Algebraic solutions of differential equations over the projective line minus three points
AALGEBRAIC SOLUTIONS OF DIFFERENTIAL EQUATIONSOVER P − { , , ∞} YUNQING TANG
Abstract.
The Grothendieck–Katz p -curvature conjecture predicts that anarithmetic differential equation whose reduction modulo p has vanishing p -curvatures for almost all p, has finite monodromy. It is known that it sufficesto prove the conjecture for differential equations on P − { , , ∞} . We prove avariant of this conjecture for P − { , , ∞} , which asserts that if the equationsatisfies a certain convergence condition for all p, then its monodromy is trivial.For those p for which the p -curvature makes sense, its vanishing implies ourcondition. We deduce from this a description of the differential Galois groupof the equation in terms of p -curvatures and certain local monodromy groups.We also prove similar variants of the p -curvature conjecture for the ellipticcurve with j -invariant minus its identity and for P − {± , ± i, ∞} . Introduction
The Grothendieck–Katz p -curvature conjecture was originally raised as a ques-tion on linear homogeneous systems of first-order differential equations (see Con-jecture (I) in [Kat72, Introduction] for more details) d y dx = A ( x ) y . Here A ( x ) is a square matrix of rational functions of x with coefficients in somenumber field K and y is a vector-valued function. For all but finitely many primes p of K , it makes sense to reduce this system modulo p and to define an invariant,the p -curvature, in terms of the resulting system. According to the conjecture, if almost all (that is, all but finitely many) p -curvatures vanish, then the originalsystem admits a full set of solutions in algebraic functions.The conjecture generalizes to a smooth variety X equipped with a vector bundlewith an integrable connection ( M, ∇ ) defined over some number field K . It isknown that the general version of the conjecture reduces to the case when X = P K − { , , ∞} . (See [Bos01, 2.4.1], [Kat82, Thm. 10.5], and [And04, 7.1.4]).In this paper, we prove a variant of the conjecture for X = P K − { , , ∞} wherethe condition for almost all p is replaced by a condition for all p . A slightly informalformulation of our main theorem is the following: Theorem. (Theorem 2.2.1) Let ( M, ∇ ) a vector bundle with a connection over X = P K − { , , ∞} . If the p -curvature of ( M, ∇ ) vanishes for all p , then ( M, ∇ ) admits a full set of rational solutions, that is, M ∇ =0 generates M as an O X -module. Let us explain the meaning of the condition of vanishing p -curvature at all primes p : at primes where p -curvature is either not defined or non-vanishing, we impose acondition on the p -adic radius of convergence of the horizontal sections of ( M, ∇ ) .When ( M, ∇ ) has an integral model at a prime p so that one can make sense of a r X i v : . [ m a t h . AG ] S e p YUNQING TANG its reduction mod p , this convergence condition is implied by the vanishing of the p -curvature.One can also extend the notion of vanishing p -curvature for all p to vector bundleswith connections over smooth algebraic curves equipped with a semistable modelover O K . However, the property of all p -curvature vanishing is not preserved underpush-forward along finite maps from the curve in question to P − { , , ∞} . There-fore, one cannot deduce from the above theorem that vanishing p -curvature for all p implies trivial monodromy in the case of arbitrary algebraic curves. Nevertheless,when X is an elliptic curve with j -invariant minus its identity point, we prove: Theorem. (Theorem 6.0.2) Let X ⊂ A Z be the affine curve defined by y = x ( x − x + 1) and let ( M, ∇ ) be a vector bundle with a connection over X . If the p -curvature of ( M, ∇ ) vanishes for all p , then ( M, ∇ ) is étale locally trivial. Namely,there exists a finite étale map f : Y → X such that f ∗ ( M, ∇ ) is isomorphic to ( O rk MY , d ) , where d is the differential operator on regular functions. Unlike the previous case, passing to a finite étale cover is necessary. We give anexample of an ( M, ∇ ) with G gal equal to Z / Z .Katz has shown in [Kat82, Thm. 10.2] that if the p -curvature conjecture holds,then for any vector bundle with an integrable connection ( M, ∇ ) on a smoothvariety X over K as above, the Lie algebra g gal of the differential Galois group G gal of ( M, ∇ ) is in some sense generated by the p -curvatures. Namely, let K ( X ) bethe function field of X . The p -curvature conjecture implies that g gal is the smallestalgebraic Lie subalgebra of gl n ( K ( X )) such that for almost all p the reduction of g gal mod p contains the p -curvature.We use Theorem 2.2.1 to prove a result analogous to Katz’s theorem when X = P K − { , , ∞} . Of course, this result (Theorem 2.2.5) involves a condition at everyprime p , but as a compensation we describe G gal and not only its Lie algebra. In thegeometric case, namely when ( M, ∇ ) is the relative de Rham cohomology with theGauss–Manin connection, this extra local condition is often vacuous. We discussthe example of the Legendre family (Remark 3.3.2) and show that a variant of ourresult implies that g gal is generated by the p -curvatures, which recovers a result ofKatz.The main tools used to prove Theorem 2.2.1 and Theorem 6.0.2 are the alge-braicity results of André [And04, Thm. 5.4.3] and Bost–Chambert-Loir [BCL09,Thm. 6.1, Thm. 7.8]. These results generalize the classical Borel–Dwork criterionfor the rationality of a formal power series. This type of results requires estimatingthe radius of convergence of solutions for ( M, ∇ ) at each place of K . These tech-niques have been used previously by André [And04, Sec. 6] and Bost [Bos01, 2.4.2]to study the Grothendieck–Katz conjecture in the case when the algebraic mon-odromy group of ( M, ∇ ) is solvable. The paper is organized as follows. In section 2 to 5, we will focus on the casewhen X = P K − { , , ∞} . In section 6, we discuss the case when X is the affineelliptic curve with j -invariant .In section 2, we formulate our main result, and in particular the condition whichsubstitutes for the vanishing of the p -curvature when it does not make sense toreduce ( M, ∇ ) mod p . We then use the main result to deduce a description of thedifferential Galois group following Katz.In section 3, we use the criterion in [And04] to prove that a vector bundle with aconnection ( M, ∇ ) , as in the theorem, is locally trivial for the étale topology of X . LGEBRAIC SOLUTIONS OF DIFFERENTIAL EQUATIONS OVER P − { , , ∞} To do this, we apply André’s criterion to the formal horizontal sections of ( M, ∇ ) centered at a specific point x . We obtain a lower bound for André’s analogueof their radii of convergence at archimedean places, using the uniformization of P C − { , , ∞} by the unit disc, which arises from its interpretation as the modulispace of elliptic curves with level 2 structure. The chosen point x correspondsto the elliptic curve with smallest stable Faltings’ height and we use the Chowla-Selberg formula to deduce the lower bound. We also discuss in this section somevariants of our main theorem and an example of the Legendre family mentionedabove.In section 4, we apply the rationality criterion in [BCL09] to prove the maintheorem. We give a lower bound for the local capacity of Ω , the image in P C −{ , , ∞} of a standard fundamental domain for Γ(2) under the uniformizationmentioned above. Together with the algebraicity of our formal solution proved insection 3, this allows us to apply the criterion in [BCL09], and deduce that thesolutions of ( M, ∇ ) are rational.Section 5 is devoted to an interpretation of our computations in section 3 in termsof the stable Faltings height, obtained by relating our estimate for archimedeanplaces to the Arakelov degree of the restriction of the tangent bundle to somepoint.In section 6, we prove our theorem when X is an affine elliptic curve with j -invariant using André’s criterion and ideas in section 3. As in section 2, wedefine the notion of p -curvature vanishing at bad primes using local convergencecondition. Using the property of theta functions and Weierstrass- ℘ functions, wededuce from a result of Eremenko [Ere] a lower bound of the archimedean radii.In section 7, we first give an example of an ( M, ∇ ) over the affine elliptic curvein section 6 such that its p -curvatures vanish for all p but its G gal is Z / Z . Moreprecisely, ( M, ∇ ) is the push-forward of ( O , d ) via the degree two self-isogeny ofthe elliptic curve. In the second half, we discuss a variant of our main theoremswhen X is A − {± , ± i } with the conclusion that ( M, ∇ ) has finite monodromy.The proof relies on the result of Eremenko used in last section. We also give anexample to show that even when ( M, ∇ ) has good reduction everywhere and all its p -curvatures vanish, it can still have local monodromies of order two around thesingular points ± , ± i, ∞ . Acknowledgement
I thank Mark Kisin for introducing this problem to me and all the enlighteningdiscussions. I thank Yves André, Noam Elkies, Hélène Esnault, and Benedict Grossfor useful comments. Moreover, I am grateful to Cheng-Chiang Tsai for conversa-tions related to this topic and to George Boxer, Kęstutis Česnavičius, Chao Li,Andreas Maurischat, Koji Shimizu, Junecue Suh, and Jerry Wang for commentson drafts of the paper.2.
