560 XIII Connections with Number Theory
The Euclidean algorithm may be used to derive the polynomial identity( 3 X^2 + 4 a)(X^4 − 2 aX^2 − 8 bX+a^2 )−( 3 X^3 − 5 aX− 27 b)(X^3 +aX+b)=d,where once againd= 4 a^3 + 27 b^2. Substitutingp/qforX, we obtain
4 dq^7 = 4 ( 3 p^2 q+ 4 aq^3 )p′−( 3 p^3 − 5 apq^2 − 27 bq^3 )q′.Similarly, the Euclidean algorithm may beused to derive the polynomial identity
f(X)( 1 − 2 aX^2 − 8 bX^3 +a^2 X^4 )+g(X)X( 1 +aX^2 +bX^3 )=d,where
f(X)= 4 a^3 + 27 b^2 −a^2 bX+a( 3 a^3 + 22 b^2 )X^2 + 3 b(a^3 + 8 b^2 )X^3 ,
g(X)=a^2 b+a( 5 a^3 + 32 b^2 )X+ 2 b( 13 a^3 + 96 b^2 )X^2 − 3 a^2 (a^3 + 8 b^2 )X^3.Substitutingq/pforX, we obtain
4 dp^7 = 4 {( 4 a^3 + 27 b^2 )p^3 −a^2 bp^2 q+( 3 a^4 + 22 ab^2 )pq^2 + 3 (a^3 b+ 8 b^3 )q^3 }p′
+{a^2 bp^3 +( 5 a^4 + 32 ab^2 )p^2 q+( 26 a^3 b+ 192 b^3 )pq^2 − 3 (a^5 + 8 a^2 b^2 )q^3 }q′.Sinced=0, it follows from these two relations that
max(|p|,|q|)^7 ≤C 1 max(|p|,|q|)^3 max(|p′|,|q′|)and hence
max(|p|,|q|)^4 ≤C 1 max(|p′|,|q′|).But the two relations also show that the greatest common divisoreofp′andq′divides
both 4dq^7 and 4dp^7 , and hence also 4d,sincepandqare coprime. Consequently
max(|p′|,|q′|)≤ 4 |d|max(|p′′|,|q′′|).Combining this with the previous inequality, we obtain
4 h(P)≤h( 2 P)+C′′for some constantC′′=C′′(a,b)>0.
This proves the result, withC=max(C′,C′′).
Proposition 9There exists a unique functionhˆ:E→Rsuch that
(i)hˆ−h is bounded,
(ii)hˆ( 2 P)= 4 hˆ(P)for every P∈E.
Furthermore, it is given by the formulahˆ(P)=limn→∞h( 2 nP)/ 4 n.