A note on $p$-rational fields and the abc-conjecture
Christian Maire Marine Rougnant





Quantum enhancement of sensitivity achieved by photon-number-resolving detection in the dark port of a two-path interferometer operating at high intensities

    Holger F. Hofmann
    Submitted 6 July 2018 / Accepted 19 March 2019
==============================================================================================================================================================




 In this short note we confirm the relation between the generalized abc-conjecture and the p-rationality of  number fields. Namely, we prove  that given  /  a real quadratic extension or an imaginary S_3-extension, if the  generalized abc-conjecture holds in , then there exist at least c log X prime numbers p ≤ X for which  is p-rational, here c is some nonzero constant depending on . The real quadratic  case was recently suggested by  Böckle-Guiraud-Kalyanswamy-Khare.








§ INTRODUCTION



Let  be a number field and let p be a prime number. To simplify, we assume p odd.
Denote by _p the maximal pro-p-extension of  unramified outside p; put G_p:=(_p/).

By class field theory, the pro-p group _p is finitely generated and one knows, since Shafarevich and Koch, that moreover _p is finitely presented (meaning that H^2(_p,_p) is finite).
In fact, _p may be pro-p free, for example when =, or when  is an imaginary quadratic field (when p>3) and p doesn't divide the class number of , or when =(ζ_p) for p regular primes, etc.

A number field  for which _p is pro-p  free is called p-rational (<cit.>).
Observe that  is p-rational if and only if the Leopoldt conjecture holds for  at p and the torsion _p of the abelianization _p^ab of _p is trivial (see <cit.>, or <cit.>).



The  study of  _p and  of the p-rationality started in the beginning of the 80's with Gras, Nguyen Quang Do, Movahhedi, Jaulent, and their students. Since the literature is rich: see for example <cit.>, <cit.>,  <cit.>,  <cit.>, <cit.>, <cit.>, <cit.>, <cit.> etc. See also  <cit.> for a well-detailed  presentation of  _p, of the  Leopoldt conjecture and of 
 p-rational fields. In the spirit of our paper, let us mention here the works of
Byeon <cit.> and Assim-Bouazzaoui <cit.> where they showed the infiniteness of 3 and 5-rational real quadratic fields.


Let us also precise at this level that a recent series of papers in different topics in number theory showed the interest of p-rational fields:  Goren <cit.>, Greenberg <cit.>,  Böckle-Guiraud-Kalyanswamy-Khare <cit.>, David-Pries <cit.>,
 Hajir-Maire <cit.>, Hajir-Maire-Ramakrishna <cit.>,  etc.
 
Assuming Leopoldt conjecture (for  at p), the p-rationality of  is therefore  equivalent to the nullity of  _p. Observe that _p ≃ H^2(_p,_p)^* for a cohomological point of view (see <cit.>).
When the  p-Sylow of the class group of  is trivial,   the quantity _p is isomorphic  to the torsion of the quotient of the units of the p-adic completions _v of  by the closure of the global units. Moreover, if we assume that no _v contains the p-roots of the unity (which is always the case when  p>[:]+1),  then the triviality of _p is equivalent to  the triviality of the normalized p-adic  regulator defined by Gras  <cit.>. 
Recently,    Gras <cit.>, <cit.>, Pitoun-Varescon <cit.>,  Barbulescu-Ray  <cit.> published a series of papers more concentrated on the computations  of _p, and on some heuristics.
In <cit.>,  Gras proposed the following conjecture:

 Let   be a number field. Then for large p,   is p-rational.  


This conjecture is in the same spirit of the Wieferich prime numbers problem. Indeed, given an odd prime number p, to compute the p-valuation of  2^p-1-1 is equivalent to compute the normalized p-adic  regulator of the   2-units of . In particular, in this case the nontriviality of the normalized p-adic  regulator  is equivalent for p to verify  the congruence 2^p-1≡ 1 ( p^2). 



In <cit.> Silverman showed how the  Wieferich prime numbers are related to the abc-conjecture. Let us be more precise.
Given an integer α∈^×\{± 1},   Silverman proved  that if the abc-conjecture holds then as X →∞

    #{ prime  number p,  p≤ X, α^p-1 1 ( p^2)}≥ c log X,

where c>0 is some absolute constant. See also <cit.>, and <cit.> for a generalization of Wieferich primes in number fields.
 


