# Diophantine Equations and Cyclotomic Fields

Boris Bartolomé

Doctoral thesis
Date of Examination:
2015-11-26

Date of issue:
2016-03-09

Advisor:
Mihailescu, Preda Prof. Dr.

Advisor:
Bilu, Yuri Prof. Dr.

Referee:
Bugeaud, Yann Prof. Dr.

Referee:
Fuchs, Clemens Prof. Dr.

Referee:
Brüdern, Jörg Prof. Dr.

Persistent Address: http://hdl.handle.net/11858/00-1735-0000-0028-86F5-5

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## Abstract

### English

This thesis examines some approaches to address Diophantine equations, specifically we focus on the connection between the Diophantine analysis and the theory of cyclotomic fields. First, we propose a quick introduction to the methods of Diophantine approximation we have used in this research work. We remind the notion of height and introduce the logarithmic gcd. Then, we address a conjecture, made by Thoralf Skolem in 1937, on an exponential Diophantine equation. For this conjecture, let K be a number field, α1 ,…, αm , λ1 ,…, λm non-zero elements in K, and S a finite set of places of K (containing all the infinite places) such that the ring of S-integers OS = OK,S = {α ∈ K : |α|v ≤ 1 pour les places v ∈/ S} contains α1 , . . . , αm , λ1 , . . . , λm α1-1 , . . . , αm-1. For each n ∈ Z, let A(n)=λ_1 α_1^n+⋯+λ_m α_m^n∈O_S. Skolem suggested [SK1] : Conjecture (exponential local-global principle). Assume that for every non zero ideal a of the ring O_S, there exists n ∈ Z such that A(n) ≡0 mod a. Then, there exists n ∈ Z such that A(n)=0. Let Γ be the multiplicative group generated by α1 ,…, αm. Then Γ is the product of a finite abelian group and a free abelian group of finite rank. We prove that the conjecture is true when the rank of Γ is one. After that, we generalize a result previously published by Abouzaid ([A]). Let F(X,Y) ∈ Q[X,Y] be an irreducible Q-polynomial. In 2008, Abouzaid [A] proved the following theorem: Theorem (Abouzaid). Assume that (0,0) is a non-singular point of the plane curve F(X,Y) = 0. Let m = degX F, n = degY F, M = max{m, n}. Let ε satisfy 0 < ε < 1. Then for any solution (α,β) ∈ Q ̅2 of F(X,Y) = 0, we have either max{h(α), h(β)} ≤ 56M8ε−2hp(F) + 420M10ε−2 log(4M), or max{|h(α) − nlgcd(α, β)|,|h(β) − mlgcd(α, β)|} ≤ εmax{h(α), h(β)}+ + 742M7ε−1hp(F) + 5762M9ε−1log(2m + 2n) However, he imposed the condition that (0, 0) be a non-singular point of the plane curve F(X,Y) = 0. Using a somewhat different version of Siegel’s “absolute” lemma and of Eisenstein’s lemma, we could remove the condition and prove it in full generality. We prove the following theorem: Theorem. Let F(X,Y) ∈ Q ̅[X,Y] be an absolutely irreducible polynomial satisfying F(0,0)=0. Let m=degX F, n=degY F and r = min{i+j:(∂^(i+j) F)/(∂^i X∂^j Y)(0,0)≠0}. Let ε be such that 0 < ε < 1. Then, for all (α, β) ∈ Q ̅2 such that F(α,β) = 0, we have either h(α) ≤ 200ε−2mn6(hp(F) + 5) or |(lgcd(α,β))/r-h(α)/n|≤1/r (εh(α)+4000ε^(-1) n^4 (h_p (F)+log(mn)+1)+30n^2 m(h_p (F)+log(mn) )) Then, we give an overview of the tools we have used in cyclotomic fields. We try there to develop a systematic approach to address a certain type of Diophantine equations. We discuss on cyclotomic extensions and give some basic but useful properties, on group-ring properties and on Jacobi sums. Finally, we show a very interesting application of the approach developed in the previous chapter. There, we consider the Diophantine equation (1) Xn − 1 = BZn, where B ∈ Z is understood as a parameter. Define ϕ∗(B) := ϕ(rad (B)), where rad (B) is the radical of B, and assume that (2) (n, ϕ∗(B)) = 