Explicit Estimates in Inter-universal Teichm¨ uller Theory (joint work w/ S. Mochizuki, I. Fesenko, Y. Hoshi, and

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Explicit Estimates in Inter-universal Teichm¨ uller Theory (joint work w/ S. Mochizuki, I. Fesenko, Y. Hoshi, and

W. Porowski)

Arata Minamide

RIMS, Kyoto University

September 7, 2021

Notations Log-volume IUTchIV Vojta ABC (ii) Lem Goal Theta Heights Auxiliary Main

Arata Minamide (RIMS, Kyoto University) Explicits Estimates in IUTch September 7, 2021 1 / 34

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Notations

Primes: the set of all prime numbers

F: a number field ⊇ OF: the ring of integers

∆F: the absolute value of the discriminant ofF V(F)^non: the set of nonarchimedean places ofF V(F)^arc: the set of archimedean places ofF V(F) ^def= V(F)^non ∪

V(F)^arc

For v∈V(F), writeF_v for the completion ofF atv

For v∈V(F)^non, writepv⊆ OF for the prime ideal corr. tov

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Let v∈V(F)^non. Writeord_v :F^×↠Z for the order def’d byv.

Then for any x∈F, we shall write

|x|v

def= ♯(OF/pv)⁻^ord^v^(x).

Let v∈V(F)^arc. Writeσ_v :F ,→Cfor the embed. det’d, up to complex conjugation, byv. Then for any x∈F, we shall write

|x|v

def= |σ_v(x)|^[F_C^v^:R].

Note: (Product formula) Forα∈F^×, it holds that

∏

v∈V(F)

|α|v = 1.

For an elliptic curveE /a field, writej(E) for the j-invariant of E

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Log-volume estimates for Θ-pilot objects (cf. [IUTchIII], Cor 3.12)

Theorem Write

−|log(Θ)| ∈ R∪ {∞}

for the (process.-normalized, mono-an.) log-volume of the “holomorphic hull” of theunion of the possible images of a Θ-pilot object, rel. to the relevant Kum. isoms, in the multira’l rep’n of [IUTchIII], Thm 3.11, (i), which we regard as sub. to (Ind1), (Ind2), (Ind3);

−|log(q)| ∈ R

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Theorem Write

−|log(Θ)| ∈ R∪ {∞}

−|log(q)| ∈ R

for the (process.-normalized, mono-an.) log-volume of the image of a q-pilot object, rel. to the relevant Kum. isoms, in the multirad’l rep’n.

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Theorem Write

−|log(Θ)| ∈ R∪ {∞}

−|log(q)| ∈ R

for the (process.-normalized, mono-an.) log-volume of the image of a q-pilot object, rel. to the relevant Kum. isoms, in the multirad’l rep’n.

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Results in [IUTchIV]

For λ ∈ Q\ {0,1},

Aλ: the elliptic curve/Q(λ) def’d by “y² =x(x−1)(x−λ)”

F_λ ^def= Q(λ,√

−1, A_λ[3·5](Q))

⇒ E_λ ^def= A_λ×_Q(λ)F_λ has at mostsplit multipl. red. at ∀ ∈V(F_λ) qλ: the arithmetic divisor det’d by theq-parameters of E_λ/F_λ fλ: the “reduced” arithmetic divisor det’d byqλ

d_λ: the arithmetic divisor det’d by the diﬀerent ofF_λ/Q

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Theorem (Vojta Conj. — in the case of P¹\ {0,1,∞}— for “K”) Let d∈Z>0,ϵ∈R>0∩R≤1,

K ⊆ Q\ {0,1}: acompactly bounded subset whose “support”∋2,∞. Then ^∃B(d, ϵ,K)∈R>0 — that depends only ond,ϵ, and K — s.t.

the function on K^≤^d^def= {λ∈ K | [Q(λ) :Q]≤d}given by λ 7→ ¹₆·deg(q_λ)−(1 +ϵ)·(deg(d_λ) + deg(f_λ)) is bounded by B(d, ϵ,K).