Observe now that the generalized abc-conjecture has already been used in  the context of Iwasawa theory. Indeed in <cit.> Ichimura gave a relationship between the Greenberg conjecture and the abc-conjecture.  A consequence of his work is that, for example,  for any real quadratic  field  if the generalized abc-conjecture holds in , then  the set of primes p for which  is p-rational, is infinite. See also  <cit.>. 



 The goal of our work is  to precise the quantity of such primes p, greatly inspired by the computations of Silverman.
 
  
Our main result involves the isotypic subspaces _p^χ of _p. 
Let us observe here that
the authors studied previously in <cit.> such cutting and the arithmetic consequences of the nullity of some _p^χ.




 Let / be a Galois extension of Galois group . Let us fix an odd prime number p ∤#. For an irreducible _p-character ψ of , let r_ψ(E_) be the ψ-rank of _p ⊗ E_, where   E_ denotes the units of the ring of integers Ø_ of .
 Let us also  cut _p by its isotypic subspaces _p^ψ, and 
 denote  by r_ψ(_p) the ψ-rank of _p.
 Observe that, assuming Leopoldt conjecture,   the number field  is p-rational if and only if r_ψ(_p)=0 for all irreducible _p-characters ψ. Moreover we will see that for p≫ 0, r_ψ(_p) ≤ r_ψ(E_) for all ψ.
 
 We will then focus on some special units u of E_: we denote by  the set of algebraic integers u ∈ having no conjugate on the unit circle.
 
 Here we prove:
 
 
Let / be a Galois extension of Galois group  and let   χ be an irreducible -character of  such that  the χ-component of  ⊗ E_ contains some unit u ∈.  If the generalized abc-conjecture holds for , then  as X→∞ 
    #{ prime  number p≤ X,  r_ψ(_p)< r_ψ(E_)    for  some   irred.  _p  -char.  ψ|χ}≥  c  log X,
 for some constant c>0 depending on .


(Of course, in Theorem <ref> one considers only prime numbers p ∤#.) 
As consequence we obtain the following result (the real quadratic case was suggested in <cit.>):



 
Let / be a real quadratic field or an imaginary S_3-extension. If the generalized abc-conjecture holds for ,  then as X →∞

    #{ prime  number p ≤ X,    is  p -rational}≥  c  log X,
 for some constant c>0 depending on .




 It is well known that Leopoldt conjecture holds in the situations of Corollary, but we don't assume Leopoldt conjecture in Theorem <ref>.



Let us add one additionnal remark about the units in  .

  
 The  following observations will be useful for us: 


  - an unit u ≠± 1 for which all the conjugates are  real is in ;

  -  every cubic field contains some unit u ∈;

  - Pisot numbers are in .

See also <cit.> on the abundance of Pisot units.



Our work contains two  sections. In the first one, we introduce the objects we need. In the second section, we give the proofs of our results.









§ THE OBJECTS
 
We start with a Galois extension / of degree m and Galois group . We denote by  the norm in /.

Let Ø_ be the ring of integers of , E_ be the units of Ø_, and μ_ be the group of the roots of the unity of .


 Let p be an odd prime number. In all that will follow, we suppose    that:
 

  (i) p∤#,

  (ii) p is unramified in /,

  (iii) p does not divide the class number h_ of .


One excludes this way only a finite set of prime numbers p. In particular, there exists an explicit prime number p_0 such that  every  p>p_0 satisfies (i), (ii) and (iii).



 §.§ p-rational fields and isotypic components




  §.§.§ 

Let S_p be the set of places of  above p. For v ∈ S_p, denote by _v the completion of  at v, by Ø_v the   ring of integers of _v, and by π_v an uniformizer of _v. 
Then the p-completion _:=_p⊗ E_K of E_ embeds diagonally, via ι,  in _p:=∏_v∈ S_p_v^1, where _v^1:=1+π_v Ø_v is the group of principal units of _v. Observe that here _p ≃_p^m.
By p-adic class field theory (and due to the fact that p∤ h_), the group _p^ab is isomorphic to _p/ι(_). Then, assuming  Leopoldt conjecture for  at p (meaning here that ι is injective), the number field  is p-rational if and only if _p/ι(_) is without torsion.



