Abstract Arakelov theory and height bounds Peter Bruin Berlin, 7 November 2009 In the work of Edixhoven, Couveignes et al. (see [5] and [4]) on computing two-dimensional Galois representations associated to modular forms over finite fields, part of the output of the algorithm is a certain polynomial with rational coefficients that is approximated numerically. Arakelov s intersection theory on arithmetic surfaces is applied to modular curves in order to bound the heights of the coefficients of this polynomial. I will explain the connection between Arakelov theory and heights, indicate what quantities need to be estimated, and give methods for doing this that lead to explicit height bounds.. Motivation The motivaton for the work I am presenting is the computation of Galois representations attached to modular forms over finite fields, and of coefficients of modular forms. S. J. Edixhoven and J.-M. Couveignes have developed (in collaboration with several others) an algorithm for this; see [5] and [4]. So far it has only been worked out for modular forms of level ; in my thesis [in preparation] I intend to give a generalisation to arbitrary levels. We begin by explaining what the Galois representation attached to a modular form is. Let n and w be positive integers, and let let f be a (classical) modular form of weight w for Γ (n), with q-expansion f = a 0 + a q + a 2 q 2 +. Assume that f is an eigenform for all the Hecke operators of level n. In this case we may assume that a =, and then we have T m f = a m f for all m and for some character d f = ɛ(d)f for all d (Z/mZ) ɛ: (Z/nZ) C. Theorem.. Let E f be the number field generated by the a m, let λ be a finite place of E f, let l be the residue characteristic of λ, and let E f,λ be the completion of E f at λ. There exists a continuous representation ρ f,λ : Gal(Q/Q) GL 2 (E f,λ ) which is unramified at all primes p nl and with the property that for every such p the characteristic polynomial of a Frobenius element at p equals x 2 a p x + ɛ(p)p w. If we require ρ f,λ to be semisimple, it is unique up to isomorphism. If f is an Eisenstein series, then ρ f,λ is reducible and straightforward to write down. The situation is much more complicated when ρ f,λ is irreducible. In this case the representations were constructed by Eichler, Shimura and Igusa for w = 2, by Deligne for w > 2, and by Deligne and Serre for w =. The construction uses étale cohomology (or Tate modules in the case w = 2). From the l-adic representations ρ f,λ, we can construct reduced representations as follows. For a suitable choice of basis, the image of ρ f,λ lies has coefficients in the ring of integers of E f,λ. We can then reduce the representation modulo λ to obtain a representation ρ f,λ : Gal(Q/Q) GL 2 (k f,λ ). Here k f,λ denotes the residue field of E f at λ. (If ρ f,λ is reducible, it depends on the choice of basis above, but it becomes unique up to isomorphism if we require in addition that ρ f,λ be semi-simple.) Remark. Serre s conjecture proved recently by Khare and Wintenberger says that in fact every continuous, odd, irreducible representation over a finite field arises from a modular form.
We assume that ρ f,λ is irreducible. Furthermore, we may assume without much loss of generality that 2 w l +, since it is known that all modular representations over finite fields are twists (by some character) of representations coming from modular forms satisfying this inequality. Let K f,λ be the fixed field of the kernel of ρ f,λ ; then the question is how to find K f,λ and the (injective) homomorphism such that ρ f,λ factors as Gal(K f,λ /Q) GL 2 (k f,λ ) Gal(Q/Q) Gal(K f,λ /Q) GL 2 (k f,λ ). One key ingredient for our strategy to compute ρ f,λ is the following theorem, which follows from work of Mazur, Ribet, Gross, and others. Theorem.2. Write n = { n if w = 2; nl if 3 w l +. Let J (n ) denote the Jacobian of the modular curve (n ) over Q, and let T (n ) denote the subring of End J (n ) generated by the Hecke operators T m for m and d for d (Z/n Z). There exists a surjective ring homomorphism T (n ) k f,λ T m a m { ɛ(d) if w = 2; d ɛ(d mod n)(d mod l) w 2 if 3 w l +. We let m f,λ denote