Proof: By applying

the finiteness of {λ∈ K^≤^d | deg(q_λ)≤γ} (γ ∈R>0) (cf. Northcott’s theorem),

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Theorem (Vojta Conj. — in the case of P¹\ {0,1,∞}— for “K”) Let d∈Z>0,ϵ∈R>0∩R≤1,

K ⊆ Q\ {0,1}: acompactly bounded subset whose “support”∋2,∞. Then ^∃B(d, ϵ,K)∈R>0 — that depends only ond,ϵ, and K — s.t.

the function on K^≤^d^def= {λ∈ K | [Q(λ) :Q]≤d}given by λ 7→ ¹₆·deg(q_λ)−(1 +ϵ)·(deg(d_λ) + deg(f_λ)) is bounded by B(d, ϵ,K).

Proof: By applying

the finiteness of {λ∈ K^≤^d | deg(q_λ)≤γ} (γ ∈R>0) (cf. Northcott’s theorem),

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♯{j(“arithmetic” elliptic curve over a field of char. zero)}= 4 (cf. Takeuchi’s list),

the prime number theorem,

the theory of Galois actions on torsion points of elliptic curves (cf. [GenEll]),

we conclude that for all butfinitely many λ∈ K^≤^d, there exists a prime number lλ such that

(i) ^∃an initialΘ-data(Q/F_λ, E_λ, l_λ, . . .) s.t. E_λ hasgood red. at every ∈V(Fλ)^good∩V(Fλ)^non that does not divide 2lλ

(In the following, we shall write

q^bad_λ : the arithmetic divisor det’d by “restricting q_λ toV^bad_mod”.)

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♯{j(“arithmetic” elliptic curve over a field of char. zero)}= 4 (cf. Takeuchi’s list),

the prime number theorem,

the theory of Galois actions on torsion points of elliptic curves (cf. [GenEll]),

we conclude that for all butfinitely many λ∈ K^≤^d, there exists a prime number lλ such that

(i) ^∃an initialΘ-data(Q/F_λ, E_λ, l_λ, . . .) s.t. E_λ hasgood red.

at every ∈V(F_λ)^good∩V(F_λ)^non that does not divide 2l_λ (In the following, we shall write

q^bad_λ : the arithmetic divisor det’d by “restricting qλ toV^bad_mod”.)

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(ii) ¹₆ ·deg(q^bad_λ ) ≤ (1 + ^20d_l ^λ

λ )·(deg(d_λ) + deg(f_λ)) + 20·δ_λ·l_λ, whered_λ := [Q(λ) :Q],δ_λ := 2¹²·3³·5·d_λ (cf. (i); “Cor 3.12”) (iii) ord_l_λ(q_□) < deg(q_λ)^1/2, whereV(F_λ)∋□|l_λ

(iv) deg(qλ)^1/2 ≤ lλ ≤ 10·δ·deg(qλ)^1/2·log(2·δ·deg(qλ)) whereδ := 2¹²·3³·5·d

Then it follows from (i), (iii) [cf. also the “compactness” of K] that λ 7→ ¹₆deg(q_λ)−¹₆deg(q^bad_λ )−deg(q_λ)^1/2log(2δdeg(q_λ)) is bounded. On the other hand, it follows from (ii), (iv) that

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(ii) ¹₆ ·deg(q^bad_λ ) ≤ (1 + ^20d_l ^λ

λ )·(deg(d_λ) + deg(f_λ)) + 20·δ_λ·l_λ, whered_λ := [Q(λ) :Q],δ_λ := 2¹²·3³·5·d_λ (cf. (i); “Cor 3.12”) (iii) ord_l_λ(q_□) < deg(q_λ)^1/2, whereV(F_λ)∋□|l_λ

(iv) deg(qλ)^1/2 ≤ lλ ≤ 10·δ·deg(qλ)^1/2·log(2·δ·deg(qλ)) whereδ := 2¹²·3³·5·d

Then it follows from (i), (iii) [cf. also the “compactness” of K] that λ 7→ ¹₆deg(q_λ)−¹₆deg(q^bad_λ )−deg(q_λ)^1/2log(2δdeg(q_λ)) is bounded. On the other hand, it follows from (ii), (iv) that

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1

6deg(q^bad_λ ) ≤ (1 +δ·deg(q_λ)⁻^1/2)(deg(d_λ) + deg(f_λ)) + 200δ²·deg(q_λ)^1/2log(2δdeg(q_λ)).

In particular, these two displays imply that λ7→ (1−²₅^(60δ)²_deg(q^log(2δ^deg(q^λ⁾⁾

λ)^1/2 )¹₆deg(q_λ)−(1 +_deg(q^δ

λ)^1/2)(deg(d_λ) + deg(f_λ)) is bounded. By enlarging our “exceptional set”, we conclude that

λ 7→ ¹₆ ·deg(q_λ)−(1 +ϵ)·(deg(d_λ) + deg(f_λ)) is bounded. This completes the proof of Theorem.