  §.§.§ 
 
 
Observe that as p is unramified in /, we also get that  p∤ |μ_|, and as p∤#, the character (as -module) of _ is equal to the character of _p⊗(⊗ E_) ≃_D_∞^, 
where D_∞ is the decomposition group  of an archimedean place in / and where  is the trivial character. In particular, _ is a submodule of the regular representation. 


To be complete,   _p is isomorphic to the regular representation (here _v^1 has no nontrivial root of unity).



  §.§.§ 
 

Let us fix an irreducible -character χ of . 
Let   []e_χ≃ M_n_χ(D) be the simple algebra of [] associated to χ, where D is a skew field of degree s_χ^2 over its center (the integer s_χ is the Schur index of χ).
Then χ=s_χ∑_ψ|χψ, where the sum is taken over irreducible _p-characters ψ dividing χ (here p∤#). 

Let E_^χ be the χ-component of the []-module ⊗ E_, then the character of  E_^χ  is written as t_χχ for some t_χ∈{0, ⋯, n_χ}.
Given an irreducible _p-character ψ|χ, the integer s_χ t_χ is then the ψ-rank r_ψ(E_)  of _p⊗ E_.



If M is a _p[]-module of finite type,  the ψ-rank r_ψ(M) of M is defined as  r_ψ(M):=1/(ψ)__p( M^ψ/(M^ψ)^ p).

As seen before r_ψ(E_)=r_ψ(_), obviously r_ψ(_) ≥ r_ψ(ι(_)), and Leopoldt conjecture is equivalent to the equality r_ψ(_)=r_ψ(ι(_)) for every  χ and ψ. 
Observe that one knows that r_ψ(ι(_))≥ 1 when r_ψ(_) ≠ 0 (see <cit.>).
 When  is abelian, one has r_ψ(_) ≤  1.
 

As seen before, with all the assumptions,  the torsion of  _p/ι(_) is isomorphic to _p. Thus, r_ψ(_p) ≤ r_ψ(_). If for every ψ|χ the ψ-rank of _p/ι(_) is maximal, meaning r_ψ(_p)=r_ψ(_), then necessarily, for every unit x ∈ E_^χ such that x≡ 1 ( )  for all |p, one must have x≡ 1 ( ^2) for all |p.

 

If there exists an unit u ∈ E_ ^χ such that u≡ 1 ( _0) but u 1 ( _0^2) for some _0|p, then r_ψ(_p) <r_ψ(_) for some ψ |χ.



Put  x=u^(_0)-1∈ E_^χ, where ()=#Ø_/. Observe that x≡ 1 ( ) for every |p (the extension / is Galois) but, easily, one also has x 1 ( _0^2). We conclude with the small discussion above.

 

 



 §.§ The generalized abc-conjecture


See  <cit.>.
If I ⊂Ø_ is an integral ideal, let us  denote by (I) the following ideal:

    (I)=∏_ | I(),

where the product is taken over prime ideal  dividing I and where as usual ()=#Ø_K/ is the absolute norm of . 

The generalized abc-conjecture for   states that for any ε>0, there exists a constant C_K,ε>0 such that the inequality :

    ∏_vmax{|a|_v,|b|_v,|c|_v}⩽ C_K,ε((abc))^1+ε

holds for all nonzero a,b,c ∈Ø_K verifying a+b=c, (a,b)=1, where the product is taken over all absolute values of  and where |·|_v denotes the normalized norm of _v (such that ∏_v|x|_v=1 for all x∈^×).

Here we use it in the case where b=u_2 and c=u_1 are two distinct units of  and a=u_1-u_2 : for every ε >0, there exists a constant C_,ε such that for all  u_1≠ u_2 ∈ E_, one has 

    |(u_1-u_2)| ≤ C_,ε((u_1-u_2))^1+ε.








 
 
 



§ PROOFS





 §.§ 
 As explained in Introduction, some part of the proof is greatly inspired by <cit.>.

Let / be a Galois extension of degree m.
Consider the number field Ł:=(ζ)  where ζ is a primitive nth-root of 1. The extension Ł/ is Galois of degree O(φ(n)).  



 
Let T_n be the set of integers j∈{1,⋯, n-1} coprime to n. We denote by Φ_n the nth cyclotomic polynomial: Φ_n(u)=∏_j∈ T_n(u-ζ^j).
The polynomial Φ_n is of degree φ(n).
Thereafter, we will focus on integer n such that φ(n)≥1/2n.
 Recall Lemma 6 of <cit.>: 
 
    #{n⩽ X , φ(n) ≥1/2 n}≥ (6/π^2-1/2) X+O(log X).