Statement of the main results
Let K be a number field and O K its ring of integers. Let X be P O K − { , , ∞} and M a vector bundle with a connection ∇ : M → Ω X K ⊗ M over X K . For a finiteplace v of K lying over a prime p , let K v be the completion of K with respect to v and denote by O v and k v the ring of integers and residue field of K v . For Σ afinite set of finite rational primes, we set O K, Σ = O K [1 /p ] p ∈ Σ ⊂ K. YUNQING TANG
The p -curvature and p -adic differential Galois groups. . For Σ , as above, sufficiently large, ( M, ∇ ) extends to a vector bundle withconnection (again denoted ( M, ∇ ) ) over X O K, Σ . In particular, if p / ∈ Σ we canconsider the pull back of ( M, ∇ ) to X ⊗ Z /p Z . If D is a derivation on X ⊗ Z /p Z ,so is D p . Let ∇ ( D ) be the map ( D ⊗ id) ◦ ∇ . Then on X ⊗ Z /p Z , the p -curvature is given by (see [Kat82, Sec. VII] for details) ψ p ( D ) := ∇ ( D p ) − ∇ ( D ) p ∈ End O X ⊗ Z /p Z ( M ⊗ Z /p Z ) . In particular, ψ p (cid:0) ddx (cid:1) = − (cid:0) ∇ (cid:0) ddx (cid:1)(cid:1) p . Since ψ p ( D ) is p -linear in D , for X = P O K − { , , ∞} , the equation ψ p ≡ is equivalent to − (cid:0) ∇ (cid:0) ddx (cid:1)(cid:1) p ≡ .In general, the ψ p depends on the choice of extension of ( M, ∇ ) over X O K, Σ . However, any two such extensions are isomorphic over X O K, Σ (cid:48) for some sufficientlylarge Σ (cid:48) . . Let L be a finite extension of K and w a place of L over v . We view L as asubfield of C p via w. Fix an x ∈ X ( L w ) . Given a positive real number r , we denoteby D ( x , r ) the open rigid analytic disc of radius r, with center x . Thus D ( x , r ) = { x ∈ X ( C p ) such that | x − x | p < r } , where | · | p is normalized so that | p | p = p − .Let M ∨ be the dual vector bundle of M . It is naturally endowed with theconnection such that for any local sections m, l of M and M ∨ respectively, d (cid:104) l, m (cid:105) = (cid:104)∇ M ∨ ( l ) , m (cid:105) + (cid:104) l, ∇ M ( m ) (cid:105) . Definition 2.1.3. If ( V, ∇ ) is a vector bundle with connection over some schemeor rigid space, we denote by (cid:104) V, ∇(cid:105) ⊗ , or simply (cid:104) V (cid:105) ⊗ , if there is no risk of confusionregarding the connection ∇ , the category of ∇ -stable sub quotients of all the tensorproducts V ⊗ m ⊗ ( V ∨ ) ⊗ n for m, n ≥ . If the scheme or rigid space over which V isa vector bundle is connected, then this is a Tannakian category. Definition 2.1.4.
Let F w be the field of fractions of the ring of all rigid analyticfunctions on D ( x , r ) and η w : Spec( F w ) → X the natural map. Consider the fiberfunctor η w : (cid:104) M | D ( x ,r ) (cid:105) ⊗ → Vec F w ; V (cid:55)→ V η w . The p -adic differential Galois group G w ( x , r ) is defined to be the automorphismgroup Aut ⊗ η w of η w .For v | p a finite place of K, we will say that ( M, ∇ ) has good reduction at v if ( M, ∇ ) extends to a vector bundle with connection on X O v . The following lemmagives the basic relation between the p -curvature and the p -adic differential Galoisgroup. Lemma 2.1.5.
Let x ∈ X ( O L w ) and suppose that ( M, ∇ ) has good reduction at v. If the p -curvature vanishes, then the local differential Galois group G w ( x , p p ( p − ) is trivial. We could have defined the p -curvatures by considering derivations on X k v for v a place of K. For primes which are unramified in K, the two definitions are essentially equivalent, and thepresent definition will allow us to formulate the inequalities which arise below in a more uniformmanner. LGEBRAIC SOLUTIONS OF DIFFERENTIAL EQUATIONS OVER P − { , , ∞} Proof.
To show that G w ( x , p p ( p − ) is trivial, we have to show that the restrictionof M to D ( x , p p ( p − ) admits a full set of solutions. It is well known that this is thecase when ψ p ≡ , but for the convenience of the reader we sketch the argument.See [Bos01, section 3.4.2, prop. 3.9] for related arguments.Assume there is an extension of ( M, ∇ ) to a vector bundle with connection ( M , ∇ ) over X O v . If m is any section of M , then a formal section in the kernelof ∇ is given by m = ∞ (cid:88) i =0 ∇ (cid:18) ddx (cid:19) i ( m ) ( x − x ) i i ! ( − i . Since ψ p ≡ (recall that this means the p -curvature vanishes on X O v ⊗ Z /p Z ), wehave ∇ ( ddx ) p ( M ) ⊂ p M . Hence ∇ ( ddx ) i ( m ) ⊂ p [ ip ] M , and one sees easily that theseries defining m converges on D ( x , p p ( p − ) . (cid:3) Remark . (1) Unlike the notion of p -curvature, the definition of G w ( x , r ) does not require ( M, ∇ ) to have good reduction. It depends only on the O v -model of X (which we of course always take to be P O v − { , , ∞} ), which is used todefine D ( x , r ) , but not on how ( M, ∇ ) is extended.(2) If ( M, ∇ ) has good reduction with respect to X O v and it admits a Frobeniusstructure with respect to some Frobenius lifting on X O v , then G w ( x , istrivial whenever x ∈ X ( O v ) . See for example [Ked10, 17.2.2, 17.2.3].From now on we set x = √ i , which corresponds to the elliptic curve withsmallest stable Faltings height. In section 5, we will give a theoretical explanation ofwhy this choice gives the best possible estimates. We set G w = G w (cid:0) √ i , p − p ( p − (cid:1) , and we take L to be a number field containing K ( √ i ) . By Lemma 2.1.5, the local differential Galois group G w is trivial when the vectorbundle with connection ( M, ∇ ) has good reduction over v , and ψ p ≡ . Thismotivates the following definition: Definition 2.1.7.
We say that the p -curvatures of ( M, ∇ ) vanish for all p if(1) ψ p ≡ for all but finitely many p ,(2) G w = { } for all primes w of L .By what we have just seen, for all but finitely many p, the condition (1) makessense, and implies (2). Thus (2) is only an extra condition at finitely many primes.As above, the definition does not depend on the extension of ( M, ∇ ) to X O K, Σ orthe choice of primes Σ . The main theorem and a Tannakian consequence.Theorem 2.2.1.
Let ( M, ∇ ) be a vector bundle with a connection over X K = P K − { , , ∞} , and suppose that the p -curvatures of ( M, ∇ ) vanish for all p. Then ( M, ∇ ) admits a full set of rational solutions. The proof of this theorem is the subject of sections 3, 4.
Remark . By varying the conditions on the radii of convergence in (2), onecan prove variants of Theorem 2.2.1, whose conclusion is that ( M, ∇ ) has finitemonodromy. See Remark 3.3.2 for details. YUNQING TANG
André has pointed out that, if one replaces (2) in Definition 2.1.7 by the conditionthat the so called generic radii of all formal horizontal sections of ( M, ∇ ) are atleast p − p ( p − , then the analogue of Theorem 2.2.1 admits an easier proof. Indeedif w | p, and the w -adic generic radius is at least p − p ( p − , then by [BS82, Sec. IV], p cannot divide the (finite by (1) and Katz’s theorem [Kat70, Thm. 13.0]) order ofthe local monodromies. If this condition holds for all w, then the local monodromiesaround , , ∞ are all trivial and hence the global monodromy is trivial.Once one uses (1) to show that the local monodromies are finite, this argumentis ‘prime by prime’. We do not know if Theorem 2.2.1 admits a similar proof,which avoids global arguments, although this seems to us unlikely. In any case, ourmethod allows us to deal with some cases when X is an affine elliptic curve or theprojective line minus more than three points. See Theorem 6.0.2 and Proposition7.2.1. The conclusion of both results is that ( M, ∇ ) has finite monodromy and wewill give examples in section 7 with nontrivial monodromy. It seems unlikely thatthese results can be proved with a ‘prime by prime’ argument.Applying Lemma 2.1.5, we have the following corollary: Corollary 2.2.3. If ( M, ∇ ) is defined over X Z and the p -curvature vanishes forall primes, then ( M, ∇ ) admits a full set of rational solutions. . As in [Kat82], we can use our main theorem to give a description of thedifferential Galois group of any vector bundle with a connection ( M, ∇ ) over X K .Let K ( X ) be the function field of X K . Let ω be the fibre functor on (cid:104) M (cid:105) ⊗ givenby restriction to the generic point of X K . Write G gal = Aut ⊗ ω ⊂ GL( M K ( X ) ) forthe corresponding differential Galois group (see [Kat82, Ch. IV] and [And04, 1.3,1.4]).Let G be the smallest closed subgroup of GL( M K ( X ) ) such that:(1) For almost all p, the reduction of Lie G mod p contains ψ p . (2) G ⊗ F w contains G w for all w, where, as above, F w is the field of fractionsof the ring of rigid analytic functions on D (cid:0) x , p − p ( p − (cid:1) .Let g be the smallest Lie subalgebra of GL( M K ( X ) ) such that for almost all p, the reduction of g mod p contains ψ p . As proved in [Kat82, Prop. 9.3], g is containedin Lie G gal . Moreover, G w is contained in G gal ⊗ F w by definition. Hence G is asubgroup of G gal . We will see from the proof of the following theorem that (in thepresence of the condition (1)), to define G we only need to impose the condition(2) at finitely many primes. Theorem 2.2.5.
