Ulmer, Douglas L. On the Fourier coefficients of modular forms. (English) Zbl 0827.11024 Ann. Sci. Éc. Norm. Supér. (4) 28, No. 2, 129-160 (1995). For a fixed prime number \(p\), non-negative integers \(k\) and \(m\), a positive integer \(N\) prime to \(p\), and a Dirichlet character \(\psi\) modulo \(p^mN\), consider the Hecke polynomial \[ E(k,m,N,\psi)=\begin{cases} \det(1-U_pT{}S_{k+2}(\Gamma_0(p^mN),\psi))&\text{if }m>0\\ \det(1-T_pT+\psi(p)p^{k+1}T^2{}S_{k+2}(\Gamma_0(N),\psi))&\text{if }m=0,\end{cases} \] where \(S_{k+2}(\Gamma_0(M),\psi)\), \(M\) any non-negative integer, denotes the space of cusp forms of weight \(k+2\) and character \(\psi:({\mathbb Z}/M{\mathbb Z})^{\times}\rightarrow{\mathbb C}\) for the congruence subgroup \(\Gamma_0(M)\) of \(\text{SL}_2({\mathbb Z})\). The Dirichlet character \(\psi\) admits a unique decomposition of the form \(\psi=\chi^a\eta\theta\), \(0\leq a\leq p-2\), where \(\chi\) may be identified with the Teichmüller character (after fixing a \(p\)-adic valuation \(v\) of \({\mathbb Q}(\mu_{p- 1})\) with normalization \(v(p)=1\), and then embedding \({\mathbb Q}(\mu_{p- 1})\hookrightarrow{\mathbb Q}_p\)), and where \(\eta:({\mathbb Z}/N{\mathbb Z})^{\times}\rightarrow{\mathbb C}\) and \(\theta:(1+p{\mathbb Z}_p)\rightarrow{\mathbb C}\) are characters of finite order. Let \(n>0\) be an integer, and define, with the notations above, the polynomial \(H(k,n,N,a)\) by \[ H(k,n,N,a)=\prod_{0\leq m\leq }\prod_{p^m{}\text{cond}(\psi)}E(k,m,N,\psi), \] where the second product runs over all characters \(\psi\) modulo \(p^mN\) with conductor \(\text{cond}(\psi)\) divisible by \(p^m\) and whose restriction to \(({\mathbb Z}/p{\mathbb Z})^{\times}\) is \(\chi^a\). Assume \(N>4\) and let \(0\leq k<p\). Also, let \(g\) be the genus of the modular curve \(X_1(N)\), and let \(c\) denote the number of cusps on \(X_1(N)\). Set \(w=g-1+c/2\), and let \(a'=0\) if \(a=0\) and \(a'=p-1-a\) if \(a\neq 0\). Define integers \(l_0,l_1,\ldots,l_{k+1}\) by: \[ l_0=\begin{cases} (p^{2n- 1}-p+2)w/2 -p^{n-1}c/2+1&\text{if }k=a=0 \\ (p^{2n-1}-p+2+2kp^{n- 1}+2a)w/2-p^{n-1}c/2&\text{otherwise},\end{cases} \]\[ l_1=l_2=\cdots=l_k=\begin{cases} (p^{2n-1}-2p^{n-1}-p+2)w&\text{if }a=0\\ (p^{2n-1}-2p^{n-1}+1)w&\text{if }a\neq 0, \end{cases} \]\[ l_{k+1}=\begin{cases} (p^{2n-1}-p+2)w/2-p^{n-1}c/2+1&\text{if }k=a=0\\ (p^{2n-1}-p+2+2kp^{n-1}+2a')w/2-p^{n-1}c/2& \text{otherwise}.\end{cases} \] Then one can make up the Newton polygon, with respect to the valuation \(v\), of \(H(k,n,N,a)\) and the Hodge polygon associated to \((l_0,l_1,\ldots,l_{k+1})\). The main result of the paper can now be stated: Theorem: The Newton polygon of \(H(k,n,N,a)\) lies on or above the Hodge polygon of\((l_0,l_1,\ldots,l_{k+1})\), and both have the same endpoints. The result (eventually a somewhat coarser form of it) may then be applied to obtain lower bounds on the \(p\)-divisibility of the Fourier coefficients of weight \(k+2\) cusp forms for \(\Gamma_1(p^nN)\). As a matter of fact what one is after is a Newton-Hodge inequality for the characteristic polynomial of \(U_p\) on the whole of \(S_{k+2}(\Gamma_1(p^nN))\).The method of attack is a motivic one: one looks for a piece of cohomology on which the Frobenius has characteristic polynomial equal to \(H(k,n,N,a)\). To this end one constructs a motive (with \(p\)-integral coefficients) \((\tilde{X},\Pi)\), where \(\tilde{X}\) is the