Then, by applying the theory of noncritical Belyi maps, we obtain (∗): the “version withK removed” of Theorem (cf. [GenEll]).

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1

In particular, these two displays imply that λ7→

(1−²₅^(60δ)²_deg(q^log(2δ^deg(q^λ⁾⁾

λ)^1/2 )¹₆deg(q_λ)−(1 + ^δ

deg(qλ)^1/2)(deg(d_λ) + deg(f_λ)) is bounded.

By enlarging our “exceptional set”, we conclude that λ 7→ ¹₆ ·deg(q_λ)−(1 +ϵ)·(deg(d_λ) + deg(f_λ)) is bounded. This completes the proof of Theorem.

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1

λ)^1/2 )¹₆deg(q_λ)−(1 + ^δ

deg(qλ)^1/2)(deg(d_λ) + deg(f_λ)) is bounded. By enlarging our “exceptional set”, we conclude that

λ 7→ ¹₆·deg(qλ)−(1 +ϵ)·(deg(dλ) + deg(fλ)) is bounded. This completes the proof of Theorem.

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1

λ)^1/2 )¹₆deg(q_λ)−(1 + ^δ

deg(qλ)^1/2)(deg(d_λ) + deg(f_λ)) is bounded. By enlarging our “exceptional set”, we conclude that

λ 7→ ¹₆·deg(qλ)−(1 +ϵ)·(deg(dλ) + deg(fλ)) is bounded. This completes the proof of Theorem.

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Theorem (Corollary of (∗) — ABC Conjecture for number fields) Let d∈Z>0,ϵ∈R>0∩R_≤1.

Then ^∃C(d, ϵ)∈R>0 — that depends only on dandϵ— s.t. for

• F: a number field — whered = [F :Q]

• (a, b, c) : a triple of elements∈F^× — wherea+b+c= 0 we have

H_F(a, b, c) < C(d, ϵ)·(∆_F ·rad_F(a, b, c))^1+ϵ

— where

HF(a, b, c) ^def= ∏

v∈V(F)max{|a|v,|b|v,|c|v}, rad_F(a, b, c) ^def= ∏

{v∈V(F)^non|♯{|a|v,|b|v,|c|v}≥2}♯(OF/p_v).

Note: We do not know the constant “C(d, ϵ)” explicitly.

For instance, it is hard to compute noncritical Belyi mapsexplicitly.

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Theorem (Corollary of (∗) — ABC Conjecture for number fields) Let d∈Z>0,ϵ∈R>0∩R_≤1.

Then ^∃C(d, ϵ)∈R>0 — that depends only on dandϵ— s.t. for

• F: a number field — whered = [F :Q]

• (a, b, c) : a triple of elements∈F^× — wherea+b+c= 0 we have

H_F(a, b, c) < C(d, ϵ)·(∆_F ·rad_F(a, b, c))^1+ϵ

— where

HF(a, b, c) ^def= ∏

v∈V(F)max{|a|v,|b|v,|c|v}, rad_F(a, b, c) ^def= ∏

{v∈V(F)^non|♯{|a|v,|b|v,|c|v}≥2}♯(OF/p_v).

Note: We do not know the constant “C(d, ϵ)”explicitly.

For instance, it is hard to compute noncritical Belyi mapsexplicitly.

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Computations concerning (ii)

For γ ∈R, we shall write ⌊γ⌋ (resp. ⌈γ⌉) for the largest integer

≤ γ (resp. the smallest integer ≥ γ).

{k_i}i∈I: a finite set of p-adic local fields (Oki: the ring of integers) ei (resp. di): the abs. ram. index (resp. the order of an gen. of δki)

a_i^def= {1

ei⌈_p^e₋ⁱ₂⌉ (p >2)

2 (p= 2) b_i^def= ⌊^log(p_log(p)^·^eⁱ^/(p⁻¹⁾⁾⌋ −_e¹_i aIdef

= ∑

i∈I

ai, bI def

= ∑

i∈I

bi, dI def

= ∑

i∈I

di

µ^log_k

I: the (nor’d) log-vol. on k_I ^def= ⊗i∈Iki s.t. µ^log_k

I (⊗i∈IOki) = 0

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Lemma

For λ∈ _e¹_iZ, write p^λOki for the fractional ideal generated by any element x∈ki s.t. ord(x) =λ. Let

ϕ:Qp⊗_Zp

⊗

i∈I

log_p(O_k^×_i) →^∼ Qp⊗_Zp

⊗

i∈I

log_p(O_k^×_i)

be an automorphismof the finite dimensional Qp-vector space that induces an automorphism of the submodule ⊗

i∈Ilog_p(O_k^×_i).

