12/10/14 06:08:27.32
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２．IUTT-IV P22
Theorem 1.10. (Log-volume Estimates for Θ-Pilot Objects) Set
Θdef = min (40ηprm/3 , 3?210)∈ R>0
? where the constant ηprm ∈ R>0 is as in Proposition 1.6. Then the constant
Θ ∈ R>0 satisfies the following property:
Fix a collection of initial Θ-data as in [IUTchI], Definition 3.1. Suppose
that we are in the situation of [IUTchIII], Corollary 3.12. Also, in the notation of
[IUTchI], Definition 3.1, let us write dmod def = [Fmod : Q] and
Fmod ⊆ Ftpd def = Fmod( EFmod [2] ) ⊆ F
for the “tripodal” intermediate field obtained from Fmod by adjoining the fields
of definition of the 2-torsion points of any model of EF over Fmod [cf. Proposition
1.8, (ii), (iii)]. Moreover, we assume that the (3・5)-torsion points of EF are defined
over F, and that
F = Fmod(√?1, EFmod [2 ・ 3 ・ 5] ) def = Ftpd(√?1, EFtpd [3 ・ 5] )
? i.e., that F is obtained from Ftpd by adjoining√?1, together with the fields of
definition of the (3 ・ 5)-torsion points of a model EFtpd of the elliptic curve EF over
Ftpd determined by the Legendre form of the Weierstrass equation [cf., e.g., the
statement of Corollary 2.2, below; Proposition 1.8, (vi)]. [Thus, it follows from
Proposition 1.8, (iv), that EF?=EFtpd Ftpd F over F.] If Fmod ⊆ F ⊆ K is any
intermediate extension which is Galois over Fmod, then we shall write