We start with the key lemma extending Lemma 5 of <cit.>.
 
  Let u ∈ E_∩. Then there exists some k ∈_>0 such that

    |(Φ_n(u^k))|≥exp(c n),

for n such that φ(n)≥1/2 n, where c>0 is a constant depending on u and k.




As u ∈, there exists an embedding σ : ↪ such that |σ(u)| ≥ a >1, for some real a.   Hence, for k∈_>0, we get |σ(u^k)|≥ a^k, and then |σ(u^k)-ζ^j| ≥ a^k-1.


Let us choose an another embedding τ. We want to give some "good" lower bound for |τ(u^k)-ζ^j|.
As u∈ there is only two situations.



∙  If |τ(u)| <1, then clearly for sufficiently  large k, we get 
    |τ(u^k)- ζ^j|≥ 1-|τ(u^k)|≥1/2.


∙ If |τ(u)| >1, for sufficiently large k, we get |τ(u^k)-ζ^j| ≥  1.




Putting all of this together, we obtain

    (Φ_n(u^k))=∏_i=1^m∏_j∈ T_n|σ_i(u^k) - ζ^j|≥((a^k-1)2^-m+1)^φ(n),

Consequently, by taking  sufficiently large k, we get that  for every n with φ(n)≥1/2 n

    (Φ_n(u^k))≥exp(c n),
 where the  σ_i's are the embeddings of  in  and where 
 c>0 is some constant (depending on u, k and m).






Suppose now that u ∈ E_ is such that 
    |(Φ_n(u))|≥exp(c n),

for every n such that φ(n)≥1/2 n 
(which is always possible by Lemma <ref>).
 
Let us write (u^n-1)=I_n J_n, with I_n and J_n relatively prime and where if |I_n, then ^2 ∤ I_n, and if |J_n then ^2 |J_n.
Then, if we write u^n-1+1=u^n, the generalized abc-conjecture implies that

    |(u^n-1)| ≪_,ε(I_n J_n)^1+ε≪_,ε((I_n) (J_n)^1/2)^1+ε.

Hence, as |(u^n-1)|=(I_n)N(J_n), we get

    (J_n)^1/2≪_,ε(I_n)^ε(J_n)^ε/2≪_,ε |(u^n-1)|^ε,

and then 
    (J_n) ≪_,ε |(u^n-1)|^2 ε.

Now let us  also write (Φ_n(u))=A_nB_n, with A_n and B_n relatively prime and  where if |A_n, then ^2 ∤ A_n, and if |B_n then ^2 |B_n. Of course, B_n | J_n, and then 

    (B_n) ≪_, ε |(u^n-1)|^2 ε.

Choose β > 1 such that |σ_i(u)| ≤β for all i. Then

    |(u^n-1)|≤∏_i=1^m(|σ_i(u)|^n+1)≤ 2^m(β^m)^n,

which implies 
    (B_n) ≪_, ε 2^2m ε(β^m)^2nε.

Hence, 
    (A_n)= (Φ_n(u))/(B_n) ≫_, εexp(n(c-2mεlogβ)).



We finally obtain:
  If the generalized abc-conjecture holds 
 then for all ε >0, one has
 
    (A_n) ≫_,εexp(n(c-2mεlogβ)),

 for every n such that φ(n) ≥1/2 n. 


Take now ε >0 such that ε < c/2mlog(β). Thanks  to Proposition <ref>, there exists n_0 ∈_>0 such that for all n≥ n_0, with φ(n) ≥1/2n, then (A_n) > n^m, where we recall that m=[:].  
Then, for each such n, there exists  a prime ideal _n ⊂Ø_K, dividing A_n but not n: indeed  if it was not the case then as A_n is square free,  A_n would divide n, which contradicts (A_n) > n^m.
Observe that _n | (u^n-1) implies  (_n) ≤ 2^m β^mn.

As _n ∤ n, the polynomial X^n-1 is separable over Ø_K/_n. Thus u is a simple root of X^n-1=∏_d| nΦ_d(X) modulo _n and, as _n divides Φ_n(u), its order in (Ø_/_n)^× is exactly n. Furthermore, _n is a divisor of A_n, so _n ^2 does not divide u^n-1 (in other words _n | I_n). 