Let ( M, ∇ ) be a vector bundle with a connection defined over X K = P K − { , , ∞} . Then G = G gal .Proof. We follow the idea of the proof of Theorem 10.2 in [Kat82]. See also [And04,Prop. 3.2.2].By a theorem of Chevalley, there exists W in (cid:104) M (cid:105) ⊗ and a line L (cid:48) ⊂ W K ( X ) such that G is the intersection of G gal with the stabilizer of L (cid:48) . Let W (cid:48) be thesmallest ∇ -stable submodule of W K ( X ) containing L (cid:48) . Then W (cid:48) has a K ( X ) -basisof the form { l, ∇ l, · · · , ∇ r − l } where l ∈ L (cid:48) , r = rk W (cid:48) , and we have written ∇ i l for ∇ ( ddx ) i ( l ) . Replacing W by W (cid:48) ∩ W, we may assume that W K ( X ) = W (cid:48) . Then L = L (cid:48) ∩ W is a line bundle in W. As above, let g be the smallest algebraic Lie subalgebra of GL( M K ( X ) ) suchthat for almost all p the reduction of g mod p contains ψ p . Let Σ be a finite set LGEBRAIC SOLUTIONS OF DIFFERENTIAL EQUATIONS OVER P − { , , ∞} of primes of Q such that ( M, ∇ ) extends to a vector bundle M with connection ∇ : M → M ⊗ Ω X O K, Σ over X O K, Σ , and g mod p contains ψ p for p / ∈ Σ . We alsoassume that Σ contains all primes p ≤ r. Let U ⊂ X O K, Σ be a non-empty open subset such that l ∈ L | U , L and W ex-tend to vector bundles with connection L and W respectively, in (cid:104)M| U (cid:105) ⊗ , and { l, ∇ l, · · · , ∇ r − l } forms a basis of W . Let N := Sym r W ⊗ (det W ∨ ) with the in-duced connection. The argument in [Kat82] implies that for p / ∈ Σ , the p -curvatureof ( N , ∇ ) vanishes. Let N := N X K ∩ U . We will use the condition (2) in the defini-tion of G to show that G w acts trivially on N η w . We already know this for p / ∈ Σ , by Lemma 2.1.5. Thus we will only need to use (2) for p ∈ Σ . Assuming this fora moment, we can apply Theorem 2.2.1 to ( N, ∇ ) and conclude that it has trivialglobal monodromy. Hence G gal acts as a scalar on W . In particular, G gal stabilizes L so, by the definition of L, G gal = G, Let D := D ( x , p − p ( p − ) . Recall that the category (cid:104) M | D ( x ,r ) (cid:105) ⊗ ⊗ F w is ob-tained from (cid:104) M | D ( x ,r ) (cid:105) ⊗ by taking the same collection of objects and tensoringthe morphisms by F w . By the definition of L , the group G w acts as a charac-ter χ on L η w . The morphism L η w → W η w is a map between G w -representations.By the equivalence of categories between (cid:104) M | D ( x ,r ) (cid:105) ⊗ ⊗ F w and the category oflinear representations of G w over F w , this morphism is a finite F w -linear combi-nation of maps L | D → W D in (cid:104) M | D ( x ,r ) (cid:105) ⊗ . In other words, there are a finitenumber of ∇ -stable line bundles W i ⊂ W D , with G w acting on W i,η w as χ suchthat L | D ⊂ (cid:80) W i . In particular, l | D = (cid:80) a i · w i , where a i ∈ F w and w i ∈ W i . Since (cid:80) W i is ∇ -stable, ∇ n l ∈ (cid:80) W i and G w acts as χ on ∇ n l | D . As W η w is generatedby { l, ∇ l, · · · , ∇ r − l }| D , the group G w acts as χ on W η w . Hence G w acts triviallyon N η w . (cid:3) Using the same idea as in the last paragraph of the proof above, we have thefollowing lemma which is of independent interest.
Lemma 2.2.6.
Let H w ⊂ G gal be the smallest closed subgroup such that G w ⊂ H w ⊗ K ( X ) F w . Then H w is normal in G gal .Proof. We need the following fact (see [And92, Lem. 1]): Assume that G is aalgebraic group over some field E . Let H ⊂ G be a closed subgroup and V an E -linear faithful algebraic representation of G . Then H is a normal subgroup of G if for every tensor space V m,n := V ⊗ m ⊗ ( V ∨ ) ⊗ n , and for every character χ of H over E , G stabilizes ( V m,n ) χ , the subspace of V m,n where H acts as χ . If G isconnected, then these two conditions are equivalent.We apply this result to H w ⊂ G gal and V = M K ( X ) . Let L ⊂ V m,n be a line,and W ⊂ V m,n the smallest ∇ -stable subspace containing L. It suffices to showthat, if H w acts via χ on L , then H w acts via χ on W. This shows that ( V m,n ) χ is ∇ -stable, and hence that G gal stabilizes ( V m,n ) χ .As in the proof of the theorem above, G w acts on W via χ. Hence H w is containedin the subgroup of G gal which acts on W via χ. (cid:3) Algebraicity: an application of André’s theorem
The main goal of this section is to prove a weaker version of Theorem 2.2.1.Namely, that if ( M, ∇ ) is a vector bundle with a connection over X K = P K −{ , , ∞} all of whose p -curvatures vanish, then ( M, ∇ ) admits a full set of algebraicsolutions. YUNQING TANG
André’s algebraicity criterion. . As the coordinate ring of X K a principal ideal domain, M is free. Hence wemay view ∇ as a system of first-order homogeneous differential equations. Thus M ∼ = O mX K and ∇ ( ddx ) y = d y dx − A ( x ) y , where y is a section of M , x is the coordinateof X , and A ( x ) is an m × m matrix with entries in O X K = K [ x ± , ( x − ± ] .As above, we set x = (1 + √ i ) . If y ∈ L m , there exists y ∈ L [[ x − x ]] m such that y ( x ) = y and ∇ ( y ) = 0 . Our goal is to show that if the p -curvaturesof ( M, ∇ ) vanishes for all p, then y is algebraic.3.1.2 . Now let y ∈ K [[ x ]] , and let v be a place of K . If v is finite, we denote by p thecharacteristic of the residue field. Let | · | v be the v -adic norm normalized so that | p | v = p − [ Kv : Q p ][ K : Q ] if v is finite, and | x | v = | x | − [ Kv : R ][ K : Q ] ∞ for x ∈ K, if v is archimedean,where | x | ∞ denotes the Euclidean norm on K v . When there is no confusion, we willalso write | · | for | · | ∞ . For a positive real number R, we denote by D v (0 , R ) therigid analytic z -disc of v -adic radius R. That is D v (0 , R ) is defined by the inequality | z | v < R. We first state the definition of v -adic uniformization and the associated radius R v defined in André’s paper ([And04, Definition 5.4.1]). Definition 3.1.3. (1) For R ∈ R + , a v -adic uniformization of y by D v (0 , R ) is a pair of meromor-phic v -adic functions g ( z ) , h ( z ) on D v (0 , R ) such that h (0) = 0 , h (cid:48) (0) = 1 and y ( h ( z )) is the germ at of the meromorphic function g ( z ) .(2) Let R v be the supremum of the set of positive real R for which a v -adicuniformization of y by D v (0 , R ) exists. We call R v the v -adic radius (ofuniformizability) .3.1.4 . In order to state the algebraicity criterion, we need to introduce two con-stants τ ( y ) , ρ ( y ) , which play similar roles as the global-boundedness condition inthe Borel–Dwork rationality criterion. Let y = (cid:80) ∞ n =0 a n x n . We define τ ( y ) = inf l lim sup n (cid:88) v, p ≥ l n sup j ≤ n log + | a j | v ,ρ ( y ) = (cid:88) v lim sup n n sup j ≤ n log + | a j | v , where log + is the positive part of log , that is log + ( a ) = log( a ) if a > and is zerootherwise. The following is a slight reformulation of André’s criterion. Theorem 3.1.5. ( [And04, Theorem 5.4.3] ) Let y ∈ K [[ x ]] such that τ ( y ) = 0 and ρ ( y ) < ∞ . Let R v be the v -adic radius of y . If (cid:81) v R v > , then y is algebraic over K ( x ) . In general the v -adic radius R v may be infinity or zero. We refer the readerto André’s paper for a precise definition of the infinite product in such situations.In our applications of this theorem, R v will always be non-zero. We remark thatwe could have also used Thm. 6.1 and Prop. 5.15 of [BCL09] in place of André’sTheorem.Suppose that y is a (component of a) formal solution of ( M, ∇ ) as above. By[And04], Corollary 5.4.5, if the p -curvatures of ( M, ∇ ) vanish for all places over a LGEBRAIC SOLUTIONS OF DIFFERENTIAL EQUATIONS OVER P − { , , ∞} set of rational primes of density one then τ ( y ) = 0 and ρ ( y ) < ∞ . Hence, in orderto prove that y is the germ of an algebraic function, we only need to prove that (cid:81) v R v > . Estimate of the radii at archimedean places.