(i) WriteI^∗^def= {i∈I |e_i > p−2}. For any λ∈ _e¹_i

0Z, i₀ ∈I, ϕ(p^λ(⊗i∈IOki)^∼) ∪

p^⌊^λ^⌋⊗

i∈I 1

2plog_p(O_k^×_i)

⊆ p^⌊^λ⁻^dÎ⁻âÎ^⌋⊗

i∈I

log_p(O^×_k_i) ⊆ p^⌊^λ⁻^dÎ⁻âÎ^⌋−^bÎ(⊗i∈IOki)^∼.

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Lemma

For λ∈ _e¹_iZ, write p^λOki for the fractional ideal generated by any element x∈ki s.t. ord(x) =λ. Let

ϕ:Qp⊗_Zp

⊗

i∈I

log_p(O_k^×_i) →^∼ Qp⊗_Zp

⊗

i∈I

log_p(O_k^×_i)

be an automorphismof the finite dimensional Qp-vector space that induces an automorphism of the submodule ⊗

i∈Ilog_p(O_k^×_i).

(i) WriteI^∗^def= {i∈I |e_i > p−2}. For any λ∈ _e¹_i

0Z, i₀ ∈I, ϕ(p^λ(⊗i∈IOki)^∼) ∪

p^⌊^λ^⌋⊗

i∈I 1

2plog_p(O_k^×_i)

⊆ p^⌊^λ⁻^dÎ⁻âÎ^⌋⊗

i∈I

log_p(O^×_k_i) ⊆ p^⌊^λ⁻^dÎ⁻âÎ^⌋−^bÎ(⊗i∈IOki)^∼.

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Moreover, we have µ^log_k

I(p^⌊^λ⁻^dÎ⁻âÎ^⌋−^bÎ(⊗i∈IOki)^∼)

≤ (−λ+d_I+ 1) log(p) +∑

i∈I^∗

(3 + log(e_i)).

(ii) Suppose that p >2 and e_i= 1 (^∀i∈I). Then ϕ((⊗i∈IOki)^∼) ⊆ ⊗

i∈I 1

2plog_p(O_k^×_i) = (⊗i∈IOki)^∼. Moreover, we have

µ^log_k

I((⊗i∈IOki)^∼) = 0.

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Moreover, we have µ^log_k

I(p^⌊^λ⁻^dÎ⁻âÎ^⌋−^bÎ(⊗i∈IOki)^∼)

≤ (−λ+d_I+ 1) log(p) +∑

i∈I^∗

(3 + log(e_i)).

(ii) Suppose that p >2 and e_i= 1 (^∀i∈I). Then ϕ((⊗i∈IOki)^∼) ⊆ ⊗

i∈I 1

2plog_p(O_k^×_i) = (⊗i∈IOki)^∼. Moreover, we have

µ^log_k

I((⊗i∈IOki)^∼) = 0.

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In the following discussion, for simplicity, write

(F /F, E, l, . . .) ^def= (Q/F_λ, E_λ, l_λ, . . .).

(⇒ K =F(E[l]) ⊇ F ⊇ F_mod: the field of moduli ofE)

d_mod ^def= [F_mod:Q] ≥ e_mod ^def= the max. ram. index of F_mod/Q d^∗_mod ^def= 2¹²·3³·5·d_mod ≥ e^∗_mod ^def= 2¹²·3³·5·e_mod

V^non_Q ^def= V(Q)^non ⊇ V^dst_Q ^def= {v_Q ∈V^non_Q |v_Q ramifies inK} Let us compute an upper boundfor the

(process.-normalized, mono-an.) log-volume of the “holomorphic hull” of the union of the possible images of aΘ-pilot object, rel. to the relevant Kum. isoms, in the multira’l rep’n of [IUTchIII], Thm 3.11, (i), which we regard as sub. to (Ind1), (Ind2), (Ind3)

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(F /F, E, l, . . .) ^def= (Q/F_λ, E_λ, l_λ, . . .).