resolution of singularities à la Deligne of the \(k\)-fold product \(X\) of the universal curve \(\mathcal E\) over the Igusa covering \(I=Ig_1(p^nN)\) of level \(p^n\) over the modular curve \(X_1(N)\) over \({\mathbb F}_p\). Away from the cusps one has an action of a group \(G\) that can be described explicitly. The projector \(\Pi\) will be a suitable idempotent in the group ring \({\mathbb Z}[1/2Nk!][G]\). One also has an action by \(({\mathbb Z}/p{\mathbb Z})^{\times}\), thus one can speak of the \(\chi^a\)- eigenspace.As a first result it can be shown that \(H(k,n,N,a)\) equals the characteristic polynomial of the Frobenius acting on the \(\chi^a\)- eigenspace of the \(\Pi\)-projection of \(H^{k+1}_{\text{cris}}(\tilde{X}/{\mathbb Z}_p\otimes{\mathbb Q}_p)\). To be able to act motivically a nice result is proved:The Newton polygon of the Frobenius on the part of crystalline cohomology cut out by a projector is bounded below by the Hodge polygon defined in terms of the dimensions of the Hodge cohomology groups cut out by the same projector.So the problem becomes the calculation of the \(\chi^a\)-eigenspaces of the \(\Pi\)-projection of the \(H^j(\tilde{X}, \Omega^i_{\tilde{X}})\). The bad reduction of the universal curve over \(I\) at the cusps leads to consider the log schemes \(I^{\times},\ldots,X^{\times},\tilde{X}^{\times}\) associated to \(I,\ldots,X,\tilde{X}\). In particular, one gets inclusions \(\Omega^i_{\tilde{X}}\hookrightarrow\Omega^i_{\tilde{X}^{\times}}\) inducing isomorphisms on the \(\Pi\)-projections of (Hodge) cohomology \(H^j(\Omega^i)\) for all \((i,j)\neq(k+1,0)\) or \((0,k+1)\). One also has isomorphic cohomology of \(X^{\times}\) and \(\tilde{X}^{\times}\) for all \((i,j)\) compatible with the actions of \(G\) and \(({\mathbb Z}/p{\mathbb Z})^{\times}\). One is left with the calculation of the cohomology of \(X^{\times}\).Putting everything together (the actual core of the paper taking about thirteen pages of proofs and calculations) one finds for the Hodge numbers of the \(\chi^a\)-eigenspaces of the motif \((\tilde{X},\Pi)\) the series \((l_0,l_1,\ldots,l_{k+1})\). The theorem follows. In the course of the various steps in the proof of the theorem it becomes clear why several restrictions on \(p\) (e.g.\(p>2\)), \(N\) and \(k\) are necessary. The penultimate section deals with the case of weight 3 and any \(p\). In particular the case \(p=2\) is resolved. In the final section the cases \(N\leq 4\) and \(p>3\) are studied by slightly modifying the projector \(\Pi\). Explicit series \((l_0,l_1,\ldots,l_{k+1})\) are given. Only three cases, \(k=2\), \(p=3\) and \(N=1\), 2 or 4 remain to be further analyzed. Reviewer: W.W.J.Hulsbergen (Haarlem) Cited in 2 ReviewsCited in 3 Documents MSC: 11F30 Fourier coefficients of automorphic forms 11F11 Holomorphic modular forms of integral weight Keywords:modular form; Fourier coefficients; Hodge polygon; Newton polygon; log scheme; Newton-Hodge inequality PDFBibTeX XMLCite \textit{D. L. Ulmer}, Ann. Sci. Éc. Norm. Supér. (4) 28, No. 2, 129--160 (1995; Zbl 0827.11024) Full Text: DOI Numdam EuDML References: [1] P. BERTHELOT and A. OGUS , Notes on Crystalline Cohomology , Princeton University Press, 1978 . MR 58 #10908 | Zbl 0383.14010 · Zbl 0383.14010 [2] H. COHEN and J. OESTERLÉ , Dimensions des espaces de formes modulaires in : J.