We begin with the followingsimple lemma.
Lemma 3.2.1.
Suppose that φ : D (0 , → P C − { , , ∞} is a holomorphic mapsuch that φ (0) = x . Then for any archimedean place w of the number field L wherethe connection and the initial conditions x , y are defined, R w ≥ | φ (cid:48) (0) | w .Proof. Let z be the complex coordinate on D (0 , . Consider the formal power series φ ∗ y . The vector valued power series g = φ ∗ y is a formal solution of the differentialequations d g dz = ( φ (cid:48) ( z )) − A ( φ ( z )) g which is associated to the vector bundle withconnection ( φ ∗ M, φ ∗ ∇ ) . Since D (0 , is simply connected, g arises from a vectorvalued holomorphic function on D (0 , which we again denote by g . Let t = φ (cid:48) (0) z, and set R = | φ (cid:48) (0) | ∞ . Then we may identify D (0 , with the t -disc D (0 , R ) = D w (0 , | φ (cid:48) (0) | w ) and the map φ with a map ˜ φ : D (0 , R ) → P C − { , , ∞} which satisfies ˜ φ (cid:48) (0) = 1 . By the definition of R w , we have R w ≥ | φ (cid:48) (0) | w . (cid:3) . Given x , the upper bound (in terms of x ) of | φ (cid:48) (0) | for all such φ in theabove lemma has been studied by Landau and other people. Based on the workof Landau and Schottky, Hempel gave an explicit upper bound (see [Hem79, Thm.4]) that can be reached when x = − √ i . For the completeness of our paper, wegive some details on the computation of | φ (cid:48) (0) | .3.2.3 . We recall the definition of θ -functions and their classical relation with theuniformization of P C − { , , ∞} . Following the notation of [Igu62] and [Igu64], let θ ( t ) = (cid:88) n ∈ Z exp( πin t ) , θ ( t ) = (cid:88) n ∈ Z exp( πi ( n t + n )) , θ ( t ) = (cid:88) n ∈ Z exp( πi ( n + 12 ) t ) These series converge pointwise to holomorphic functions on H , which we denoteby the same symbols. Lemma 3.2.4. ( [Igu64, p. 243] ) These holomorphic functions θ , θ , θ are mod-ular forms of weight and level Γ(2) . Moreover, there is an isomorphism from thering of modular forms of level
Γ(2) to C [ X, Y, Z ] / ( X − Y − Z ) given by sending θ , θ and θ to X, Y and Z respectively. We need the following basic facts mentioned in [Igu62, p. 180] and [Igu64, p. 244]in this section and section 5:
Lemma 3.2.5. (1)
Let η be the Dedekind eta function defined by η = q / (cid:81) (1 − q n ) , where q = e πit . We have η = ( θ θ θ ) . In particular, the holomorphicfunctions θ , θ , θ are everywhere nonzero on the upper half plane. (2) The derivative λ (cid:48) ( t ) = πi ( θ ( t ) θ ( t ) θ ( t ) ) . (3) The holomorphic function ( θ + θ + θ ) is the weight Eisenstein formof level SL ( Z ) with constant term in its Fourier expansion; the holomorphicfunction ( θ + θ )( θ + θ )( θ − θ ) is the weight Eisenstein form oflevel SL ( Z ) with constant term in its Fourier expansion. . Let λ = θ ( t ) θ ( t ) : H → P ( C ) and t = ( − √ i ) . Then λ : H → P ( C ) −{ , , ∞} is a covering map with Γ(2) as the deck transformation group ([Cha85],VII, §7). In particular, the projective curve defined by v = u ( u − u − λ ( t )) isan elliptic curve. Moreover, it is isomorphic to the elliptic curve C / ( Z + t Z ) (see loc. cit. ). Lemma 3.2.7.
The map λ sends t to x .Proof. Since the automorphism group of the lattice Z + t Z , hence that of the ellipticcurve C / ( Z + t Z ) is of order , the automorphism group of the elliptic curve v = u ( u − u − λ ( t )) must also be of order . In particular, λ must send t to either (1 + √ i ) or (1 − √ i ) (the roots of j ( t ) = 2 λ ( t ) − λ ( t )+1) λ ( t ) ( λ ( t ) − ). Moreover,from the definition of θ , we can easily see that λ ( t ) has positive imaginary part. (cid:3) Proposition 3.2.8.
Let y be a component of the formal solution of the differentialequations. Then R [ L : Q ][ Lw : R ] w ≥ / / π = 5 . · · · .Proof. Consider the map λ ◦ α : D (0 , → X C , where α : D (0 , → H is a holomor-phic isomorphism such that α (0) = t , that is, α : z (cid:55)→ − + √ i z +11 − z . We wouldlike to apply Lemma 3.2.1 to the map λ ◦ α , which maps ∈ D (0 , to x since λ ( t ) = λ ( ( − √ i )) = x by Lemma 3.2.7.Note that | x | = | − x | = 1 , so we have | θ ( t ) | = | θ ( t ) | = | θ ( t ) | . ByLemma 3.2.5, we have | λ (cid:48) ( t ) | = | πi ( θ ( t ) θ ( t ) θ ( t ) ) | = π | θ ( t ) | = π | η ( t ) | / . We now apply the Chowla–Selberg formula (see [SC67]) to Q ( √ i ) : | η ( t ) | (cid:61) ( t ) = 14 π √ (cid:18) Γ(1 / / (cid:19) . Then we have | λ (cid:48) ( t ) | = π | η ( t ) | / = π / π √ (cid:61) ( t ) (cid:18) Γ(1 / / (cid:19) . We get | ( λ ◦ α ) (cid:48) (0) | = | λ (cid:48) ( t ) | · | α (cid:48) (0) | = π / π √ (cid:61) ( t ) (cid:18) Γ(1 / / (cid:19) · (cid:61) ( t ) = 3Γ(1 / / π by the fact Γ(1 / /
3) = π √ . (cid:3) Algebraicity of the formal solutions.Proposition 3.3.1.
Let ( M, ∇ ) be a vector bundle with a connection over P K −{ , , ∞} , and assume that the p -curvatures of ( M, ∇ ) vanish for all p . Then ( M, ∇ ) is locally trivial with respect to the étale topology of P K − { , , ∞} .Proof. Consider y ∈ L [[( x − x )]] . By Proposition 3.2.8, we have (cid:89) w |∞ R w ≥ . · · · . LGEBRAIC SOLUTIONS OF DIFFERENTIAL EQUATIONS OVER P − { , , ∞} If w | p is a finite place of L, then since G w is trivial, ( M, ∇ ) has a full set ofsolutions over D ( x , | p | p ( p − ) . In particular, y is analytic on D ( x , | p | p ( p − ) . Hence (cid:89) w | p R w ≥ (cid:89) w | p | p | − p ( p − w = p − p ( p − . and log( (cid:89) w R w ) ≥ log 5 . · · · − (cid:88) p log pp ( p − > . · · · . Applying Theorem 3.1.5, we have that y is algebraic. Hence ( M, ∇ ) is étalelocally trivial. (cid:3) Remark . It is possible to define G w using different radii such that the proofof the above proposition continues to hold. Here are two examples:(1) Set G (cid:48) w := G w ( x , ) for all primes w | and G (cid:48) w = G w ( x , for other w . Wecan define G (cid:48) in the same way as G in section 2.2.4 but replacing G w by G (cid:48) w . Inthis situation, we have log( (cid:81) w R w ) ≥ log 5 . · · · − log 4 > . · · · . Applyingthe same argument as in Theorem 2.2.5, we have
Lie G (cid:48) = Lie G gal . In particular, if ( M, ∇ ) is a vector bundle with connection on X K such that ψ p ≡ for almost all p, and G (cid:48) w = { } for all w, then ( M, ∇ ) has finite monodromy.This result cannot be proved ‘prime by prime’ because the condition at w | is tooweak to imply that does not divide the order of the local monodromies.The equality Lie G (cid:48) = Lie G gal fails in general, if one drops condition (1) insection 2.2.4, and defines G (cid:48) using just the analogue of condition (2) (that is with G w replaced by G (cid:48) w ). (The condition (1) is used to guarantee the assumption that τ ( y ) = 0 , ρ ( y ) < ∞ in Theorem 3.1.5.)To see this, we consider the Gauss–Manin connection on H of the Legendrefamily of elliptic curves. Since the Legendre family has good reduction at primes w (cid:45) , H admits a Frobenius structure at such primes, so that G w = { } (see Remark2.1.6). For w | we have G w (cid:0) x , (cid:1) = { } by a direct computation: as in section 5.2below, we see that the matrix giving the connection lies in End( M O K ) ⊗ Ω X O K anda formal horizontal section of a general differential equation of this form will haveconvergence radius . Hence, the smallest group containing all p -adic differentialGalois groups is trivial while Lie G gal = sl . In particular, G (cid:48) (defined with thecondition (1)) is the smallest group containing almost all ψ p and we recover aspecial case of [Kat82, thm. 11.2].(2) We now consider a variant of our result when X equals to P minus morethan three points. Let D be the union of { } and all -th roots of unity andlet X = A − D . Let u be one of the preimages of x of the covering map f : X → P − { , , ∞} , u (cid:55)→ x = − ( u + u − − . We may assume that thenumber field L contains u .We consider the following weaker version of p -curvature conjecture: Proposition . Let ( N, ∇ ) be a vector bundle with connection over X . Assumethat the p -curvatures vanish for almost all p and that for any finite place v , all theformal horizontal sections of ( N, ∇ ) converges over the largest disc around u in X L w . Then ( N, ∇ ) must be étale locally trivial. By direct calculation, the w -adic distance from u to D is | | w when w is fi-nite. Then our assumption means that all the formal horizontal sections of ( N, ∇ ) centered at u converge over D ( u , | | w ) . Proof of the proposition.
