aa r X i v : . [ m a t h . N T ] M a y THE ABC THEOREM FOR MEROMORPHIC FUNCTIONS
MACHIEL VAN FRANKENHUIJSEN
Abstract.
Using a ‘height-to-radical’ identity, we define the archimedeancontribution to the radical, r arch , and we give a new proof of the abc the-orem for the field of meromorphic functions. The first step of the proof iscompletely formal and yields that the height is bounded by the radical, h ≤ r ,where r = r na + r arch is the radical completed with the archimedean con-tribution. The second step is analytic in nature and uses the lemma on thelogarithmic derivative to derive a bound for r arch . Introduction
Although the abc theorem for meromorphic functions was first formulated andproved in [12], it has essentially been known since the beginning of the twenti-eth century. For example, it is an easy consequence of Nevanlinna’s second maintheorem. Since Mason [5], the point of view has changed, especially when it be-came known that the abc conjecture for numbers implies an effective version ofMordell’s conjecture (see [1, 13, 15, 16]; the proofs of Faltings and Vojta [2, 3, 17]are ineffective).Here, we prove this theorem again, organizing the proof in two steps to revealthe main structure. In the first step, we use a ‘height-to-radical’ identity to obtainin a completely formal way h ( ρ ) ≤ r ( ρ ) for every radius ρ ≥
1, where h and r arerespectively the (logarithmic) height and radical of an abc sum a ( z )+ b ( z )+ c ( z ) = 0of meromorphic functions. We will see how to define a natural contribution of thearchimedean valuations to the radical, r ( ρ ) = r na ( ρ ) + r arch ( ρ ). Then in the secondstep, we apply the lemma on the logarithmic derivative to obtain the inequality r arch ( ρ ) ≤ h ( ρ ) + O (log log h ( ρ )) for every ρ ≥ E of[1 , ∞ ) of finite total length.To explain the motivation from number theory, let k be a number field. Thevaluations of k satisfy the Artin–Whaples sum formula: for every x ∈ k ∗ , X v v ( x ) = 0 . (1.1)Here, we fix the archimedean valuations so that v (2) = [ k v : R ] log 2, and thenonarchimedean valuations so that v ( p ) = − [ k v : Q p ] log p , where p is the rationalprime such that v ( p ) <
0. Note that we define our valuations with a minus sign ascompared to [8].
Date : May 12, 2008.1991
Mathematics Subject Classification.
Primary 30D35; Secondary 11Axx.
Key words and phrases.
ABC theorem, Nevanlinna theory, error term in the abc conjecture,archimedean contribution to the radical.
For a point P = ( a : b : c ) ∈ P ( k ), we define the contribution of the valuation v to the height by h v ( a, b, c ) = ( max { v ( a ) , v ( b ) , v ( c ) } if v is nonarchimedean, [ k v : R ]2 log (cid:0) | a | + | b | + | c | (cid:1) if v is archimedean , where we write | a | = √ a ¯ a for the usual absolute value of a complex number. The height of P is defined by h ( P ) = X v h v ( a, b, c ) . By (1.1), the height does not depend on the choice of coordinates for P , even thoughthe local contributions do depend on this choice. Remark . The height depends on the number field, but the ‘canonical height’ h ( P ) / [ k : Q ] does not depend on k . On the other hand, for the radical, to bedefined below, there is no good definition of a ‘canonical radical’.The degree of a valuation is defined bydeg( v ) = ( log k ( v ) if v is nonarchimedean,0 if v is archimedean,where k ( v ) denotes the residue class field of v . The contribution of the valuation v to the radical is defined by r v ( P ) = ( v ( a ) = v ( b ) = v ( c ) , deg( v ) otherwise,which is clearly independent of the choice of coordinates for P . We define the incomplete radical of P by r na ( P ) = X v r v ( P ) . Thus the radical has a contribution from every nonarchimedean valuation wherethe orders of a , b and c