Let p_n be the prime number such that p_n =_n ∩. 

In conclusion, we obtain:

 
Take u∈ E_ as before. For each n≥ n_0 such that  φ(n) ≥1/2 n,  there exists a prime ideal _n ⊂Ø_ such that 


  (i) _n | Φ_n(u) and u^n 1 ( _n^2),

  (ii) u is of order n in (Ø_/_n)^×,

  (iii) (_n) ≤γ^n, for some γ depending only on .



By (ii) of Proposition <ref>, it follows that _n=_n' if and only if n=n'.
 Observe that a set of primes _n of size Y gives at least Y/m primes p_n.
 
Now given X ≥ 1,  let n_1 be the largest integer such that γ^n_1≤ X. Assume X sufficiently large to ensure  n_0 ≤ n_1. Then, for each n ∈ [n_0, n_1] 
such that φ(n)≥1/2n,
there exists a prime ideal _n ⊂Ø_   for which u^n≡ 1 ( _n) and u^n 1 ( _n^2).  Note that p_n ≤(_n) ≤γ^n ≤γ^n_1≤  X.
Thereby:

    1/m#{n,  n_0≤ n⩽ n_1 ,  φ(n)≥1/2 n}
          ≤#{p_n ≤ X, p_n  prime | ∃ _n∈Ø_k, _n|p_n , u^n≡ 1 ( _n)  and  u^n 1 ( _n^2)}.


In conclusion, one has found at least  c  log X  prime numbers p_n ≤ X satisfying  (i) of Proposition <ref>  for some _n|p_n.




 §.§ 
 Proof of Theorem <ref>. 
Let χ be an irreducible -character of  such that there exists some u∈ E_^χ∩.
By the previous section, there exists k≥ 1 such that u^kn≡ 1 ( _n) and u^kn 1 ( _n^2) for at least c log X prime numbers  p_n ≤ X (where _n|p_n). We conclude with Lemma <ref> (after forgetting the prime numbers smaller than p_0).





Proof of the Corollary.

Observe first that, in the two cases, the Leopoldt conjecture holds and the field  contains some unit in  (see Remark <ref>). Take p>p_0.
The choice of the character is the following : if  is real quadratic, let χ=ψ be  the  nontrivial character of  ; if / is an imaginary S_3-extension, let χ be the irreducible -character of  of degree 2 (observe that  χ=ψ is also _p-irreducible).
Then  ⊗ E_ =E_^χ,   r_ψ(E_)=1, and _p=_p^ψ. Therefore by Theorem <ref>,  _p={1} for at least c log X prime numbers p≤ X.











99
Assim-BouazzaouiJ. Assim, Z. Bouazzaoui, Half-integral weight modular forms and real quadratic p-rational fields, 2018, arXiv:1906.03344.
Barbulescu-Ray R. Barbulescu and J. Ray, Some remarks and experiments on Greenberg's p-rationality conjecture, 2017,  arXiv:1706.04847.
BertinM.-J. Bertin, T. Zaïmi, Complex Pisot numbers in algebraic number fields, C. R. Acad. Sci. Paris, Ser. I 353 (2015), 965-967.
Boeckle-andall G. Böckle, D.-A. Guiraud, S. Kalyanswamy, C. Khare, Wieferich Primes and a mod p Leopoldt Conjecture, 2018,  arXiv:1805.00131.
Byeon D. Byeon, Indivisibility of special values of Dedekind zeta functions of real quadratic fields, Acta Arithmetica 109 (2003), no. 3, 231-235. 
David-Pries R. David, R. Pries, Cohomology groups of Fermat curves via ray class fields of cyclotomic fields, 2018, arXiv:1806.08352.
GorenE. Z. Goren, Hasse invariants for Hilbert modular varieties, Israel Journal of Mathematics 122 (2001), 157-174. 
Gras-Sram G. Gras, Practice of incomplete p-ramification over a number field – History of abelian p-ramification, 2019,  arXiv:1904.10707.
 Gras-BS G. Gras, Heuristics in direction of a p-adic Brauer–Siegel theorem, Mathematics of  Computation 88 (2019), no. 318, 1929-1965.
 Gras-algo G. Gras, On p-rationality of number fields. Applications – PARI/GP programs, Publ. Math. Besançon (Théorie des Nombres), Années 2017/2018, to appear, arXiv:1709.06388.
 Gras-IJNTG. Gras, The p-adic Kummer-Leopoldt Constant - Normalized p-adic Regulator, International Journal of Number Theory 14 (2018), no. 2, 329-337. 
 Gras-CJMG. Gras, Les Θ-régulateurs locaux d'un nombre algébrique : Conjectures p-adiques, Canadian Journal of Mathematics 68 (2016), 571-624.
Gras-livreG. Gras, Class Field Theory, From Theory to practice,  corr. 2nd ed., Springer Monographs in Mathematics, 
Springer (2005), xiii+507 pages.

