain th rduction of th algbra problm to on in gomtry, whil th lattr ons dscrib th solution to th gomtric problm. 2.1. A cohomological intrprtation of th invariants ν a () and fpt(a): Givn any lmnt f R and intgrs, r 1, w hav f r m [p] if and only if f r F : H 3 m(r) H 3 m(r) has a non-trivial krnl, whr F is th Frobnius ndomorphism of R. Th ndomorphism f r F : R R sits in a morphism of xact squncs: 0 R f R R/f f r F F Fr 0 R f p r R R/f p r 0 Hr Fr dnots th lift of th Frobnius map R/f R/f to a map R/f R/f p r (followd by th canonical projction), which xists bcaus a = b mod f implis a p = b p mod f p. Standard rsults on local cohomology rsults about rgular local rings thn stablish an idntification Thus, w s that kr(h 2 m(f r )) kr(h 3 m(f r F )). ν a (p ) = max{r kr(h 2 m(f r )) = 0}. 2.2. Going from th singularity (R, a) to th gomtry of E: If L dnots th ampl lin bundl usd to mbd E P 2, thn w hav a canonical idntification H 2 m(r/f) n Z H 1 (E, L n ). An asy way to s this is to ralis th blowup of Spc(R/f) at {m} as th total spac of L 1 ovr E with th xcptional divisor corrsponding to th 0 sction. Th map Fr considrd abov is dscribd gomtrically via th following diagram: E F (p r)e p E P 2 Fr F0 E P 2. Hr th squar is a pullback, and ne dnots th ordr n infinitsimal nighbourhood of E. This diagram inducs a map n Z H 1 (E, L n ) n Z H 1 ((p r)e, Fr L n ) which is idntifid with th map Hm(F 2 r ) considrd in th local dicussion abov. Elmntary curv thory can b usd to show that (F ) : H 1 (E, L n ) H 1 (E, L np ) is injctiv for n 0. Thus, it follows that ν a (p ) = max{r kr(fr : H 1 (E, O E ) H 1 ((p r)e, O (p r)e)) = 0}. 2.3. Th ordinary cas: If E is ordinary, thn th Frobnius map H 1 (E, O E ) H 1 (E, O E ) is injctiv, and th sam is tru for its itrats. Hnc, th map on cohomology inducd by Fp 1 is injctiv, showing that ν a (p ) p 1. On th othr hand, it follows from th (algbraic) dfinition that ν a (p ) < p. Hnc, w find ν a (p ) = p 1. Taking a larg limit thn shows that fpt(a) = 1 whn E is ordinary. 2 0 F
2.4. Th dformation-thortic contnt of ν a (p) p 2 in th suprsingular cas: W will rstrict ourslvs to th cas = 1 until th nd of th argumnt; th passag to th cas of gnral is straightforward in th prsnt stup, and is xplaind in th last stp. In th cas = 1, our goal is to show that ν a (p) p 2. W now xplain how to intrprt this condition in trms of th gomtry of E, spcially its dformation thory. Lt I O P 2 to b th shaf of idals dfining E. Th prcding gomtric intrprtation of ν a (p) shows th following: w hav ν a (p) p 2 if and only if th map Fp 2 1 : 2E E inducs an injctiv map on cohomology. To as notation, w dnot by c : H 1 (E, O E ) H 1 (2E, O 2E ) th map on cohomology inducd by th map lablld Fp 2 1 abov. In ordr to undrstand th injctivity of c, considr th following commutativ diagram H 1 (2E, I/I 2 ) H 1 (E, I/I 2 ) s H 1 c (E, O E ) H 1 (2E, O 2E ) Frob=0 H 1 (2E, O E ). Hr th scond column is an xact squnc, and th map s xists bcaus of th suprsingularity condition Frob = 0. In ordr to show that c is injctiv, it suffics to show that s is injctiv. Th Calabi-Yau condition allows us to idntify th group H 1 (E, I/I 2 ) with th dual of th tangnt spac to th spac of plan cubics in P 2 at th point dfind by E. Morovr, this association satisfis crtain compatibilitis with th diagram abov which w now xplain. Givn a functional λ : H 1 (E, I/I 2 ) k, th