V^non_Q ^def= V(Q)^non ⊇ V^dst_Q ^def= {v_Q ∈V^non_Q |v_Q ramifies in K}

Let us compute an upper boundfor the

(process.-normalized, mono-an.) log-volume of the “holomorphic hull” of the union of the possible images of aΘ-pilot object, rel. to the relevant Kum. isoms, in the multira’l rep’n of [IUTchIII], Thm 3.11, (i), which we regard as sub. to (Ind1), (Ind2), (Ind3)

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(F /F, E, l, . . .) ^def= (Q/F_λ, E_λ, l_λ, . . .).

V^non_Q ^def= V(Q)^non ⊇ V^dst_Q ^def= {v_Q ∈V^non_Q |v_Q ramifies in K} Let us compute an upper boundfor the

(process.-normalized, mono-an.) log-volume of the “holomorphic hull” of the union of the possible images of aΘ-pilot object, rel.

to the relevant Kum. isoms, in the multira’l rep’n of [IUTchIII], Thm 3.11, (i), which we regard as sub. to (Ind1), (Ind2), (Ind3)

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(A) Let v_Q∈V^dst_Q . Fix j (∈ {1,2, . . . , l^⋇}) and the collection {v_i}_i_∈S^±

j+1

of [not necessarily distinct] elements of V(Fmod)v_Q. Write v_i ∈V

→∼ V(F_mod) for the elem’t corr. to v_i.

Then, by applying Lem, (i), we obtain an upper boundon the component of the log-volume in question corresponding to the tensor product of the Q-spans of the log-shells associated to the collection {v_i}_i_∈S^±

j+1 as follows: (−λ+d_I+ 1) log(p_v_Q) + 4(j+ 1)ι_v_Qlog(e^∗_mod·l)

— whereλ = {_j2

2lord(qv_j) (v_j ∈V^bad)

0 (v_j ∈V^good) ι_v_Q = {

1 (pv_Q ≤e^∗_modl) 0 (p_v_Q > e^∗_modl)

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j+1

→∼ V(F_mod) for the elem’t corr. to v_i. Then, by applying Lem, (i), we obtain an upper boundon the component of the log-volume in question corresponding to the tensor product of the Q-spans of the log-shells associated to the collection {v_i}_i_∈S^±

j+1 as follows:

(−λ+d_I+ 1) log(p_v_Q) + 4(j+ 1)ι_v_Qlog(e^∗_mod·l)

— whereλ = {_j2

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j+1

→∼ V(F_mod) for the elem’t corr. to v_i. Then, by applying Lem, (i), we obtain an upper boundon the component of the log-volume in question corresponding to the tensor product of the Q-spans of the log-shells associated to the collection {v_i}_i_∈S^±

j+1 as follows:

(−λ+d_I+ 1) log(p_v_Q) + 4(j+ 1)ι_v_Qlog(e^∗_mod·l)

— whereλ = {_j2

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(B) Let v_Q∈V^non_Q \V^dst_Q . Fix j (∈ {1,2, . . . , l^⋇}) and {vi}_i_∈S± j+1

as in (A).

Then, by applying Lem, (ii), we obtain an upper boundon the comp. of the log-vol. in question corr. to the tensor prod. of the Q-spans of the log-shells assoc. to {vi}_i_∈S±

j+1 as follows: 0

(C) Let v_Q ∈V^arc_Q . Fix j (∈ {1,2, . . . , l^⋇}) and {v_i}_i_∈S^±

j+1 as in (A). Then we obtain an upper boundon the comp. of the log-vol. in question corr. to the tensor prod. of the Q-spans of the log-shells assoc. to {v_i}_i_∈S^±

j+1 as follows:

(j+ 1)·log(π)

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as in (A). Then, by applying Lem, (ii), we obtain an upper boundon the comp. of the log-vol. in question corr. to the tensor prod. of the Q-spans of the log-shells assoc. to {vi}_i_∈S±

j+1 as follows:

0

j+1 as follows:

(j+ 1)·log(π)

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j+1 as follows:

0

j+1 as follows:

(j+ 1)·log(π)

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j+1 as follows:

0

j+1 as in (A). Then we obtain an upper boundon the comp. of the log-vol.