By applying Theorem 3.1.5 to the formal horizontal sec-tions around u , one only need to show that (cid:81) w |∞ R w ≥ / . Since the uni-formization λ ◦ α : D (0 , → P ( C ) − { , , ∞} factors through f : A ( C ) − D → P ( C ) − { , , ∞} , then for the formal horizontal sections of ( N, ∇ ) , we have R w ≥ | . · · · | w / | f (cid:48) ( u ) | w by the chain rule and Lemma 3.2.1. A direct com-putation shows that (cid:81) w |∞ | f (cid:48) ( u ) | w = 4 and then (cid:81) w |∞ R w ≥ / by the fact . ... > · / . (cid:3) We now formulate another possible proof of this proposition. The idea is toreduce the problem for ( N, ∇ ) over X to f ∗ ( N, ∇ ) over P − { , , ∞} . Over P −{ , , ∞} , the assumption on ( N, ∇ ) shows that for f ∗ ( N, ∇ ) , the p -adic differentialgroup G (cid:48) w := G w ( x ,
1) = 1 for w (cid:45) . Although for w | , the -adic differentialgroup G (cid:48) w := G w ( x , − / ) is not trivial, we still have R w ≥ | | / w by consideringthe uniformization h ( z ) = − (( z + u ) + ( z + u ) − − . More precisely, by theassumption on ( N, ∇ ) , we can take R = | | w · | | / w in Definition 3.1.3 and checkthat | h (cid:48) (0) | w = 1 and h (0) = x . Then we apply André’s theorem and concludethat f ∗ ( N, ∇ ) and hence ( N, ∇ ) admit a full set of algebraic solutions.If one replaces the assumption in Proposition 3.3.3 by that the generic radii ofall formal horizontal sections of ( N, ∇ ) are at least | | w for all w finite, the resultsin [BS82] does not apply directly due to the fact that the points in D are too closeto each other in L w when w | . However, one may modify the argument there,especially a modified version of eqn. (3) in loc. cit. , to see that the condition ongeneric radii would imply trivial monodromy of ( N, ∇ ) .4. Rationality: an application of a theorem of Bost andChambert-Loir
In this section, we will first review the rationality criterion due to Bost andChambert-Loir for an algebraic formal function using capacity norms. Then wewill use the moduli interpretation of X to compute the capacity norm and verifythat in our situation this theorem is applicable.4.1. Review of the rationality criterion.
We will review the definition of adélictube adapted to a given point, the definition of capacity norms for the special casewe need, and the rationality criterion in [BCL09].
Definition 4.1.1. ([BCL09, Definition 5.16]) Let Y be a smooth projective curveover K, and let ( x ) be the divisor corresponding to a given point x ∈ Y ( L ) forsome number field L ⊃ K . For each finite place w of L , let Ω w be a rigid analyticopen subset of Y L w containing x . For each archimedean place w , we choose oneembedding σ : L → C corresponding to w and we let Ω w be an analytic open set of Y σ ( C ) containing x . The collection (Ω w ) is an adélic tube adapted to ( x ) if thefollowing conditions are satisfied:(1) for an archimedean place, the complement of Ω w is non-polar (e.g. a finitecollection of closed domains and line segments); if w is real, we further assumethat Ω w is stable under complex conjugation. LGEBRAIC SOLUTIONS OF DIFFERENTIAL EQUATIONS OVER P − { , , ∞} (2) for a finite place, the complement of Ω w is a nonempty affinoid subset;(3) for almost all finite places, Ω w is the tube of the specialization of x in thespecial fiber of Y. That is, Ω w , is the open unit disc with center at x .We call (Ω w ) a weak adélic tube if we drop the condition that Ω w is stable undercomplex conjugation when w is real.4.1.2 . Now let Y = P O K . The weak adélic tube that we will use can be describedas follows:(1) For an archimedean place, Ω w will be an open simply connected domain inside P C − { , , ∞} .(2) For a finite place, Ω w will be chosen to be an open disc of form D ( x , ρ w ) .(3) For almost all finite places, ρ w = 1 .4.1.3 . For Ω w as above, Bost and Chambert-Loir have defined the local capacitynorms || · || cap w (see [BCL09, Chapter 5]). These are norms on the line bundle T x X over Spec( O L ) . The Arakelov degree of T x X with respect to these norms playsthe same role as log( (cid:81) R w ) in section 3. This degree can be computed as a localsum after choosing a section of this bundle. We will use the section ddx , in whichcase one has the following simple description of local capacity norms:(1) For an archimedean place, let φ : D (0 , R ) → Ω w be a holomorphic isomorphismthat maps to x , then || ddx || cap w = | Rφ (cid:48) (0) | − w (see [Bos99, Example 3.4]).(2) For a finite place, || ddx || cap w = ρ − w (see [BCL09, Example 5.12].Now, we can state the rationality criterion: Theorem 4.1.4. ( [BCL09, Theorem 7.8] ) Let (Ω w ) be an adélic tube adapted to ( x ) . Suppose y is a formal power series over X centered at x satisfying thefollowing conditions: (1) For all w , y extends to an analytic meromorphic function on Ω w ; (2) The formal power series y is algebraic over the function field K ( X ) . (3) The Arakelov degree of T x X defined as (cid:88) w − log( || ddx || cap w ) is positive.Then y is rational. Corollary 4.1.5.
The theorem still holds if we only assume that (Ω w ) is a weakadelic tube.Proof. The idea is implicitly contained in the discussion in [Bos99, section 4.4]. Weonly need to prove that y is rational over X L (cid:48) , where L (cid:48) /L is a finite extensionwhich we may assume does not have any real places. Let w be a place of L and w (cid:48) a place of L (cid:48) over w .For w is archimedean, choose the embedding σ (cid:48) : L (cid:48) → C corresponding to w (cid:48) which extends the chosen embedding σ : L → C corresponding to w. We have anatural identification Y σ (cid:48) ( C ) = Y σ ( C ) , and we take Ω w (cid:48) := Ω w . If w is a finite place,we set Ω w (cid:48) = Ω w ⊗ L w L w (cid:48) . Since L (cid:48) does not have any real places, the weak adélic tube (Ω w (cid:48) ) is an adélictube. The first two conditions in Theorem 4.1.4 still hold and the Arakelov degreeof T x X with respect to (Ω (cid:48) w ) is the same as that of T x X with respect to (Ω w ) .We can apply Theorem 4.1.4 to y over X L (cid:48) and conclude that y is rational. (cid:3) Proof of the main theorem.
Let y be the algebraic formal function whichis one component of the formal horizontal section y of ( M, ∇ ) over X K . Lemma 4.2.1.
Let y be as above. Then this formal power series centered at x has convergence radius equal to for almost all finite places.Proof. Since the covering induced by y is finite étale over X L , by Proposition 3.3.1,it is étale over X O w at x for almost all places. For such places, we have ρ w = 1 bylifting criterion for étale maps. (cid:3) . We now define an adélic tube ( Ω w ) adapted to x . For an archimedean place w , we choose the embedding σ : L → C corresponding to w such that σ ( x ) =(1 + √ i ) / . Let (cid:101) Ω be the open region in the upper half plane cut out by thefollowing six edges (see the attached figure): (cid:60) t = − , | t + 2 | = 1 , | t + | = , | t + | = , | t − | = 1 , and (cid:60) t = . This is a fundamental domain of the arithmeticgroup Γ(2) ⊂ SL ( Z ) .We define Ω w to be λ ( (cid:101) Ω) .For w finite, we choose Ω w to be D ( x , if y is étale over X O w at x ; otherwise,we choose Ω w to be D ( x , p − p ( p − ) .The collection (Ω w ) is a weak adélic tube and y extends to an analytic (inparticular meromorphic) function on each Ω w by Lemma 4.2.1, Lemma 3.2.1, andLemma 2.1.5. Lemma 4.2.3.
