Ka- I A RESULT ON MODULAR FORMS IN CHARACTERISTIC p Nicholas M. Katz ABSTRACT The action of the erivation e = q ~ on the q-expansions of moular forms in characteristic p is one of the funamental tools in the Serre/Swinnerton-Dyer theory of mo p moular forms. In this note, we exten the basic results about this action, alreay known for P > 5 an level one, to arbitrary p an arbitrary prime-to-p level.!. Review of moular forms in characteristic p We fix an algebraically close fiel K of characteristic p > 0, an integer N > 3 prime to p, an a primitive N'th root of unity ~ ~ K. The mouli problem "elliptic curves E over K-algebras with level N structure ~ of eterminant {" is represente by (Euniv i ~univ) M N with M N a smooth affine irreucible curve over K. In terms of the invertible sheaf on M N I = ~, fleuniv/m N ' the grae ring R~ of (not necessarily holomorphic at the cusps) level N moular forms over K is O HO(MN,@f k) k~ Given a K-algebra B, a test object (E,~) over B, an a nowhere-vanishing invariant ifferential ~ on E, any element
Ka-2 54 f e R N (not necessarily homogeneous) has a value f(e,~,~) e B, an f is etermine by all of its values (cf. [2]). Over B = K((ql/N)), we have the Tate curve Tare(q) with its "canonical" ifferential C0ca n (viewing Tare(q) as ~m/q ~, ~can "is" t/t from ~m ). By evaluating at the level N structures as o of eterminant ~ on Tare(q), all of which are efine over K((ql/n)), we obtain the q-expansions of elements f e R N at the corresponing cusps: fm0(q) fn f(tate(q),~can,~ 0) e K((ql/N)) A homogeneous element f e R k N is uniquely etermine by its weight k an by any one of its q-expansions. A form f c R k N is sai to be holomorphic if all of its q-expansions lie in K[[ql/N]], an to be a cusp form if all of its q-expansions lie in q I/NK[[ql/N]] The holomorphic forms constitute a subring R~,holo of RN, an the cusp forms are a grae ieal in RN,holo. The Hasse invariant A c R is efine moularly as olo follows. Given (E,~,(~) over B, let r i c HI(E,OE ) be the basis ] ual to ~ c H0(E,CE/B). The p'th power enomorphism x -~ x p of O E inuces an enormorphism of HI(E,OE ), which must carry r; to a multiple of itself. So we can write ~P = A(E,~,~)-,~ in HI(E,OE ), for some A(E,~,~) c B, which is the value of A on (E,~,~). All the q-expansions of the Hasse invariant are ientically ]: A o(q ) = l in K((q~/N)) for each s o. For each level N structure s o on Tare(q), the corresponing q-expansion efines ring homomorphisms whose kernels are precisely the principal ieals (A-I)R~ an
55 ~-3 (A-l)R~,holo respectively ([4], [5]). A form f ~ R N k is sai to be of exact filtration k if it is not ivisible by A in R~, or equivalently, if there is no form f' k~ k' R N with ( k which, at some cusp, has the same q-expansion that f oes. I!. Statment of the theorem, an its corollaries The following theorem is ue to Serre an Swinnerton-Dyer ([4], [5]) in characteristic P > 5, an level N = I. Theorem. (I) There exists a erivation A0:R~ ~ RN+P+~ which increases egrees by p + I, an whose effect upon each q-expansion is q~ : (A0f) (q) = q _ (fso(q)) for each S O So q (2) If f c R k N has exact filtration k, an p oes not ivie k, then A0f has exact filtration k + p + ~, an in particular A0f ~ O. k g c R N. (3) If f c R~ k an A0f = 0, then f = gp for a unique Some Corollaries (]) The operator A0 maps the subring of holomorphic forms to the ieal of cusp forms. (Look at q-expansions.) (2) If f is non-zero an holomorphic, of weight I ~ k ~ p - 2, then f has exact filtration k. (For if f = Ag, then g is holomorphic of weight k - (p-l) ~ O, hence g = 0.) injective. (3) If ] ~ k ~ p 2, the map 0 k ~ Rk+P+1 - A :RN,holo N,holo is (This follows from (2) above an the theorem.) (4) If f is non-zero an holomorphic of weight p - l, an vanishes at some cusp, then f has exact filtration p - ~. (For if f = Ag, then g is holomorphic of weight O, hence constant; as g vanishes at one cusp, it must be zero.)