corrsponding infinitsimal dformation of E can b (uniquly) intgratd to a lin L in th spac of cubics dfining a diagram of th form: E π 0 E ɛ Spc(k) = [0] Spc(k[ɛ]) = 2[0] L Hr [0] is th point in L dfining E, and 2[0] dnots its first ordr infinitsimal nighbourhood. Gomtrically, on can think of L as a pncil of plan cubics spannd by E and a diffrnt cubic E satisfying th condition that th lin L joining E and E in th spac of cubics dfins th tangnt vctor λ at E; this intrprtation allows us to think of E as th blowup of P 2 along th bas locus of this pncil. Sinc th strict transform of th divisor E P 2 is xactly th fibr π 0 of π, th map φ Eɛ factoriss as E ɛ 2E P 2. Th inducd map on cohomology dfins th following morphism of short xact squncs: π ɛ E π φ P 2 H 1 (E, I/I 2 ) a ɛ H 1 (E, O E ) k H 1 (2E, O 2E ) b H 1 (E ɛ, O Eɛ ) H 1 (E, O E ) d H 1 (E ɛ, O Eɛ ) k[ɛ] k H 1 (E, O E ) 3
Hr th horizontal maps ar maps inducd by φ Eɛ. Th compatibility rfrrd to abov is th following: th map a is idntifid with functional λ dfining th original dformation, and th map d is th idntity 1. It rmains to intrprt th composit map g : H 1 (E, O E ) c H 1 (2E, O 2E ) b H 1 (E ɛ, O Eɛ ). As th map c was dfind by rstricting th Frobnius morphism on th ambint projctiv spac P 2 to E, it is not hard to chck that th map g is th map on cohomology inducd by rlativ Frobnius map E E (p) dfind by th family L whn rstrictd to 2[0]. In mor dtail, using th usual diagram E F π π E (p) F E π (p) F L E L F L L w can us th rlativ Frobnius morphism F π to obtain a 2[0]-morphism E ɛ E L 2[0] E (p) L 2[0]. Sinc F L (2[0]) = [0], th targt of th prcding morphism is idntifid with th trivial dformation E 2[0] of E ovr 2[0]. Th inducd morphism on cohomology is idntifid with th map g considrd abov. Thus, in ordr to show ν a (p) p 2, it suffics to show th following: thr xists an infinitsimal dformation of E in P 2 such that th corrsponding map g considrd abov is injctiv. 2.5. Finding th dsird dformations to prov ν a (p) p 2: W pick λ : H 1 (E, I/I 2 ) k to b a functional such that th inducd dformation E ɛ 2[0] (following notation abov) is vrsal (w can do this bcaus plan cubics modl all lliptic curvs). Concrtly this mans that th infinitsimal proprtis of E ɛ 2[0] bhav lik thos of th univrsal lliptic curv ovr th moduli spac of lliptic curvs. Igusa s thorm that th suprsingular locus on th moduli spac (as dfind by th vanishing locus of Hass invariant) is rducd thn implis that th rlativ Frobnius morphism for th family E ɛ 2[0] is nonconstant. In particular, it inducs a non-zro map H 1 (E, O E ) H 1 (E, O E ) k k[ɛ] H 1 (E ɛ, O ɛ ), whr th first map is th xtnsion by scalars map. Sinc w idntifid this map with th map g abov, it follows from th on-dimnsionality of H 1 (E, O E ) that th map g is injctiv, as dsird. 2.6. Computing ν a (p ) and fpt(a): W may assum that E is suprsingular. Our goal now is to show that ν a (p ) p p 1 1. Th prcding discussion shows that th map F : pe E (dfind by th Frobnius on P 2 ) givs ris to a diagram H 1 (pe, I/I p ) h H 1 (E, O E ) h H 1 (pe, O pe ) π Frob=0 H 1 (pe, O E ). with h and h injctiv. In ordr to show that ν a (p ) p p 1 1, w nd to vrify that F p p 1 1 : H1 (E, O E ) H 1 (p E, O p E/I p 1 +1 ) 1 Th idntification of th targt of a with th trivial vctor spac ariss from th isomorphism of H 1 (E, I/I 2 ) with th tangnt spac to th moduli spac of plan cubics at E implicit in th prcding discussion. Indd, this lattr idntification rquirs us to pick a trivialisation of th canonical lin bundl of E which, in turn, givs a trivialisation of H 1 (E, O E) via th trac map. 4