The Arakelov degree of T x X with respect to the adélic tube (Ω w ) defined above is positive.Proof. We want to give a lower bound of ( || ddx || cap w ) − , the capacity of Ω w . Let a = − + √ i . On the line (cid:60) ( t ) = − , the point a is the closest point to t = ( − √ i ) with respect to Poincaré metric. The stabilizer of t in SL ( Z ) hasorder , and permutes the geodesics (cid:60) t = − , | t + | = , | t − | = 1 , and this actionpreserves the Poincaré metric. Using this, together with the fact that the distanceto t is invariant under z (cid:55)→ − − ¯ z, one sees that the distance from any point onthe boundary of (cid:101) Ω to t is at least that from a to t . Since α : D (0 , → H (definedin the proof of Prop. 3.2.8) preserves the Poincaré metrics, α − ( (cid:101) Ω) contains a discwith respect to the Poincaré radius equal to the distance from t to a . LGEBRAIC SOLUTIONS OF DIFFERENTIAL EQUATIONS OVER P − { , , ∞} In D (0 , , a disc with respect to Poincaré metric is also a disc in the Euclideansense. Hence α − ( (cid:101) Ω) contains a disc of Euclidean radius | α − ( a ) | = | ( a − t ) / ( a − ¯ t ) | = 0 . · · · . Since λ maps the fundamental domain (cid:101) Ω isomorphically onto Ω w , by 4.1.3, the localcapacity ( || ddx || cap w ) − is at least | ( a − t ) / ( a − ¯ t ) | · | λ (cid:48) ( ( − √ i )) | .By 4.1.3, we have − log( || ddx || cap w ) ≥ − log pp ( p − when w | p . Recall in Proposition3.2.8 we have | λ (cid:48) ( ( − √ i )) | = 5 . · · · , hence the Arakelov degree of T x X is (cid:88) w − log( || ddx || cap w ) > log(5 . · · · × . · · · ) − (cid:88) p log pp ( p − > . · · · . (cid:3) Now we are ready to prove Theorem 2.2.1:
Proof.
Applying Proposition 3.3.1, we have a full set of algebraic solutions y .Choosing the weak adélic tube as in 4.2.2 and applying Corollary 4.1.5 (the as-sumptions are verified by 4.2.2 and Lemma 4.2.3), we have that these algebraicsolutions are actually rational.This shows that ( M, ∇ ) has a full set of rational solutions over X L . Since for-mation of ker( ∇ ) commutes with the finite extension of scalars ⊗ K L, this impliesthat ( M, ∇ ) has a full set of rational solutions over X K . (cid:3) Interpretation using the Faltings height
In this section, we view X Z [ ] as the moduli space of elliptic curves with level structure. Let λ ∈ X ( ¯ Q ) and E the corresponding elliptic curve. Using theKodaira–Spencer map, we will relate the Faltings height of E with our lower boundfor the product of radii of uniformizability (see section 3) at archimedean places ofthe formal solutions in (cid:98) O X K ,λ . We will focus mainly on the case when λ ∈ X (¯ Z ) and sketch how to generalize to λ ∈ X ( ¯ Q ) at the end of this section. In thissection, unlike the previous sections, we will use λ as the coordinate of X .5.1. Hermitian line bundles and their Arakelov degrees. . Let K be a number field, and O K its ring of integers. Recall that an Hermit-ian line bundle ( L, ||·|| σ ) over Spec( O K ) is a line bundle L over Spec( O K ) , togetherwith an Hermitian metric || · || σ on L ⊗ σ C for each archimedean place σ : K → C . Given an Hermitian line bundle ( L, || · || σ ) , its (normalized) Arakelov degree isdefined as: (cid:100) deg( L ) := 1[ K : Q ] (cid:32) log( L/s O K )) − (cid:88) σ : K → C log || s || σ (cid:33) , where s is any section.For a finite place v over p , the integral structure of L defines a norm || · || v on L K v . More precisely, if s v is a generator of L O Kv and n is an integer, we define || p n s v || v = p − n [ K v : Q p ] . We obtain a norm on O v by viewing it as the trivial line bundle. We will use || · || v for the norms on different line bundle as no confusionwould arise. We may rewrite the Arakelov degree using the p -adic norms: (cid:100) deg( L ) = 1[ K : Q ] (cid:32) − (cid:88) v log || s || v (cid:33) , where v runs over all places of K . It is an immediate corollary of the productformula that the right hand side does not depend on the choice of s .5.1.2 . Let E be an elliptic curve over a number field K, and denote by e : Spec K → E and f : E → Spec K the identity and structure map respectively. For each σ : K → C , we endow e ∗ Ω E/K = f ∗ Ω E/K with the Hermitian norm given by || α || σ =( π ´ σE | α ∧ ¯ α | ) (cid:15)σ , where (cid:15) σ is for real embeddings and otherwise.This can be used to define the Faltings’ height of E , which we recall precisely onlyin the case when E has good reduction over O K . Denote by f : E →
Spec O K theelliptic curve over O K with generic fibre E, and again write e for the identity sectionof E . The norms || α || σ make e ∗ Ω E / Spec( O K ) = f ∗ Ω E / Spec( O K ) into a Hermitian linebundle, and we define the (stable) Faltings height by h F ( E λ ) = (cid:100) deg( f ∗ Ω E / Spec( O K ) ) . Notice that h F ( E λ ) does not depend on the choice of K . Here we use Deligne’sdefinition for convenience [Del85, 1.2]. This differs from Faltings’ original definition(see [Fal86]) by a constant log( π ) .In general, the elliptic curve E would have semi-stable reduction everywhereafter some field extension. We assume this is the case and E has a Neron model f : E →
Spec O K which endows f ∗ Ω E / Spec( O K ) a canonical integral structure. Withthe same Hermitian norm defined as above, we have a similar definition of Faltingsheight in the general case. See [Fal86] for details. As in the good reduction case,this definition does not depend on the choice of K .5.1.3 . We will assume that λ and λ − are both units at each finite place.Given such a λ , consider the elliptic curve E λ over Q ( λ ) defined by the equation y = x ( x − x − λ ) . Then E λ has good reduction at primes not dividing , andpotentially good reduction everywhere, since its j -invariant is an algebraic integer.Let K be a number field such that ( E λ ) K has good reduction everywhere. Wedenote by E λ the elliptic curve over O K with generic fiber E λ . . To express our computation of radii in terms of Arakelov degrees, we endowthe O K -line bundle T λ ( X O K ) , the tangent bundle of X O K at λ , with the structureof an Hermitian line bundle as follows. For each archimedean place σ : K → C , wehave the universal covering λ : H → σX, introduced in 3.2.6. The SL ( R ) -invariantmetric dt (cid:61) ( t ) on the tangent bundle of H induces the desired metric on the tangentbundle via push-forward. As in the proof of Proposition 3.2.8, our lower bound onthe radius of the formal solution is | (cid:61) ( t ) λ (cid:48) ( t ) | (cid:15) σ = || ddλ || − σ , where t is a pointon H mapping to λ . It is easy to see the left hand side does not depend on thechoice of t . Under the assumptions in 5.1.3, the tangent vector ddλ is an O K -basisvector for the tangent bundle T λ ( X O K ) , and we have (cid:100) deg( T λ X ) = 1[ K : Q ] ( − (cid:88) σ : K → C log || ddλ || σ ) ≤ K : Q ] log( (cid:89) σ R σ ) , where the R σ are the radius of uniformization discussed in section 3.2. LGEBRAIC SOLUTIONS OF DIFFERENTIAL EQUATIONS OVER P − { , , ∞} The Kodaira–Spencer map.
Consider the Legendre family of elliptic curves E ⊂ P Z [ ] × X Z [ ] over X Z [ ] given by y = x ( x − x − λ ) . We have the Kodaira–Spencer map ([FC90, Ch. III,9],[Kat72, 1.1]): KS : ( f ∗ Ω E/X Z [ 12 ] ) ⊗ → Ω X Z [ 12 ] , α ⊗ β (cid:55)→ (cid:104) α, ∇ β (cid:105) , (5.2.1)where ∇ is the Gauss–Manin connection and (cid:104)· , ·(cid:105) is the pairing induced by thenatural polarization.5.2.2 . Following Kedlaya’s notes ([Ked, Sec. 1,3]), we choose { dx y , xdx y } to be anintegral basis of H dR ( E/X ) | λ and compute the Gauss–Manin connection: ∇ dx y = 12(1 − λ ) dx y ⊗ dλ + 12 λ ( λ − xdx y ⊗ dλ. The Kodaira–Spencer map then sends ( dx y ) ⊗ to λ ( λ − dλ .This computation shows: Lemma 5.2.3.
Given v a finite place not lying over , the Kodaira–Spencer map (5.2.1) preserves the O v -generators of ( f ∗ Ω E/X Z [ 12 ] ) ⊗ | λ and Ω X Z [ 12 ] | λ when λ and λ − are both v -units. . For the archimedean places σ , we consider f ∗ Ω σE/ Spec C with the metrics || α || σ defined in section 5.1, and we endow Ω X Z | λ the Hermitian line bundle struc-ture as the dual of the tangent bundle.To see the Kodaira–Spencer map preserves the Hermitian norms on both sides,one may argue as follows. Notice that the metrics on ( f ∗ Ω σE/ Spec C ) ⊗ and Ω X Z are SL ( R ) -invariant (see for example [ZP09, Remark 3 in Sec. 2.3]). Hence theyare the same up to a constant and we only need to compare them at the cusps. Todo this, one studies both sides for the Tate curve. See for example [MB90, 2.2] fora related argument and Lemma 3.2.5 (2) for relation between θ -functions and Ω X .Here we give another argument: Lemma 5.2.5.
The Kodaira–Spencer map preserves the Hermitian metrics: || ( dx y ) ⊗ || σ = || dλ λ ( λ − || σ . Proof.