Ka-4 56 k has Aef = 0, (5) (etermination of Ker(Ae)). If f ~ R N then we can uniquely write f = Ar.g p with 0 ~ r < p - I, r + k ~ O(mo p), an g ~ R~ with pf + r(p-]) = k. (This is proven by inuction on r, the case r = 0 being part (3) of the theorem. If r ~ O, then k @ O(p), but Aef = 0. Hence by part (2) of k+]-p the theorem f = Ah for some h ~ R N Because f an h have the same q-expansions, we have Aeh = O, an h has lower r.) (6) In (5) above, if f is holomorphic (resp. a cusp form, resp. invariant by a subgroup of SL2(Z/NZ)), so is g (by unicity of g).!i!. Beginning of the proof: efining e an A~, an proving part (1) The absolute Frobenius enomorphism F of M N inuces an F-linear enomorphism of H 1 (E /M ~ as follows. The pull- DR univ" N/' (F) of is obtaine by iviing Euniv by back Euniv Euniv its finite flat rank p subgroup scheme Ker Fr where Fr:Euniv ~(F) > ~univ is the relative Frobenius morphism. The esire map is Fr * 1 (F) Fr: HDR(Euniv /MN) >4R(Euniv/MN) ~R(Euniv/MN) (F) Lemma ]. The image U of Fr is a locally free submoule of H~(Eunl~n "v/m-)n of rank one, with the quotient H~R(Euniv/MN)/U,Hasse locally free of rank one. The open set M N C M N where A is invertible is the largest open set over which U splits the Hoge filtration, i.e., where ~@U > R(Euniv/MN). Proof. Because F~ kills HO(Euniv, ~Euniv/MN )(F) it factors through the quotient H](Euniv,O) (F), where it inuces the inclusion map in the "conjugate filtration" short exact sequence
57 Ke-~ (cf [~], 2.3) 0 --> H I (F) Fr nl (Euniv,@) -->4R(Euniv/MN) --> HO(Euniv, Euniv/MN) --> O. This proves the first part of the lemma. To see where U splits the Hoge filtration, we can work locally on M N. Choose a basis ~, rl of H~R aapte to the Hoge filtration, an satisfying <~,TI>DR = ] Then ~ projects to a basis of HI(E,OE ) ual to ~, an so the matrix of Fr on R is (remembering Fr(~ ( ) = 0) * 4 * ~) (0 A where A is the value of the Hasse invariant. Thus U is spanne by Be + A~, an the conition that e an Be + A~ together span H~R is precisely that A be invertible. Q.E.D. Remark. Accoring to the first part of the lemma, the func- tions A an B which occur in the above matrix have no common zero. This will be crucial later. We can now efine a erivation e of RN[I/A] as follows. (Compare [2], A1.4.) Over M~ asse, we have the ecomposition which for each integer k > I inuces a ecomposition Symmk~R ~ _m k(~)(e k-1 ( ~ )(~U) _... ~ U k The Gauss-Manin connection ] I inuces, for each k > I a connection Using the Koaira-Spencer isomorphism ([2], A.].3.]7) ~ HN/K
- Ka- 6 58 we can efine a mapping of sheaves k k+ 2 as the composite > Sym:L = (~ -- Iil DR -- "'" f~m R KS Pr 1 } ~ k+ 2 MHasse Passing to global sections over -N ' we obtain, for k > I, a map 0,~Hasse Hasse,,@k+ 2 ) e " > H0(mN,m :H ~N "-- " Lemma 2. The effect of @ upon q-expansions is Proof. qt@" Consier Tate(q) with its canonical ifferential co 2 can over k((ql/n)) Uner the Koaira-Spencer isomorphism, can correspons to q/q, the ual erivation to which is q ~. By the explicit calculations of ([2], A.2.2.7), U is spanne by v(q ~)(~can). Thus given an element k ~O,~Hasse~ _.~, _ k), its local expression as a section of _ f C on (Tate(q), some ~0) is f 0(q).~ @kcan. Thus v(f c~ 0 k q k ) V(q ~--~)((q)'~can) q (q)'ecan = f~0 k 2 (q)'~can) ~can : v(q T4)(f% k+ 2 : q -q (f~0 (q))" ~can k+ I. V( q --~) (~can) ' + k~f~0(q)" can Because ~ (q ~q) (~can) lies in U, it follows from