the kernel of this homomorphism; this is a maximal ideal of the Hecke algebra T (n ). We define a closed subscheme J (n )[m f,λ ], finite over Spec Q, as the intersection of the kernels of the elements of m f,λ. Note that k f,λ acts on J (n )[m f,λ ]. Theorem.3. The k f,λ [Gal(Q/Q)]-module J (n )[m f,λ ](Q) is, up to semi-simplification, a direct sum of copies of ρ f,λ. Remark. In the vast majority of cases, J (n )[m f,λ ](Q) is in fact isomorphic to ρ f,λ. assume for simplicity that this is the case. We will The question is now how to give an explicit description of ρ f,λ, or equivalently of the finite k f,λ -vector space scheme J (n )[m f,λ ] over Q. The strategy is as follows. We abbreviate = (n ), J = J (n ), g = genus() = dim(j), m = m f,λ. Fix a point O (Q) (for example a rational cusp) and consider the surjective morphism Sym g J D [D go]. This map is an isomorphism above a dense open subset of J, and we assume again for simplicity that it is an isomorphism above J[m]. Then J[m] is isomorphic via the above map to a closed subscheme D of Sym g. In other words, every point x J[m](Q) corresponds to a unique divisor D x of degree g on Spec Q such that x = [D x go]. We choose a function ψ: P Q 2
such that the map ψ : Sym g Sym g P Q P g Q is a closed immersion on D; the last isomorphism comes from the symmetrisation map Σ g : (P ) g P g given by the elementary symmetric functions. Furthermore, we choose a rational map γ: P g Q P Q that is well defined and a closed immersion on the image of D in P g Q. We have now constructed a closed immersion Ψ: J[m] P Q. Let F Ψ Q[u, v] be the homogeneous polynomial (defined up to multiplication by an element of Q ) whose zero locus V is the image of Ψ. The idea is now to approximate F Ψ to sufficient precision, and to reconstruct F Ψ from this approximation. Remark. There are various ways to obtain such an approximation of F Ψ. J. G. Bosman [] has done computations using computations on the complex modular curve (n )(C). Using algorithms for computing in Jacobians of curves over finite fields invented by K. Khuri-Makdisi [8], J.-M. Couveignes [3] and myself [2], F Ψ can also be computed modulo many small prime numbers. Remark. Of course, to describe J[m] as a k f,λ -vector space scheme, we need some extra structure. Let A denote the affine coordinate ring of V. Then the extra structure consists of a Q-algebra homomorphism m # : A A Q A describing addition, as well as Q-algebra endomorphisms α # : A A for all α k f,λ, describing scalar multiplication. It is possible to compute this extra structure together with A, but we will not describe this. To be able to reconstruct F Ψ from an approximation, we need a bound on the height h(f Ψ ) of F Ψ. This is simply the maximum of the absolute values of the coefficients of F Ψ when these coefficients are chosen to be coprime integers. 3
2. First estimates We first apply some preliminary estimates for the height of F Ψ. For this it is useful to talk about heights in slightly greater generality. Let r be a non-negative integer and let x = (x 0 : x :... : x r ) be a point of P r (Q). The height of x is defined as follows. If K is a number field over which x is defined, then h P r(x) = [K : Q] log max{ x 0 v,..., x r v }, v where v runs over all places of K and t v is the corresponding absolute value, normalised such that multiplication by t on the local field K v scales the Haar measure by a factor t v. We use the following results without proof; they follow from basic considerations about valuations. Lemma 2.. Let Σ r : (P ) r P r be the symmetrisation map. Then for all p,..., p r in P r (Q), have r h P r(σ r (p,..., p r )) r log 2 + h P (p i ). Lemma 2.2. Let γ: P r P s be a rational map given by a non-zero (s + ) (r + )-matrix over Q. Let h(γ) be the height of this matrix, viewed as an element of P rs+r+s (Q). Then for any p P r (Q) such that γ is defined at p, i= h P s(γ(p)) log r + h(γ) + h P r(p). For every x J[m](Q) we decompose the corresponding divisor D x as D x = P x, + + P x,g with P x,i (Q). Then a computation using the above two lemmata gives the inequality h(f Ψ ) g h P (ψ(p x,i )) x J[m](Q) i= + #J[m](Q) ((g + ) log 2 + log g + h(γ) ). (2.) The next step is to study h P (ψ(p x,i )) using Arakelov theory. 4