Let dz be an invariant holomorphic differential of C / ( Z ⊕ t Z ) , where λ ( t ) = λ . By the theory of the Weierstrass- ℘ function, we have a map from the complextorus to the elliptic curve u = 4 v − g ( t ) v − g ( t ) such that dz maps to dvu . Here g is the weight modular form of level SL ( Z ) with π as the constant term in its Fourier series and g is the weight modularform with π as the constant term. Using Lemma 3.2.5 (3), we see that the righthand side has three roots: π ( θ ( t ) + θ ( t )) , − π ( θ ( t ) + θ ( t )) , π ( θ ( t ) − θ ( t )) . Hence this curve is isomorphic to y = x ( x − x − λ ) via the map(5.2.6) x = v − π ( θ ( t ) + θ ( t )) − π θ ( t ) , y = u − π θ ( t )) / , and we have dx y = πiθ ( t ) dvu = πiθ ( t ) dz. Hence || ( dx y ) ⊗ || σ = | π θ ( t ) · ( 12 π ˆ E ( C ) | dz ∧ d ¯ z | ) | (cid:15) σ = | πθ ( t ) (cid:61) ( t ) | (cid:15) v . On the other hand, using Lemma 3.2.5 (2), we have || dλ λ ( λ − || /(cid:15) σ σ = (cid:12)(cid:12)(cid:12)(cid:12) (cid:61) ( t ) | λ (cid:48) ( t ) | λ ( λ − (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) (cid:61) ( t ) πθ ( t ) θ ( t ) θ ( t ) λ ( λ − (cid:12)(cid:12)(cid:12)(cid:12) = | πθ ( t ) (cid:61) ( t ) | . (cid:3) Proposition 5.2.7. If λ and λ − are both units at every finite places, we have (cid:100) deg( T λ X ) = − h F ( E λ ) + log 23 .Proof. By lemma 5.2.3 and lemma 5.2.5, we have − (cid:100) deg( T λ X ) = (cid:100) deg(Ω X OK | λ )= 1[ K : Q ] ( − (cid:88) v log || dλ λ ( λ − || v )= 1[ K : Q ] ( − (cid:88) v |∞ log || dλ λ ( λ − || v − (cid:88) v finite log || dλ λ ( λ − || v )= 1[ K : Q ] ( − (cid:88) v |∞ log || ( dx y ) ⊗ || v − (cid:88) v not divides , ∞ log || ( dx y ) ⊗ || v − (cid:88) v | log || / || v )=2 h F ( E λ ) + 1[ K : Q ] (cid:88) v | log || ( dx y ) ⊗ || v − log 2 . (5.2.8)Now we study || ( dx y ) ⊗ || v given v | . The sum K : Q ] (cid:80) v | log || ( dx y ) ⊗ || v does notchange after extending K , hence we may assume that E λ over O v has the Deuringnormal form u + auw + u = w (see [Sil09] Appendix A Prop. 1.3 and the proofof Prop. 1.4 shows in the good reduction case, a is a v -integer). An invariantdifferential generating f ∗ Ω E λ / Spec O K [ ] is dw u + aw +1 .Because both dw u + aw +1 and dx y are invariant differentials, we have || dx y || v = || ∆ / ∆ || v || dw u + aw +1 || , where ∆ and ∆ are the discriminant of the Deuring nor-mal form and that of the Legendre form respectively. Since E has good reduction, || ∆ || v = 1 (see the proof of loc. cit. ). Hence || dx y || v = || dw u + aw + b || v · || / || / v = || || − / v .Hence (cid:100) deg( T λ X ) = − h F ( E λ ) − log 2 + log 2 = − h F ( E λ ) + log 23 . (cid:3) . As pointed out by Deligne ([Del85, 1.5]), the point √ i corresponds tothe elliptic curve with smallest height. Hence, our choice √ i gives the largest (cid:100) deg( T λ X ) among those λ such that λ and λ − are units at every prime. LGEBRAIC SOLUTIONS OF DIFFERENTIAL EQUATIONS OVER P − { , , ∞} The general case.
For the general case when λ ∈ X ( ¯ Q ) , using a similarargument as in section 5.2, we have K : Q ] ( − (cid:88) σ : K → C log || ddλ || σ ) ≤ − h F ( E λ ) + log 23+ 1[ K : Q ] (cid:16) (cid:88) v finite log + || λ || v + log( | Nm λ ( λ − | ) (cid:17) (5.3.1)and equality holds if and only if λ ∈ X (¯ Z ) . As discussed in 5.1.4, the left handside is the sum of the logarithms of our estimates of the radii of uniformizability atarchimedean places.We also need to modify the estimate of the radii at finite places in Lemma 2.1.5.A possible estimate for R v is p − p ( p − · min {|| λ || v , || λ − || v , } . One explanationof the factor min {|| λ || v , || λ − || v , } is that we cannot rule out the possibility thatone has local monodromy at , , ∞ merely from the information of p -curvature at v . Compared to the case when λ ∈ X (¯ Z ) , our estimate for the sum of the loga-rithms of the archimedean radii increases by at most K : Q ] ( (cid:80) v finite log + || λ || v +log( | Nm λ ( λ − | )) , while the estimate for the sum of logarithms of the radiiat finite places becomes smaller by (cid:80) v max { log + || λ − || v , log + || ( λ − − || v } . Anexplicit computation shows that the later is larger than the former. Hence theestimate for the product of the radii does not become larger than the case when λ ∈ X (¯ Z ) . 6. The affine elliptic curve case
Let X ⊂ A Z be the affine curve over Z defined by the equation y = x ( x − x + 1) . The generic fiber X Q is an elliptic curve (with j -invariant ) minusits identity point. Given a vector bundle with connection over X K , we will definethe notion of vanishing p -curvature for all finite places along the same lines as insection 2.1. The main result of this section is: Theorem 6.0.2.
Let ( M, ∇ ) be a vector bundle with connection over X K . Supposethat the p -curvatures of ( M, ∇ ) vanish for all p. Then ( M, ∇ ) is étale locally trivial.Remark . This theorem cannot be deduced from applying Theorem 2.2.1 tothe push-forward of ( M, ∇ ) via some finite étale map from an open subvariety ofthe affine elliptic curve to P K − { , , ∞} . Unlike the P K − { , , ∞} case, theconclusion here allows the existence of ( M, ∇ ) with finite nontrivial monodromy.See section 7.1.6.1. Formal horizontal sections and p -curvatures. . We fix x = (0 , ∈ X ( Z ) and denote by ( x ) K and ( x ) k v the images of x in X ( K ) and X ( k v ) . Let y : X → A Z be the projection to the y -coordinate. It is easyto check that this map is étale along x and hence induces isomorphisms between thetangent spaces T x X ∼ = T A Z and between the formal schemes (cid:100) X K/ ( x ) K ∼ = (cid:100) A K/ .In particular, we have an analytic section s v of the projection y from D (0 , ⊂ A ( K v ) to X ( K v ) such that s v (0) = x for any finite place v by the lifting criterion for étale maps. By definition, the image s v ( D (0 , is the open rigid analytic disc in X ( K v ) which is the preimage of ( x ) k v under the reduction map X ( K v ) → X ( k v ) .By choosing a trivialization of M in some neighborhood of ( x ) K , we can viewa formal horizontal section m of ( M, ∇ ) around ( x ) K as a formal function in (cid:92) O X K , ( x ) K r ∼ = (cid:92) O A K , r , where r is the rank of M . We denote f ∈ (cid:92) O A K , r to bethe image and the goal of this section is to prove that the formal power series f isalgebraic.Let U be X − { (0 , , (0 , − } . It is a smooth scheme over Z . Our chosen point x is a Z -point of U and s v ( D (0 , ⊂ U ( K v ) . For v | p a finite place of K , wesay that ( M, ∇ ) has good reduction at v if ( M, ∇ ) extends to a vector bundle withconnection on U O v . Similar to Lemma 2.1.5, we have: Lemma 6.1.2.
Suppose that ( M, ∇ ) has good reduction at v . If the p -curvature ψ p vanishes , then the formal function f is the germ of some meromorphic functionon the disc D (0 , p − p ( p − ) ⊂ A .Proof. Let ( M , ∇ ) be an extension of ( M, ∇ ) over X O v . Since y is étale, thederivation ∂∂y is regular over some Zariski open neighborhood ¯ V of x ∈ X ⊗ Z /p Z . Let V ⊂ X ( K v ) be the preimage of ¯ V under reduction map. Since the p -curvature vanishes, we have ∇ ( ∂∂y ) p ( M| V ) ⊂ p M| V . Notice that s v ( D (0 , ⊂ V .Then the proof of Lemma 2.1.5 shows the existence of horizontal sections of M on s v ( D (0 , p − p ( p − )) . Via a local trivialization of M and the isomorphism of formalneighborhoods of x and , we see that f is meromorphic over D (0 , p − p ( p − ) . (cid:3) This lemma motivates the following definition:
Definition 6.1.3.
We say that the p -curvatures of ( M, ∇ ) vanish for all p if(1) the p -curvature ψ p vanishes for all but finitely many p ,(2) all formal horizontal sections around x , when viewed as formal functions in (cid:92) O A K , r , are the germs of some meromorphic functions on D (0 , p − p ( p − ) for allfinite places v . Remark . The second condition does not depend on the choice of local trivi-alization of M . Moreover, for each v , this condition remains the same if we replacethe projection y by any map g : W O v → A O v such that W O v is a Zariski openneighborhood of ( x ) O v in X O v and that g is étale.6.2. Estimate at archimedean places and algebraicity.