in question corr. to the tensor prod. of the Q-spans of the log-shells assoc. to {v_i}_i_∈S^±

j+1 as follows:

(j+ 1)·log(π)

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After computing a “weighted average upper bound”, i.e., (_[F ¹

mod:Q])^j+1 ∑

v0,... ,vj∈V(Fmod)vQ

∏

0≤i≤j

[(F_mod)_v_i :Qv_Q](−)

and then a “procession-normalized upper bound”, i.e.,

1 l^⋇

∑

1≤j≤l^⋇

(−)

for eachv_Q ∈V_Q, by summing over v_Q ∈V_Q these estimates, we obtain anupper boundon −|log(Θ)| as follows:

l+1 4

{

(1 + ^12d_l^mod)·(deg(d_λ) + deg(f_λ)) + 10·e^∗_mod·l

−¹₆ ·(1−¹²_l2)·log(q^bad_λ )

}− _2l¹ ·deg(q^bad_λ )

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After computing a “weighted average upper bound”, i.e., (_[F ¹

mod:Q])^j+1 ∑

v0,... ,vj∈V(Fmod)vQ

∏

0≤i≤j

[(F_mod)_v_i :Qv_Q](−)

and then a “procession-normalized upper bound”, i.e.,

1 l^⋇

∑

1≤j≤l^⋇

(−)

for eachv_Q ∈V_Q, by summing over v_Q ∈V_Q these estimates, we obtain anupper boundon −|log(Θ)| as follows:

l+1 4

{

(1 +^12d_l^mod)·(deg(d_λ) + deg(f_λ)) + 10·e^∗_mod·l

−¹₆ ·(1−¹²_l2)·log(q^bad_λ )

}−_2l¹ ·deg(q^bad_λ )

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On the other hand, since −|log(Θ)| ≥ −|log(q)|=−_2l¹ ·deg(q^bad_λ ), we conclude that

1

6 ·(1− ¹²_l2)·deg(q^bad_λ ) ≤

(1 +^12d_l^mod)·(deg(d_λ) + deg(f_λ)) + 10·e^∗_mod·l, hence that

1

6 ·deg(q^bad_λ ) ≤ (1 +^20d^mod_l )·(deg(d_λ) + deg(f_λ)) + 20·d^∗_mod·l.

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1

6 ·(1− ¹²_l2)·deg(q^bad_λ ) ≤

(1 +^12d_l^mod)·(deg(d_λ) + deg(f_λ)) + 10·e^∗_mod·l,

hence that

1

6 ·deg(q^bad_λ ) ≤ (1 +^20d^mod_l )·(deg(d_λ) + deg(f_λ)) + 20·d^∗_mod·l.

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1

6 ·(1− ¹²_l2)·deg(q^bad_λ ) ≤

(1 +^12d_l^mod)·(deg(d_λ) + deg(f_λ)) + 10·e^∗_mod·l, hence that

1

6 ·deg(q^bad_λ ) ≤ (1 +^20d_l^mod)·(deg(d_λ) + deg(f_λ)) + 20·d^∗_mod·l.

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Goal of this joint work:

Under certain conditions, we prove (∗) directly [i.e., without applying the theory of noncritical Belyi maps] to compute the constant “C(d, ϵ)” explicitly.

Technical Diﬃculties of Explicit Computations

(i) We cannot use the compactness of “K” at the place2

⇒ We develop the theory of´etale theta functions so that it functions properly at the place 2

(ii) We cannot use the compactness of “K” at the place∞

⇒ By restricting our attention to special number fields, we

“bound” thearchimedeanportion of the “height” of the elliptic curveE_λ

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Goal of this joint work: Under certain conditions, we prove (∗) directly [i.e., without applying the theory of noncritical Belyi maps] to compute the constant “C(d, ϵ)” explicitly.

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⇒ We develop the theory of´etale theta functions so that it functions properly at the place2

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⇒ We develop the theory of´etale theta functions so that it functions properly at the place2

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Etale Theta Functions´

p,l: distinct prime numbers — wherel≥5 k: a p-adic local field ⊇ Ok: the ring of integers X: an elliptic curve/k which has split multipl. red. /Ok

q ∈ Ok: the q-parameter of X

X^log ^def= (X,{o} ⊆X): the smooth log curve/k assoc. toX In the following, we assume that

√−1 ∈ k

X[2l](k) = X[2l](k) [X^log/{±1}]is a k-core