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On Some Norm Equations Over Cyclotomic Fields On Some Norm Equations over Cyclotomic Fields Peter Lombaers Tese de Doutoramento apresentada à Faculdade de Ciências da Universidade do Porto Matemática 2019 Peter Lombaers: On Some Norm Equations Over Cyclotomic Fields , July 2019, Porto supervisor: António Machiavelo ABSTRACT For an odd prime p, the equation xp + yp = zp can be rewritten as p (x + y)Np(x + yz p) = z , where z p is a primitive p-th root of unity and Np is the norm map of the field Q(z p). That this norm equation does not have non-trivial integer solutions is known since Andrew Wiles’ proof of Fermat’s last theorem. However, very little is known about two natural generalisations of this equation given by the equations p k Np(x + yz p) = z , and (a0 + a1 + ··· + ak)Np(a0 + a1z p + ··· + akz p) = zp, for some integer 1 ≤ k ≤ p − 3. In this thesis, we look at the cyclotomic methods that were used to attack the equation xp + yp = zp, and we apply them to those two gen- eralisations. Naturally, this leads to a number of new observations and problems. First of all, we fix the coefficients a0,..., ak and investigate their behaviour as p goes to infinity. This leads to an upper bound k on p in certain cases. Next, we show that Np(a0 + ··· + akz p) can be expressed in terms of linear recurrence sequences depending only on the coefficients ai. This allows us to show that, for fixed a, b, there 2 p are only finitely many primes p such that Np(az p + bz p − a) = z has non-trivial integer solutions. We review Kummer’s method of attack to the equation Np(x + p yz p) = z by the use of a logarithmic derivative, and we show that the Iwasawa p-adic logarithm can be used to obtain the same results. We review the theory of unramified extensions of degree p of Q(z p), and we see how this leads to a proof of Herbrand’s theorem. Combining knowledge of the units of Z[z p] with computational results, we show p 8 that if Np(x + yz p) = z , then max(jxj, jyj) > 5 × 10 , for regular primes p ≥ 5. Finally, we use the p-adic logarithm to give a unified way of proving congruence identities related to Wolstenholme’s the- orem. With the same method, we can also prove a number of other congruences, which at first sight seem quite unrelated to each other. iii RESUMO Para um primo p ímpar, a equação xp + yp = zp pode ser reescrita p como (x + y)Np(x + yz p) = z , em que z p é uma raiz primitiva p- ésima da unidade e Np é a norma do corpo Q(z p). Que esta equação não tem soluções inteiras não-triviais é sabido desde a prova de Andrew Wiles do último teorema de Fermat. No entanto, pouco se sabe acerca das duas generalizações seguintes dessa equação: Np(x + p k p yz p) = z e (a0 + a1 + ··· + ak)Np(a0 + a1z p + ··· + akz p) = z , para algum inteiro 1 ≤ k ≤ p − 3. Nesta tese, olhamos para os métodos ciclotómicos que foram usados para atacar a equação xp + yp = zp, e aplicamo-los a essas duas gener- alizações. Naturalmente, isso conduz a uma série de novas observações e problemas. Em primeiro lugar, fixamos os coeficientes a0,..., ak e investigamos o seu comportamento quando p tende para infinito. Isto conduz a um limite superior para p, em certos casos. Em seguida, k mostramos que a norma Np(a0 + ··· + akz p) pode ser expressa em termos de sequências de recorrência linear que dependem apenas dos coeficientes ai. Isso permite-nos mostrar que, para a, b fixos, há apenas 2 p um número finito de primos p tais que Np(az p + bz p − a) = z tem soluções inteiras não-triviais. Revemos o método de Kummer para atacar a equação Np(x + p yz p) = z usando derivadas logarítmicas e mostramos que o log- aritmo p-ádico de Iwasawa pode ser usado para alcançar os mesmos resultados. Revemos a teoria das extensões não-ramificadas de grau p de Q(z p), e vemos como isso conduz a uma demonstração do teo- rema de Herbrand. Combinando conhecimento das unidades de Z[z p] p com resultados computacionais, mostramos que se Np(x + yz p) = z , então max(jxj, jyj) > 5 × 108, para primos regulares p ≥ 5. Final- mente, usamos o logaritmo p-ádico para dar uma maneira unificada de demonstrar identidades envolvendo congruências relacionadas com o Teorema de Wolstenholme. Com o mesmo método podemos também demonstrar