-P. Serre and D.B. Zagier, Eds. Modular Functions of One Variable VI, (Lect. Notes in Math., Vol. 627, 1977 , pp. 69-73). MR 57 #12396 | Zbl 0371.10020 · Zbl 0371.10020 [3] P. DELIGNE , Formes modulaire et représentations \ell -adiques (Séminaire Bourbaki 1968 - 1969 , Lect. Notes in Math., Vol. 179, 1969 , pp. 139-172). Numdam | Zbl 0206.49901 · Zbl 0206.49901 [4] P. DELIGNE , Théorie de Hodge, II (Inst. Hautes Études Sci. Publ. Math., Vol. 40, 1972 , pp. 5-57). Numdam | Zbl 0219.14007 · Zbl 0219.14007 [5] R. HARTSHORNE , Algebraic Geometry , Berlin Heidelberg New York, Springer, 1977 . MR 57 #3116 | Zbl 0367.14001 · Zbl 0367.14001 [6] J.-I. IGUSA , On the algebraic theory of elliptic modular functions (J. Math. Soc. Japan, Vol. 20, 1968 , pp. 96-106). Article | MR 39 #1457 | Zbl 0164.21101 · Zbl 0164.21101 [7] L. ILLUSIE , Complexe de de Rham-Witt et cohomologie cristalline (Ann. Sci. Éc. Norm. Sup., Vol. 12, 1979 , pp. 501-661). Numdam | MR 82d:14013 | Zbl 0436.14007 · Zbl 0436.14007 [8] K. KATO , Logarithmic structures of Fontaine-Illusie in : J.-I. Igusa, Ed. Algebraic Analysis, Geometry and Number Theory (Proceedings of the JAMI inaugural conference, 1989 , pp. 191-224). MR 99b:14020 | Zbl 0776.14004 · Zbl 0776.14004 [9] N. KATZ , Slope Filtration of F-crystals (Astérisque, Vol. 63, 1979 , pp. 113-164). MR 81i:14014 | Zbl 0426.14007 · Zbl 0426.14007 [10] N. KATZ , p-adic properties of modular schemes and modular forms in : W. Kuyk and J.-P. Serre, Ed. Modular Functions of One Variable III (Lect. Notes in Math., Vol. 350, 1973 , pp. 69-190). MR 56 #5434 | Zbl 0271.10033 · Zbl 0271.10033 [11] N. KATZ and B. MAZUR , Arithmetic Moduli of Elliptic Curves , Princeton University Press, 1985 . MR 86i:11024 | Zbl 0576.14026 · Zbl 0576.14026 [12] N. KATZ and W. MESSING , Some consequences of the Riemann hypothesis for varieties over finite fields (Invent. Math., Vol. 23, 1974 , pp. 73-77). MR 48 #11117 | Zbl 0275.14011 · Zbl 0275.14011 [13] G. KEMPF , F. KNUDSEN , D. MUMFORD and B. SAINT-DONAT , Toroidal Embeddings I (Lect. Notes in Math., Vol. 339, 1973 ). MR 49 #299 | Zbl 0271.14017 · Zbl 0271.14017 [14] B. MAZUR , Frobenius and the Hodge filtration (Bull. Amer. Math. Soc., Vol. 78, 1972 , pp. 653-667). Article | MR 48 #8507 | Zbl 0258.14006 · Zbl 0258.14006 [15] J. S. MILNE , On a conjecture of Artin and Tate (Annals of Math., (2), Vol. 102, 1975 , pp. 517-533). MR 54 #2659 | Zbl 0343.14005 · Zbl 0343.14005 [16] J. S. MILNE , Etale Cohomology , Princeton University Press, 1980 . MR 81j:14002 | Zbl 0433.14012 · Zbl 0433.14012 [17] T. MIYAKE , Modular Forms , Berlin Heidelberg New York, Springer, 1989 . MR 90m:11062 | Zbl 0701.11014 · Zbl 0701.11014 [18] N. Nygaard , Slopes of powers of Frobenius on crystalline cohomology (Ann. Sci. Éc. Norm. Sup., Vol. 14, 1981 , pp. 369-401). Numdam | MR 84d:14011 | Zbl 0519.14012 · Zbl 0519.14012 [19] T. ODA , Convex Bodies and Algebraic Geometry , Berlin Heidelberg New York, Springer, 1985 . [20] A. J. SCHOLL , Motives for modular forms (Invent. Math., Vol. 100, 1990 , pp. 419-430). MR 91e:11054 | Zbl 0760.14002 · Zbl 0760.14002 [21] D. L. ULMER , On universal elliptic curves over Igusa curves (Invent. Math., Vol. 99, 1990 , pp. 377-391). MR 90m:11092 | Zbl 0705.14024 · Zbl 0705.14024 [22] D. L. ULMER , L-functions of universal elliptic curves over Igusa curves (Amer. J. Math., Vol. 112, 1990 , pp. 687-712). MR 91j:11050 | Zbl 0731.14013 · Zbl 0731.14013 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.