3. Basics of Arakelov theory We start with some analytic definitions. Let be a Riemann surface of genus g, and let (α,..., α g ) be an orthonormal basis of Ω () with respect to the inner product α, β = i 2 The canonical (, )-form on is defined as µ = i 2g α β. g α j ᾱ j. j= The canonical Green function of is the unique smooth function gr outside the diagonal on such that πi gr (, y) = µ δ y and gr (, y)µ = 0 for every y. Let K be a number field, and let Z K be its ring of integers. We write K fin and K inf for the sets of finite and infinite places of K. For every v K inf the completion K v (which is isomorphic to R or C) we choose an algebraic closure K v ; we do not need to choose an isomorphism K v = C. Definition. An arithmetic surface over Z K is a projective flat scheme ZK whose geometric fibres are semi-stable curves and whose generic fibre K is smooth. Remark. We have adopted the definition of Moret-Bailly [9]; it implies that ZK is normal. (Faltings has the stronger requirement that ZK be regular.) This definition has the advantage that is stable under any base change of the form Spec Z L Spec Z K, where L is a finite extension of K. The assumption that the geometric fibres are semi-stable is to ensure that the relative dualising sheaf Ω /ZK exists and is a line bundle. For each f K inf, we define v to be the Riemann surface K ( K v ). A metrised line bundle on is a line bundle L together with a Hermitean metric v on the line bundle L v on the Riemann surface v for each v K inf. Now assume that the fibres of ZK are of genus g. Then for each v K inf we have the canonical (, )-form µ v on v and the canonical Green function gr v on f v f v ; these are independent of an identification of Kv with C. A metrised line bundle L is admissible if v is smooth for each v and for some (hence any) local generating section s we have πi log s v = (deg L)µ v. Given L, there is a one-dimensional family of admissible metrics on each of the L v. Furthermore, if D is a Cartier divisor on ZK, there is a natural admissible metric on the line bundle O (D), given by log v (x) = P v n P gr v (x, P ) for x outside the support of D v = P v n P P. Finally, the dualising sheaf Ω /ZK, which is a line bundle since ZK is semi-stable, has a canonical admissible metric. Let Pic ZK denote the group of isomorphism classes of admissible line bundles on ZK. Then we have a symmetric bilinear intersection pairing Pic ZK Pic ZK R (L, M) (L. M). 5
If D and E are Cartier divisors without common components, we have (O (D). O (E)) = v K fin log #k(v) (O (D). O (E)) v + v K inf [K v : R](O (D). O (E)) v, where (O (D). O (E)) v is the sum of the usual local intersection numbers at points above v if v is finite, and (O (D). O (E)) v = P,Q v m P n Q gr v (P, Q) if v is an infinite place and D v = P v m P P, E v = Q v n Q Q. 4. Applying Arakelov theory We return to estimating the height of the polynomial F Ψ, starting from (2.). Let K be a number field such that all the points P x,i are K-rational and such that has a semi-stable model ZK over Spec Z K. We extend each P x,i to a section After blowing up ZK P x,i : Spec Z K ZK. if necessary, the morphism ψ: P Q extends to a morphism ψ: ZK P Z K. By composing these morphisms, we extend ψ(p x,i ) to a section ψ(p x,i ): Spec Z K P Z K. We endow the line bundle O P ( ) on P Z K with the metric defined by log OP ( )(z) = log max{, z }. Then it follows from the definition of local intersection numbers (at the finite and infinite places of K) that h P (ψ(p x,i )) = [K : Q] (O P ( ). ψ(p x,i)). Now we apply the projection formula; here we have to be careful since the pull-back of the given metric on O P ( ) differs from the canonical admissible metric on O P (ψ ). After compensating for this, we obtain ( h P (ψ(p x,i )) = (O (P x,i ). O (ψ )) +, (4.) [K : Q] ) [K v : R]φ v ((P x,i ) v ) v K inf where φ v is the real-valued function on v defined by φ v (x) = i gr v (x, y)d log ψ(y) + log max{, ψ(y) }µ v (y); y v y v ψ(y) = this can be estimated (independently of x) by φ v (x) (deg ψ) sup v v gr v + y v log max{, ψ(y) }µ v (y). Remark. Equivalently, we have φ v (x) = gr v (x, y)ψ µ P (y) + log max{, ψ(y) }µ v (y), y v y v where µ P is the current on P (C) given by µ P (χ) = α=0 χ(exp(it))dt. The function log{, z } on P (C) is a Green function for this current. 6