59 Ka-7 (q)). the efinition of e that we have (ef)~o(q) = q -q (f~0 Q.E.D. k ~k+p+1 Lemma 3. For k ~ ], there is a unique map Ae:R N > m N such that the iagram below commutes HO ( 4asse, )_ ~ k U 9_~_> re'o". Hassek~N, k+2~ A > n"o'kmn~asse,~ k+ p+ ] ) RNk = H0(MN, @k) A0 > ~k+p+l ~N = HO(MN,fk+P+?) Proof. Again we work locally on M N. Let @ be a local I basis of, ~ the local basis of ~MN/K corresponing to by the Koaira-Spencer isomorphism, D the local basis of DerNN/K ual to ~, an ~' = V(D)@ ~ ~R" Then <@'~'>DR = I, (this characterizes D), so that @ an m' form a basis of ] HDR, aapte to the Hoge filtration, in terms of which the matrix of Fr is 0B) (0 A ~asse with A = A(E,e). Let u e U be the basis of U over M N which is ual to e. Then u is proportional to Be + Ae', an satisfies <e'~'>dr = I, so that B u = K ~ + e l In terms of all this, we will compute 9f for f e RN, k an show that it has at worst a single power of A in its enominator. k k Locally, f is the section f]-@ of ~, with fl holomorphic. v(f~ ~k) = v(d)(f1~ k).~ = V(D)(flco k).@ 2 k+ 2 k+ 1, = D(f] ).co + kf1@.co B =D(f I) k+2 + kflco k+1(u _ K ~)
Ka-8 60 B k+2!<+i : (D(fl) -kf 1 K) + kfl "u Thus from the efinition of e it follows that the local expression of ~(f) is 0(f) = (D(f~) - kf I ~--) k+2 Q.E.D. We can now conclue the proof of Part (1) of the theorem. Up to now, we have only efine A on elements of R~ of positive egree. But as R~ has units which are homogeneous of positive egree (e.g., A), the erivation Ae extens uniquely to all of RN by the explicit formula Aef A~(f'aPr) for r >> 0 AP r The local expression for Ae(f) k Ae(f) = (AD(fl)-kfiB)~ k+p+1 for f e R N remains vali. IV. Conclusion of the proof: Parts (2) an (3) Suppose f e R k N has exact filtration k. This means that f is not ivisible by A in R~, i.e., that at some zero of A, f has a lower orer zero (as section of k) than A oes (as section of p-l). (In fact, we know by Igusa [3] that A has simple zeros, so in fact f must be invertible at some zero of A. Rather surprisingly, we will not make use of this fact.) Locally on MN, we pick a basis ~ of ~. Then f becomes Ok f].~, an A~(f) is given by A~(f~- ~k) = (kd(fl)_kbfl) Suppose now that k is not ivisible by p. Recall that B is invertible at all zeros of A (of the remark following Lemma ]). Thus if x s M N is a zero of A where orx(f]) < Orx(A),
61 Ka-9 we easily compute orx(ad(f l)-kbf I) = orx(f I) < orx(a). This proves Part (2) of the theorem. To prove Part (3), let expression for Ae(f) gives f c R pk N have A@(f) = 0= The local An(f I ) k+p+1 = 0 an hence D(fi) = 0. Because M N is a smooth curve over a perfect fiel of characteristic p, this implies that fl is a pth power say fl = (gl)p" Thus fl ~ kp = (g1~ k) p, so that f, as section of _~kp, is, locally on M N, the pth power of a k (necessarily unique) section g of ci~ By unicity, these local g's patch together. Q.E.D. REFERE NCE S 171] [2] [3] [4] N. Katz, Algebraic solutions of ifferential equations, Inventiones Math. 18 (1972), 1-118. P-aic properties of moular schemes an moular forms, Proceeings of the 1972 Antwerp International Summer School on Moular Functions, Springer Lecture Notes in Mathematics 350 (1973) 70-189. J. Igusa, Class number of a efinite quaternion with prime iscriminant, Proe. Nat'l. Aca. Sci. 44 (1958) 312-314. J. -P. Serre Congruences et formes moulaires ('apres H. P. F. Swinnerton-Dyer) Expos@ 416, S4minaire N. Bourbaki, 1971/72, Springer Lecture Notes in Mathematics 317 (1973), 319-338. H. P. F. Swinnerton-Dyer, On ~-aic representations an congruences for coefficients of moular forms, Proceeings of the 1972 Antwerp International Su~mmer School on Moular Forms Springer Lecture Notes in Mathematics 350 (1973), 1-55. N. Katz Departement of Mathematics Fine Hall Princeton University Princeton, N.J. 08540