várias outras congruências que, à primeira vista, parecem não estar relacionadas entre si. iv ACKNOWLEDGMENTS This thesis would not have been possible without the help and support of a number of people. First of all I would like to thank my supervisor. António, it’s been a great pleasure to work with you. You’ve always been friendly, enthousiastic and helpful, and I feel lucky to have had you as my supervisor. Next, I would like to thank my friends here in Porto and the friends from the Netherlands that came to visit. You made my time here a great experience, whether it was to enjoy the good things, or complain about the bad things. Special mention should go to Atefeh, without you I would have been taking the stairs for the past four years, and to Nikos, you were always there, listening to my explanations of what I was working on, so that I could understand it better myself. I also want to thank my family for their love and support while I was far away from home. Finally, Saul, without you I would not survived the final year, but you turned it into a great year. I would like to acknowledge the financial support by the FCT (Fundação para a Ciência e a Tecnologia) through the PhD grant PD/BD/128063/2016, and by CMUP (Centro Matemática - Universi- dade do Porto) (UID/MAT/00144/2013), which is funded by FCT (Por- tugal) with national (MEC) and European structural funds (FEDER), under the partnership agreement PT2020. v CONTENTS 1 introduction1 1.1 The main problem 1 1.2 Kummer’s approach 2 1.3 Description of the chapters 4 1.4 Notation 6 2 limits of N-th roots9 2.1 The Mahler measure 9 2.2 An upper bound on p 11 3 recurrence sequences 15 3.1 Norms in terms of a recurrence sequence 15 3.2 A general recurrence theorem 21 4 from norms to singular integers 27 4.1 Singular integers 27 4.2 Self-prime integers 28 5 singular integers 35 5.1 Idempotents and the Herbrand-Ribet theorem 35 5.2 Kummer’s logarithmic derivative 37 5.3 The p-adic logarithm 40 6 kummer extensions 47 6.1 Kummer extensions 47 6.2 Ramification 49 6.3 Units in cyclotomic fields 52 6.4 Unramified extensions 55 7 regular primes 61 7.1 Theoretical results 61 7.2 Computational results 63 8 wolstenholme’s theorem 67 8.1 Logarithms and binomial coefficients 68 8.2 Logarithms and cyclotomic integers 73 bibliography 81 vii INTRODUCTION 1 1.1 the main problem In this thesis we look at two generalisations of Fermat’s last theorem. They arise naturally when we approach Fermat’s last theorem via the classical, cyclotomic method. Let us first recall the statement of Fermat’s last theorem: Theorem 1.1 (FLT). For each integer n ≥ 3 there are no non-zero integers x, y, z such that xn + yn = zn.(1.1) The first succesful proof was given by Andrew Wiles, using a con- nection with elliptic curves. Before that, the traditional method to attack FLT was via cyclotomic fields. We first note that we may assume in Theorem 1.1, without loss of generality, that n = 4 or n = p is an odd prime. Fermat himself already wrote a proof for the case n = 4 (see chapter 1 of [29] for a good account of the early history of FLT). Therefore we can restrict our attention to the case n = p is an odd prime. p p Since p is odd, we know that x + y is divisible by x + y. Let z p be a primitive p-th root of unity. Then we have the equality p−1 p p i x + y = ∏(x + yz p). i=0 = Let us denote by Np NQ(z p)/Q the norm function of the p-th cy- p−1 i clotomic field Q(z p), hence we have ∏i=1 (x + yz p) = Np(x + yz p). Proving FLT is thus equivalent to showing that there are no non-zero integers x, y, z such that p (x + y)Np(x + yz p) = z .(1.2) The first generalisation we shall consider is a slightly less general version of a conjecture by George Gras in [12]. He referred to it as ‘Strong Fermat’s last theorem’. Conjecture 1.2 (SFLT). For each prime p ≥ 5, the only solutions in pairwise coprime integers x, y, z to the equation p Np(x + yz p) = z (1.3) are given by z = 1 and x + yz p 2 f±1, ±z p, ±(1 + z p)g. 1 2 introduction Note that in [12], Conjecture 1.5 states that also Np(x + yz p) = pzp only has the trivial solutions. The second generalisation we will consider is: Problem 1.3. Let p ≥ 5 be a prime and let 1 ≤ k ≤ p − 3 be an integer.
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