where Combining (4.), (2.) and the estimate for φ v, we get h(f Ψ ) [K : Q] (O (D). O (ψ )) + #J[m](Q) ( (g + ) log 2 + log g + h(γ) + g(deg ψ) sup gr + g log max{, ψ }µ ), and where D is the divisor defined by D = = (n )(C) x #J[m](Q) 5. Sketch of what remains to be done We take ψ of the form ψ = f 2 /, where f 2 is some cusp form of weight 2 and is the usual discriminant modular form. Let us pretend that (n ) has a smooth model Z over Spec Z (even though we know that this is not the case). Then we can have h(f Ψ ) (O (D). O (ψ )) + #J[m](Q) ( (g + ) log 2 + log g + h(γ) + g(deg ψ) sup gr + g log max{, ψ }µ ), where the intersection number is now taken on the (imaginary) smooth model of (n ) over Spec Z. We assume in addition that we do not have to blow up Z in order to make ψ well defined. Then ψ is an effective linear combination of cusps. Using a lot of Arakelov theory (the adjunction formula, Faltings s arithmetic Riemann Roch theorem, the Faltings Hriljac formula relating intersection numbers to Néron Tate heights) one can then derive the estimate ( (O (D). O (ψ )) #J[m](Q)(deg ψ) Here h Faltings () is the Faltings height of, defined by D x. 3πg(g ) sup gr + g h Faltings () + h Faltings () = deg det Rπ(, O ), sup α,α = sup α Ω g(g ) (O. Ω /Z ) 2 where π: Spec Z is the structure morphism. By an unpublished result of Bost, there is a constant B such that the Faltings height of any curve of genus g over a number field is at least Bg. Again, all of the above is under the assumption that the map from Sym g to J Q is injective above J[m]. If this is not the case, we need in addition Zhang s bound for the Néron Tate height below which there exist infinitely many algebraic points. Furthermore, we still have to take into account the fact that (n ) is not smooth over Spec Z and that we may need to blow up (n ) in order to make ψ well defined. One also has to show that γ can be taken such that h(γ) is small, and that ψ can be taken such that log max{, ψ }µ is small. In the end, everything is reduced to estimating the following quantities related to (n ): () the canonical Green function; (2) suprema of cusp forms of weight two; (3) integrals of the form log max{, ψ }µ, where ψ is of the form f 2 / with f 2 a cusp form of weight 2. I am currently working on estimates that are as explicit as possible, using the spectral theory for the Laplace operator on (n )(C). In part, this work builds on ideas of Jorgenson and Kramer; see [6] and [7]. 7 ).
References [] J. G. Bosman, Explicit computations with modular Galois representations. Proefschrift, Universiteit Leiden, 2008. [2] P. J. Bruin, Computing in Picard groups of curves over finite fields. Notes of a talk held at the Institut für Experimentelle Mathematik, Essen, 0 November 2009. Available online: http://www.math.leidenuniv.nl/~pbruin/. [3] J.-M. Couveignes, Linearizing torsion classes in the Picard group of algebraic curves over finite fields. Journal of Algebra 32 (2009), 2085 28. [4] J.-M. Couveignes and S. J. Edixhoven (editors), Computational aspects of modular forms and Galois representations. Princeton University Press, to appear. [5] S. J. Edixhoven (with J.-M. Couveignes, R. S. de Jong, F. Merkl and J. G. Bosman), On the computation of coefficients of a modular form. Preprint, 2006/2009. Available online: http://arxiv.org/abs/math.nt/0605244. [6] J. Jorgenson and J. Kramer, Bounding the sup-norm of automorphic forms. Geometric and Functional Analysis 4 (2004), no. 6, 267 277. [7] J. Jorgenson and J. Kramer, Bounds on canonical Green s functions. Compositio Mathematica 42 (2006), no. 3, 679 700. [8] K. Khuri-Makdisi, Asymptotically fast group operations on Jacobians of general curves. Mathematics of Computation 76 (2007), no. 260, 223 2239. [9] L. Moret-Bailly, Métriques permises. Dans: L. Szpiro, Séminaire sur les pinceaux arithmétiques : la conjecture de Mordell, Astérisque 27 (985), 29 87. 8