Gras-Jaulent G. Gras and J.-F. Jaulent,
Sur les corps de nombres réguliers, Mathematische Zeitschrift 202 (1989), 343-
365.
Graves-Murty H. Graves, R. Murty, The abc conjecture and non-Wieferich primes in arithmetic progressions,
Journal of Number Theory 133 (2013), no. 6, 1809–1813. 
greenbergR. Greenberg, Galois representations with open image, Annales Mathématiques du Québec 40  (2016), no. 1, 83-119. 
Hajir-Maire F. Hajir, C. Maire, Prime decomposition and the Iwasawa mu-invariant, Mathematical Proceedings of the Cambridge Philosophical Society, to appear.
Hajir-Maire-Ramakrishna F. Hajir, C. Maire, R. Ramakrishna, Cutting towers of number fields, 2019, arXiv:1901.04354.
Ichimura H. Ichimura, A note on Greenberg's conjecture and the abc conjecture, Proceedings of the American Math. Society 126 (1998), no. 5,  1315-1320. 
Jaulent J.-F. Jaulent,  Sur l'indépendance ℓ-adique de nombres algébriques, Journal of Number Theory 20 (1985), 149-158.
Jaulent-NguyenJ.-F. Jaulent and T. Nguyen Quang Do,
Corps p-réguliers, corps p-rationnels et ramification restreinte, Journal de  Théorie des  Nombres de Bordeaux 5 (1993), 343-363.
Jaulent-Sauzet J.-F. Jaulent, O. Sauzet,  Pro-ℓ-extensions de corps de nombres ℓ-rationnels, Journal of  Number Theory 65 (1997), 240-267.
Maire-Rougnant C. Maire, M. Rougnant, Composantes isotypiques de pro-p-extensions de corps de nombres et p-rationalité, Publicationes Mathematicae Debrecen 94 1/2 (2019), 123-155.

Mova-NguyenA. Movahhedi and T. Nguyen Quang Do, Sur l’arithmétique des corps de nombres
p-rationnels, Séminaire de Théorie des Nombres, Paris 1987–88, 155–200, Progr. Math., 81, Birkhäuser Boston, Boston, MA, 1990.
Movahhedi-phdA. Movahhedi, Sur les p-extensions des corps p-rationnels,  PhD Université Paris VII, 1988.
Movahhedi  A. Movahhedi,  Sur les p-extensions des corps p-rationnels, Mathematische Nachrichten 149 (1990), 163–176.
NSW J. Neukirch, A. Schmidt and K. Wingberg, Cohomology of Number Fields, GMW 323, Springer-Verlag Berlin Heidelberg, 2000. 
Nguyen2 T. Nguyen Quang Do, Sur la structure galoisienne des corps locaux et de la théorie d'Iwasawa, Compositio Mathematica 46  (1982), no. 1, 85-119.
Nguyen T. Nguyen Quang Do, Sur la _p-torsion de certains modules
galoisiens, Annales Institut Fourier 36 (1986), no. 2, 27-46.
Pitoun-VaresconF. Pitoun and F. Varescon, Computing the torsion of the p-ramified module of a number field, Mathematics of  Computation 84 (2015), no. 291, 371-383. 
Sauzet O. Sauzet, Théorie d'Iwasawa des corps p-rationnels et p-birationnels, Manuscripta Math. 96 (1998), no. 3, 263–27.
SilvermanJ.H. Silverman, Wieferich's Criterion and the abc-Conjecture, Journal of Number Theory 30 (1988), 226-237.
SSK. Sirinas, M. Subramani, Non-Wieferich primes in number fields and abc-conjecture, Czechoslovak Mathematical Journal 68 no 2 (2018), 445-453.
Vojta P. Vojta, A more general ABC conjecture, International Math. Research Notices 21 (1998),  1103–1116.







