A RESULT ON MODULAR FORMS IN CHARACTERISTIC p. Nicholas M. Katz ABSTRACT. d The action of the derivation e = q ~ on the q-expansions of

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1 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 k) k~ Given a K-algebra B, a test object (E,~) over B, an a nowhere-vanishing invariant ifferential ~ on E, any element

2 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

3 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.)

4 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 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 > 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

5 57 Ke-~ (cf [~], 2.3) 0 --> H I (F) Fr nl -->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

6 - Ka 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 2 ) e " > H0(mN,m :H ~N "-- " Lemma 2. The effect upon q-expansions is Proof. 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 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

7 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 = A0 > ~k+p+l ~N = HO(MN,fk+P+?) Proof. Again we work locally on M N. 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 ~' = ~ ~R" Then = I, (this characterizes D), so 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 of ~, with fl holomorphic. v(f~ ~k) = v(d)(f1~ k).~ = V(D)(flco 2 k+ 2 k+ 1, = D(f] ).co + B =D(f I) k+2 + kflco k+1(u _ K ~)

8 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),

9 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 = 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), 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) J. Igusa, Class number of a efinite quaternion with prime iscriminant, Proe. Nat'l. Aca. Sci. 44 (1958) J. -P. Serre Congruences et formes moulaires ('apres H. P. F. Swinnerton-Dyer) 416, S4minaire N. Bourbaki, 1971/72, Springer Lecture Notes in Mathematics 317 (1973), 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), N. Katz Departement of Mathematics Fine Hall Princeton University Princeton, N.J

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