are not all equal to each other. In particular, r na ( P ) = ∞ ifone of the coordinates of P vanishes. It has no contribution from the archimedeanvaluations, since we have put deg( v ) = 0 for these valuations. Conjecture ψ [6, 16, 18]) . There exists a function ψ with ψ ( h ) = o ( h ) such that for every point P = ( a : b : c ) ∈ P ( k ) on the line a + b + c = 0, h ( P ) ≤ r na ( P ) + ψ ( h ( P )) + log | disc( k ) | . One may further conjecture that ψ/ [ k : Q ] is independent of the number field k . Remark . This conjecture can be interpreted as a weak form of Hurwitz’ formula.See Remark 4.4 and [9].The abc conjecture is known with ψ ( h ) = h − C log h for a constant C (thisfunction is not o ( h )) [10, 11], and if the abc conjecture holds for some function ψ ,then we must have ψ ( h ) ≥ . √ h/ log h [10, 14]. Numerical evidence suggests thatthe abc conjecture may hold with ψ ( h ) = 4[ k : Q ] √ h . Thus we obtain the strongerconjecture HE ABC THEOREM FOR MEROMORPHIC FUNCTIONS 3
For every point P = ( a : b : c ) ∈ P ( k ) on the line a + b + c = 0, h ( P ) ≤ r na ( P ) + 4[ k : Q ] p h ( P ) + log | disc( k ) | . In Section 6, we give a possible interpretation of the error term ψ ( h ).It is rather surprising that this conjecture implies an effective version of bothVojta’s height inequality [15] and the radicalized Vojta heigh inequality [16] (see [18,Conjectures 2.1 and 2.3]). In particular, it implies Roth’s theorem with an effectiveerror term and an effective version of Mordell’s conjecture. These implications usethe construction of a Bely˘ı function, for which there is no analogue for functionfields or the field of meromorphic functions.2. The Height-to-Radical Identity
Let M be the field of meromorphic functions on C , and let ( f ) ℓ denote thelogarithmic derivative, ( f ) ℓ ( z ) = f ′ ( z ) f ( z ) . It is easily verified that ( f ) ℓ ( z ) is a meromorphic function with only simple poles.We call a point P ( z ) = ( a ( z ) : b ( z ) : c ( z )) ∈ P ( M ) nonconstant if at least oneof the functions a/b , b/c or c/a is not constant. Lemma 2.1 (Height-to-Radical Identity) . For a nonconstant point P = ( a : b : c ) on the line a + b + c = 0 in P ( M ) , we have the identity ( a : b : c ) = (cid:0) ( b/c ) ℓ : ( c/a ) ℓ : ( a/b ) ℓ (cid:1) . (2.1) Proof.
The right-hand side does not change if we replace ( a, b, c ) by ( a/c, b/c, P is constant if c = 0). Hence we need to check that for f + g + 1 = 0,( f : g : 1) = ( g ′ /g : − f ′ /f : f ′ /f − g ′ /g ) . Now f ′ + g ′ = 0 and g ′ does not vanish identically (since P is nonconstant), hencethis identity follows after multiplying the left-hand side by g ′ and the right-handside by f g . (cid:3) Remark . The height-to-radical identity provides a canonical way to choosemeromorphic coordinates for a point ( a : b : c ) on the line a + b + c = 0. Indeed,replacing ( a, b, c ) by ( λa, λb, λc ), for a meromorphic function λ ( z ), does not changethe right-hand side of (2.1).3. The Valuations of the Field of Meromorphic Functions
The valuations of M satisfy the Poisson–Jensen formula X | x | <ρ v x ( f, ρ ) + Z | z | = ρ v z ( f, ρ ) dz πiz − v ∞ ( f ) = 0 , (3.1)where the nonarchimedean valuations are parametrized by x with | x | < ρ , v x ( f, ρ ) = ( − ord( f, x ) log ρ | x | for 0 < | x | < ρ, − ord( f,
0) log ρ for x = 0;the archimedean valuations are parametrized by z on the circle of radius ρ , v z ( f, ρ ) = log | f ( z ) | , MACHIEL VAN FRANKENHUIJSEN and v ∞ ( f ) is the absolute value of the first coefficient in the Laurent series of f around 0: for f ( z ) = f n z n + f n +1 z n +1 + . . . , v ∞ ( f ) = log | f n | , where n = ord( f,