1. For a fixed B ∈ N_(>1)we let N(B) = {n ∈ N_(>1) | ∃ k > 0 such that n|ϕ∗(B)}. If p is an odd prime, we shall denote by CF the combined condition requiring that I The Vandiver Conjecture holds for p, so the class number h+ of the maximal real subfield of the cyclotomic field Q[ζp ] is not divisible by p. II We have ir>(p) < √p − 1, in other words, there is at most √p − 1 odd integers k < p such that the Bernoulli number Bk ≡ 0 mod p. Current results on Equation (1) are restricted to values of B which are built up from two small primes p ≤ 13 [BGMP] and complete solutions for B < 235 ([BBGP]). If expecting that the equation has no solutions, – possibly with the exception of some isolated examples – it is natural to consider the case when the exponent n is a prime. Of course, the existence of solutions (X,Z) for composite n imply the existence of some solutions with n prime, by raising X, Z to a power. The main contribution of our work has been to relate (1) in the case when n is a prime and (2) holds, to the diagonal Nagell – Ljunggren equation (X^n-1)/(X-1)=n^e Y^n, e={(O si X≢1[n]@1 sinon)┤ This way, we can apply results from [M] and prove the following: Theorem. Let n be a prime and B > 1 an integer with (ϕ∗(B), n) = 1. Suppose that equation (1) has a non-trivial integer solution different from n = 3 and (X,Z;B) = (18,7;17). Let X ≡ u mod n, 0 ≤ u < n and e = 1 if u = 1 and e = 0 otherwise. Then: 1. n > 163106. 2. X − 1 = ±B/ne and B < nn. 3. If u ∉ {−1,0,1}, then condition CF (II) fails for n and 2n−1 ≡ 3n−1 ≡ 1 mod n2 , and rn−1 ≡ 1 mod n2 for all r|X(X2 − 1). If u ∈ {−1, 0, 1}, then Condition CF (I) fails for n. Based on this theorem, we also prove the following: Theorem. If equation (1) has a solution for a fixed B verifying the conditions (2), then either n ∈ N(B) or there is a prime p coprime to ϕ∗(B) and a m ∈ N(B) such that n = p.m. Moreover, Xm, Ym are a solution of (1) for the prime exponent p and thus verify the conditions of the previous Theorem. This is a strong improvement of the currently known results. As we have made heavy use of [M], at the end of this thesis we have added an appendix to expose some new result that allows for a full justification of Theorem 3 of [M]. Keywords Diophantine Equations, Cyclotomic Fields, Nagell-Ljunggren Equation, Skolem, Abouzaid, Exponential Diophantine Equation, Baker’s Inequality, Subspace Theorem. References [A] M. Abouzaid, Heights and logarithmic gcd on algebraic curves, Int. J. Number Th. 4, pp. 177–197 (2008). [BBGP] A.Bazso, A.Bérczesc K.Györy and A.Pintér, On the resolution of equations Axn − Byn = C in integers x, y and n ≥ 3, II, Publicationes Mathematicae Debrecen 76, pp. 227 – 250 (2010). [BGMP] M. A. Bennett, K. Györy, M. Mignotte and A. Pintér, Binomial Thue equations and polynomial powers, Compositio Math. 142, pp. 1103–1121 (2006). [M] P. Mihăilescu Class Number Conditions for the Diagonal Case of the Equation of Nagell and Ljunggren, Diophantine Approximation, Springer Verlag, Development in Mathematics 16, pp. 245–273 (2008). [S1] T. Skolem, Anwendung exponentieller Kongruenzen zum Beweis der Unlosbarkeit gewisser diophantischer Gleichungen, Avhdl. Norske Vid. Akad. Oslo I 12, pp. 1–16 (1929). [S2] T. Skolem, Lösung gewisser Gleichungssysteme in ganzen Zahlen oder ganzzahligen Polynomen mit beschränktem gemeinschaftlichen Teiler, Oslo Vid. Akar. Skr. I, 12 (1929).**Keywords:**Diophantine Equations; Cyclotomic Fields; Nagell-Ljunggren Equation; Skolem; Abouzaid; Exponential Diophantine Equation; Baker’s Inequality; Subspace Theorem