is injctiv. Writing F = F 1 F and using th prcding diagram, w find that it suffics to vrify that F 1 : p E pe inducs an injctiv map H 1 (pe, I/I p ) H 1 (p E, I p 1 /I p ). Filtring both sids with th natural filtrations and looking at th map on associatd gradd pics (which multiplis dgrs by p 1 ), th injctivity ndd abov follows from th fact that Frobnius acts injctivly on th cohomology of a ngativ dgr lin bundl on an lliptic curv (s Rmark 2.2). Hnc, w hav sn that ν a (p ) p p 1 1. On th othr hand, th fact that E is suprsingular can b sn to imply ν a (p ) < p 1 (p 1) = p p 1 as follows: sinc f p 1 m [p] (following th algbraic notation), taking Frobnius twists shows f p p 1 m [p]. Thus, w s that ν a (p ) = p p 1 1 in th suprsingular cas. Dividing by p and taking a larg limit shows that fpt(a) = 1 1 p, as dsird. Rmark 2.2. Th prcding proof usd th following fact multipl tims: if E is an lliptic curv, and I is a ngativ dgr lin bundl on E, thn th Frobnius map F : E E inducs an injctiv map H 1 (I) H 1 (I p ). On way to s this is as follows: aftr passing to an appropriat fild xtnsion if ncssary, w may choos a dgr 0 lin bundl M that is not p-torsion, and a non-zro sction s H 0 (I 1 M). Viwing s as a map I M, w obtain an xact squnc 0 I M Q 0 whr Q M Z is a lin bundl supportd on a zro-dimnsional subschm Z E with dfinining idal I M 1. Taking cohomology and using that M has no cohomology, w gt an isomorphism a s : H 0 (Q) H 1 (I). As M is not p-torsion, th sam analysis applis to th Frobnius pullback of th prcding xact squnc to giv b s : H 0 (F Q) H 1 (I p ). Morovr, ths isomorphisms sit in a commutativ diagram H 0 (Q) F (Q) H 0 (F Q) a s H 1 (I) F (I) b s H 1 (I p ) with th vrtical maps inducd by pulling back along F. As a s and b s ar isomorphisms, to show that F (I) is injctiv, it suffics to show that F (Q) is injctiv. Th map F (Q) is inducd by pullback of sctions of a lin bundl along a finit flat morphism F Z = pz Z and is thrfor injctiv, and so th claim follows. Rmark 2.3. It sms rasonabl to guss that th abov mthod can b gnralisd to show th following: if X P n is a smooth Calabi-Yau hyprsurfac and a is th corrsponding idal in th homognous coordinat ring of P n, thn fpt(a) = 1 ord(h) p, whr ord(h) is th ordr of vanishing of th Hass invariant on th vrsal dformation spac of X. Rsults du to Arthur Ogus (s [Og]) giv uppr bounds bounds on ord(h) for p > n, which givs lowr bounds on fpt(x). Morovr, w bliv ths bounds can b shown to b sharp by xhibiting highly suprsingular xampls. Rmark 2.4. A natural qustion suggstd by th prcding calculation is th following: what about th F -pur thrsholds of homognous forms in kx, y, z dfining highr gnus curvs? Following th mthod xplaind abov lads on naturally to qustions about th infinitsimal bhaviour of th loci dfind by rank conditions imposd on Hass-Witt matrix on th subspsac of th vrsal dformation spac of th 5
curv spannd by planar dformations. W do not know how tractabl thss qustions ar, spcially if on is intrstd in sharp answrs. Howvr, it dos not sm unrasonabl to xpct lowr bounds. REFERENCES [Og] A. Ogus, On th Hass locus of a Calabi-Yau family, Math. Rs. Ltt. 8 (2001), no. 1-2, 35 41. 6