0) and f n = lim z → f ( z ) z − n . We count this function among the archimedean valuations, even though, strictlyspeaking, it is not a valuation. Note that v ( f, ρ ) is only a valuation for ρ ≥ P = ( a : b : c ) ∈ P ( M ), the local contributions to the height are h x ( a, b, c )( ρ ) = max { v x ( a, ρ ) , v x ( b, ρ ) , v x ( c, ρ ) } , for v x , | x | < ρ,h z ( a, b, c )( ρ ) = log p | a ( z ) | + | b ( z ) | + | c ( z ) | , for v z , | z | = ρ,h ∞ ( a, b, c ) = log p | a m | + | b m | + | c m | , for v ∞ , where m = min { ord( a, , ord( b, , ord( c, } . Then the height of P is defined by h ( P, ρ ) = X | x | <ρ h x ( a, b, c )( ρ ) + Z | z | = ρ h z ( a, b, c )( ρ ) dz πiz − h ∞ ( a, b, c ) . (3.2)By the Poisson–Jensen formula (3.1), the height does not depend on the choice ofcoordinates for P .The degree of a valuation is defined bydeg( v x , ρ ) = ( log ρ | x | for 0 < | x | < ρ, log ρ for x = 0 , deg( v z , ρ ) = 0 for | z | = ρ, deg( v ∞ , ρ ) = 0 . Note that deg( v , ρ ) ≥ ρ ≥
1. All other nonarchimedean valuations havea positive degree, and the archimedean valuations have vanishing degree.The local contributions to the radical are defined by r v ( P, ρ ) = ( v ( a ) = v ( b ) = v ( c ) , deg( v, ρ ) otherwise,independent of the choice of coordinates for P . Thus the archimedean valuations v z and v ∞ do not contribute to the radical. We define the incomplete radical of apoint P = ( a : b : c ) ∈ P ( M ) by r na ( P, ρ ) = X v r v ( P, ρ ) . Except if one of the coordinates of P vanishes identically, this sum is finite for every ρ > a , b and c form a discrete set.4. The Formal ABC Theorem for Meromorphic Functions
Theorem 4.1 ([12]) . Let P = ( a : b : c ) be a nonconstant point in P ( M ) suchthat a + b + c = 0 . Then, for every ρ ≥ ,h ( P, ρ ) ≤ r na ( P, ρ ) + Z | z | = ρ h z (cid:0) ( bc ) ℓ , ( ca ) ℓ , ( ab ) ℓ (cid:1) ( ρ ) dz πiz − h ∞ (cid:0) ( bc ) ℓ , ( ca ) ℓ , ( ab ) ℓ (cid:1) . HE ABC THEOREM FOR MEROMORPHIC FUNCTIONS 5
Proof.
Since P = (cid:0) ( b/c ) ℓ : ( c/a ) ℓ : ( a/b ) ℓ (cid:1) by the height-to-radical identity, we usethese coordinates to compute the height of P . Let x be a point with | x | < ρ suchthat ord( a, x ) > ord( b, x ) = ord( c, x ). Then c/a has a pole and a/b has a zero at x , hence ( c/a ) ℓ and ( a/b ) ℓ have a simple pole at x . Moreover, ( b/c ) ℓ has no poleat x . Therefore, v x contributes equally to the height and the radical, and the sameholds for all points where either b or c has a larger order of vanishing. We obtain h x ( P, ρ ) = r x ( P, ρ ) whenever v x ( a ) , v x ( b ) and v x ( c ) are not all equal.For the points x with | x | < ρ where v x ( a ) = v x ( b ) = v x ( c ), none of the coordi-nates of P has a pole, hence h x ( P, ρ ) ≤ r x ( P, ρ ) . (For x = 0, we need that ρ ≥ (cid:3) Definition 4.2.
We define the archimedean contribution to the radical of a non-constant point P = ( a : b : c ) on the line a + b + c = 0 in P ( M ) by r arch ( P, ρ ) = Z | z | = ρ h z (cid:0) ( b/c ) ℓ , ( c/a ) ℓ , ( a/b ) ℓ (cid:1) ( ρ ) dz πiz − h ∞ (cid:0) ( b/c ) ℓ , ( c/a ) ℓ , ( a/b ) ℓ (cid:1) . The completed radical is defined by r ( P, ρ ) = r na ( P, ρ ) + r arch ( P, ρ ) . Note that r arch ( P, ρ ) does not depend on the choice of coordinates for P . Withthese definitions, Theorem 4.1 reads Theorem 4.3 (Formal ABC) . For every nonconstant point P = ( a : b : c ) on theline a + b + c = 0 in P ( M ) , h ( P, ρ ) ≤ r ( P, ρ ) , for every ρ ≥ .Remark . (See [13, 15].) By Hurwitz’ formula for a function f : C → P froman algebraic curve C of genus g to the projective line,deg f = f − { , , ∞} + 2 g − − X x : f ( x ) =0 , , ∞ (ord( f, x ) − ≤ f − { , , ∞} + 2 g − . We interpret Theorem 4.3 as a weak form of Hurwitz’ formula, where deg f is theheight of ( f : 1 − f : − f − { , , ∞} is the radical of ( f : 1 − f : − The ABC Theorem for Meromorphic Functions
Lemma 5.1.