Let σ : K → C bean archimedean place. Let φ : D (0 , → X ( C ) be a uniformization map such that φ (0) = x . We have the following lemma whose proof is the same as that of Lemma3.2.1: Lemma 6.2.1.
The σ -adic radius R σ (see Definition 3.1.3) of the formal functions f in 6.1.1 would be at least | ( y ◦ φ ) (cid:48) (0) | σ . Let t = i . A direct manipulation of the definition shows λ ( t ) = − , where λ is defined in 3.2.6. Let F : D (0 , → C − ( Z + t Z ) be a uniformization map suchthat F (0) = . Lemma 6.2.2. (Eremenko) The derivative | F (cid:48) (0) | = 2 − / π − / Γ(1 / = 0 . .. . This means ψ p ≡ on X O v ⊗ Z /p Z as in section 2.1.1. LGEBRAIC SOLUTIONS OF DIFFERENTIAL EQUATIONS OVER P − { , , ∞} Proof.
From [Ere, Sec. 2], we have F (cid:48) (0) = / B (1 / , / | ( λ − ) (cid:48) ( i ) | , where B is Betafunction. By Lemma 3.2.5, the Chowla-Selberg formula ([SC67])(6.2.3) | η ( i ) | = 2 − π − / Γ(1 / , and θ ( i ) = 2 θ ( i ) = 2 θ ( i ) , we have | ( λ − ) (cid:48) ( i ) | = | πi ( θ ( i ) θ ( i ) θ ( i ) ) | = π | η ( i ) | = Γ(1 / π . We obtain the desired formula by noticing that B (1 / , /
4) = π − / Γ(1 / . (cid:3) Lemma 6.2.4.
Let α be the constant − π θ ( t )) / and ℘ be the Weierstrass- ℘ function. We have y ◦ φ = α − ℘ (cid:48) ◦ F , up to some rotation on D (0 , .Proof. The map g := ( ℘, ℘ (cid:48) ) maps C − ( Z + t Z ) to the affine curve u = 4 v − g ( t ) v − g ( t ) . Let s be the isomorphism from this affine curve to X ( C ) givenby (5.2.6). Since both s ◦ g (1 / and x are the unique point fixed by the fourautomorphisms of X ( C ) , we have s ◦ g (1 /
2) = x . Hence s ◦ g ◦ F (0) = x = φ (0) and then the uniformizations s ◦ g ◦ F and φ are the same up to some rotation. Wehave y ◦ φ = y ◦ s ◦ g ◦ F = α − ℘ (cid:48) ◦ F by (5.2.6). (cid:3) Proposition 6.2.5.
The σ -adic radius R [ K : Q ][ Kσ : R ] σ ≥ − / π − Γ(1 / = 3 . · · · . Proof.
Differentiate both sides of ( ℘ (cid:48) ( z )) = 4( ℘ ( z )) − g ( t ) ℘ ( z ) − g ( t ) , we have ℘ (cid:48)(cid:48) (1 /
2) = 6 ℘ (1 / − g ( t ) / − g ( t ) / , where the second equality follows from that ℘ (1 /
2) = π ( θ ( t ) + θ ( t )) / π θ ( t )( λ ( t ) + 1) / . By Lemma 3.2.5 and θ ( t ) = − θ ( t ) = θ ( t ) / , we have | g ( t ) | = 4 π · | θ ( t ) + θ ( t ) + θ ( t ) | = 4 π | θ ( t ) | . Then by Lemma 6.2.4 the absolute value of the derivative of y ◦ φ at would be | α − ℘ (cid:48)(cid:48) (1 / · F (cid:48) (0) | = 2 − π − | θ ( t ) | − · π | θ ( t ) | · | F (cid:48) (0) | = π | θ ( t ) | · − / π − / Γ(1 / (by Lemma 6.2.2) = 2 π · − π − / Γ(1 / · − / π − / Γ(1 / = 2 − / π − Γ(1 / = 3 . · · · , (6.2.6)where the third equality follows from | θ ( t ) | = 2 − / | θ ( t ) θ ( t ) θ ( t ) | / = 2 / | η ( t ) | = 2 / | η ( i ) | , and (6.2.3). (cid:3) Proof of Theorem 6.0.2.
By Proposition 6.2.5, we have (cid:81) v |∞ R v ≥ . · · · . ByDefinition 6.1.3, we have log( (cid:81) v (cid:45) ∞ R v ) ≥ − (cid:80) p log pp ( p − = − . · · · . Hence log( (cid:89) v R v ) ≥ log 3 . · · · − . · · · = 0 . · · · > . We conclude by applying Theorem 3.1.5. (cid:3) The choice of λ there is different. We have λ ( i ) = 2 here. Examples
In this section, we first give an example of ( M, ∇ ) with p -curvature vanishing forall p but with nontrivial global monodromy over the affine elliptic curve in section6. Then we discuss a variant of our main theorems with X being the affine lineminus all -th roots of unity.7.1. An example with vanishing p -curvature for all p and nontrivial G gal . Let K be Q ( √− , X ⊂ A Z be the affine curve defined by y = x ( x − x + 1) , E be the elliptic curve defined as the compactification of X K , and f : E → E bea degree two self isogeny of E . We will also use f to denote the restriction of f to X K \{ P } , where P is the non-identity element in the kernel of f .Let ( M, ∇ ) be f ∗ ( O X K \{ P } , d ) . By definition, G gal is Z / Z . Proposition 7.1.1.
The p -curvature of ( M, ∇ ) vanishes for all p .Proof. Notice that f extends to a degree two étale cover from E to E over Z [ i ] .Then for p (cid:45) , the p -curvature of ( M, ∇ ) coincides with that of f ∗ ( M, ∇ ) by the factthat p -curvatures remain the same under étale pull back . Hence the p -curvatureof ( M, ∇ ) vanishes as f ∗ ( M, ∇ ) is trivial.For p | , we write ( M, ∇ ) out explicitly. Without loss of generality, we mayassume that f from the curve y = x ( x − x + 1) to the curve s = t ( t − t + 1) is given by t = − i ( x − x ) and s = i yx ( x + x ) . Locally around ( t, s ) = (0 , , , x is an O X K basis of f ∗ O X K and this basis gives rise to a natural Zariski localextension of ( M, ∇ ) over X O p . Direct calculation shows that ∇ (1) = 0 , ∇ ( x ) = 2 s ( t − t − ds + 2 st (1 + 2 i )( t − t − xds. Therefore, ∇ ( f + f x ) ≡ df + xdf (mod 2) and the p -curvature of ( M, ∇ ) vanishes. (cid:3) Remark . In the above proof, we show that ( M, ∇ ) has all p -curvatures van-ishing in the strict sense: there is an extension of ( M, ∇ ) over X O K such that its p -curvatures are all vanishing. However, given the argument for p (cid:45) , in orderto to apply Theorem 6.0.2, we do not need to construct an extension of ( M, ∇ ) but only need to check that x , locally as a formal power series of s , converges on D (0 , − / ) for v | . This is not hard to see: x , as a power series of t , convergeswhen | t | v < | | v ; and t , as a power series of s , converges when | s | v < | | / v and theimage of | s | v < | | / v is contained in | t | v < | | v .7.2. A variant of the main theorems.
In this section, we will prove a variantof the main theorems when X = A Q − {± , ± i } . Similar to Theorem 6.0.2, theconclusion is that ( M, ∇ ) has finite monodromy and we give an example withnontrivial finite monodromy.To define the local convergence conditions for bad primes, we take x = 0 . Proposition 7.2.1.
Let ( M, ∇ ) be a vector bundle with connection over X with p -curvature vanishes for all p . We further assume that the formal horizontal sectionsaround x converge over D ( x , for v | . Then ( M, ∇ ) is étale locally trivial. Because p (cid:54) = 2 is unramified in K and ( M, ∇ ) has good reduction at p , the notion of p -curvaturehere is classical. LGEBRAIC SOLUTIONS OF DIFFERENTIAL EQUATIONS OVER P − { , , ∞} Proof.
By Lemma 6.2.2, we have R ∞ ≥ · . · · · . By the assumptions on finiteplaces, we have log( (cid:81) v (cid:45) ∞ R v ) ≥ − (cid:80) p (cid:54) =3 , pp ( p − = − . · · · . Then we concludeby applying Theorem 3.1.5. (cid:3) Example . Let s be (1 − x ) / . It is the solution of the differential equation dsdx = − x − x . Consider the connection on O X given by ∇ ( f ) = df + x − x dx . It has p -curvature vanishing for all p : ∇ ( f ) ≡ df (mod ) and ∇ ( f ) ≡ df + ( p + 1) x − x dx (mod p ) with solution s ≡ (1 − x ) ( p +1) / (mod p ) when p (cid:54) = 2 . In conclusion, ( O X , ∇ ) satisfies the assumptions in the above proposition while it has nontrivialmonodromy of order two. Remark . If we replace our assumption by similar conditions on generic radii,the above example shows that one could have order two local monodromy around ± , ± i . The reason is [BS82, III eqn. (3)] does not hold in this situation and amodification of their argument would show that an order two local monodromy ispossible. References [And04] Yves André,
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Department of Mathematics, Harvard University
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