For a nonconstant point P = ( a : b : c ) on the line a + b + c = 0 in P ( M ) , there exists an ‘exceptional set’ E ⊂ (0 , ∞ ) of finite total length such that r arch ( P, ρ ) ≤ h ( P, ρ ) + O (log log h ( P, ρ )) for all ρ > , ρ E. Proof.
We first recall the lemma on the logarithmic derivative [4, Theorem 6.1, p.48]. Let the proximity function of a meromorphic function f be defined by m ( f, ρ ) = Z | z | = ρ log p | f ( z ) | dz πiz , MACHIEL VAN FRANKENHUIJSEN and the height by h ( f, ρ ) = h (( f : 1 : 0) , ρ ), as defined in (3.2). Then, for anonconstant meromorphic function f , there exists a subset E ⊂ (0 , ∞ ) of finitetotal length such that m (( f ) ℓ , ρ ) ≤ log h ( f, ρ ) + O (log log h ( f, ρ )) for all ρ > , ρ E. Choose now a , b and c such that ( a, b, c ) = (cid:0) ( b/c ) ℓ , ( c/a ) ℓ , ( a/b ) ℓ (cid:1) . Since c = − a − b , we have | a | + | b | + | c | ≤ | a | )(1 + | b | ). Hence Z | z | = ρ h z ( a, b, c )( ρ ) dz πiz ≤ log √ m ( a, ρ ) + m ( b, ρ ) . Since a = ( b/c ) ℓ and b = ( c/a ) ℓ , we find a subset E of (0 , ∞ ) of finite total lengthsuch that Z | z | = ρ h z ( a, b, c )( ρ ) dz πiz ≤ log h ( b/c, ρ ) + log h ( c/a, ρ )+ O (log log h ( b/c, ρ ) + log log h ( c/a, ρ ))for all ρ > ρ E . Since h ( b/c, ρ ) and h ( c/a, ρ ) ≤ h ( P, ρ ), the lemma follows. (cid:3)
Remark . Using an ultrafilter on (0 , ∞ ), one can suppress the dependence on ρ ,see [7]. In this formalism, there is no exceptional set E .Applying the bound for r arch ( P, ρ ) of Lemma 5.1, we obtain from Theorem 4.3,
Theorem 5.3 (Function Theoretic ABC) . Let P = ( a : b : c ) be a nonconstantpoint on the line a + b + c = 0 in P ( M ) . Then there exists an open set E ⊂ (1 , ∞ ) of finite total length such that h ( P, ρ ) ≤ r na ( P, ρ ) + 2 log h ( P, ρ ) + O (log log h ( P, ρ )) , for every ρ ≥ , ρ E . Conclusion
From Lemma 5.1, we see that the term 2 log h ( P, ρ )+ O (log log h ( P, ρ )) is a boundfor the archimedean contribution to the radical. This suggests that the function ψ in Conjecture 1.2 should be viewed as a bound for the archimedean contribution tothe radical, and not as this contribution itself. Therefore, we propose for a numberfield k ,1. to define a completed radical r ( P ) = r na ( P ) + r arch ( P );2. to show that h ( P ) ≤ r ( P ) for every point on the line a + b + c = 0 in P ( k );3. to obtain a bound of the type r arch ( P ) ≤ ψ ( h ( P )).Presumably, step 2 will follow from formal properties of the definition in step 1. Step3 will be the hard step. The strongest possible result will have ψ ( h ) = O ( √ h/ log h ),but even a result where ψ ( h ) = (1 − /C ) h for some C > h ≤ Cr . This would settle, for example, Fermat’s Last Theoremfor all exponents greater than 3 C . HE ABC THEOREM FOR MEROMORPHIC FUNCTIONS 7
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