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Poster summarizing ”The abc and some of its consequences” Razvan Barbulescu, Michel Waldschmidt

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Razvan Barbulescu, Michel Waldschmidt. Poster summarizing ”The abc conjecture and some of its consequences”. 6th World Conference on 21st Century Mathematics 2015, Oct 2017, Lahore, India. ￿hal-01626155v3￿

HAL Id: hal-01626155 https://hal.archives-ouvertes.fr/hal-01626155v3 Submitted on 8 Jan 2018

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The abc conjecture and some of its consequences

The abc conjecture The Fermat-Wiles theorem (1621, 1994) Wieferich’s theorem (1909) The Waring-Hilbert theorem (1770, 1909) Vojta’s height conjecture (1987) In the quest for examples

Oesterle´ and Masser (1985) Let p be a prime and x, y, z positive such that xp + yp = zp and p doesn’t divide xyz. Then Bosman, Broberg, Browkin, Brzezinski, Dokchitser, Elkies, Kanapka, Frey, Gang, Hegner, Nitaj, The equation p has the property that For any k there exists g(k) such that each Vojta stated a conjectural inequality on the Reyssat, te Riele, P. Montgomery, Schulmeiss, Rosenheinrich, Visser, de Weger. positive is a sum of at most g(k) kth height which implies the abc conjecture. 2 p−1 For any relatively prime positive integers a, b, c such that a + b = c we set For any ε > 0, there exists κ(ε) such that, if a, xn + yn = zn p divides 2 − 1. powers. Another consequence of this inequality is the b and c are relatively prime positive integers following. Let K be a number field and S a Such a prime is called a . An effective bound on the set of Wieferich primes would which satisfy a + b = c, then finite set of absolute values of K. If X is a log c has no integer solutions (x, y, z, n) with yield a new proof of the Fermat-Wiles theorem in the first case (p does not divide xyz). variety with trivial canonical bundle and D is λ(a, b, c) = . Rad(abc) > κ(ε)c1−ε, x, y, z > 0 and n > 2. log(Rad(abc)) an effective ample normal crossing divisor, where for any positive integer n, Rad(n) is the Infinitely many non-Wieferich primes then the S-integral points on the affine variety product of its distinct prime factors. The abc conjecture implies asymptotic Fermat-Wiles A conjecture on g(k) X\D are not Zariski dense. Baker’s effective abc implies that λ < 1, 75. The largest known examples are Silverman (1988) Assume xn + yn = zn with gcd(x, y, z) = 1. Then abc applied to (xn, yn, zn) implies a + b = c λ(a, b, c) author J. A. Euler (1772): For all k ≥ 1, g(k) ≥ I(k) The Lang-Faltings theorem (1991) 2 + 310 · 109 = 235 1.6299 ... Reyssat The abc conjecture implies that there are where I(k) = 2k + b(3/2)kc − 2. Indeed, the 112 + 32 · 56 · 73 = 221 · 23 1.6259 ... de Weger 3 n n n n(1−ε) infinitely many non-Wieferich primes. integer 2b(3/2)kc − 1 is less than 3k so it must If X is an abelian variety then the above statement holds. Remarks z > xyz = Rad(x y z ) > κ(ε)z . 19 · 1307 + 7 · 292 · 318 = 28 · 322· 1.6234 ... Browkin, Brzezinski be written so that only powers of 2 and 1 occur, 2k 2k • The set {(a, b, c) = (1, 3 − 1, 3 ) | k ∈ N} shows that one cannot put ε = 0. Nothing is known about the finitness of the set and the most economic expression uses I(k) When n ≥ 4 we set ε = 1 and obtain a bound on zn. terms. • If up + vq = w then the triple (a, b, c) = ((uwq)p, (vwp)q, wpq+1) shows than one cannot drop 5 of Wieferich primes, the only two known Demeyer, Nitaj, de Weger, de Smit, H. Lenstra, Palenstijn, Rubin, Calvo, Wrobenski, Bretschneider’s conjecture (1853): g(k) = I(k) the condition that gcd(a, b, c) = 1. examples being 1093 and 3511. For any relatively prime positive integers a, b, c such that a + b = c we set for any k ≥ 2. The Fermat-Catalan conjecture Further consequences of the abc conjecture log(abc) • Lang’s : lower bounds for heights, number of integral points on elliptic curves. ρ(a, b, c) = · Brun (1914) log(Rad(abc)) Evaluations of g(k) for k = 2, 3, 4,... (Frey 1987) (Hindry, Silverman 1988) Best unconditional result g(2)=4 Lagrange 1770 • Squarefree and powerfree values of . (Browkin, Filaseta, Greaves, Schinzel 1995) The equation The largest known examples are g(3)=9 Kempner 1912 Stewart and Kunrui Yu (1991, 2001) • Bounds for the order of the Tate–Shafarevich group. (Goldfeld, Szpiro 1995) xp + yq = zr g(4)=19 Balusubramanian, Dress, Deshouillers 1986 The Erdos˝ -Woods conjecture (1981) g(5)=37 Chen Jingrun 1964 • Frey proved in 1987 the equivalence between the height conjecture and the abc conjecture, a + b = c %(a, b, c) author For all a, b, c triple of coprime positive integers g(6)=73 Pillai 1940 while Mai and Murty in 1996 proved the equivalence with the degree conjecture. 6 30 13 2 has a finite set of solutions (x, y, z, p, q, r) in 13 · 19 + 2 · 5 = 3 · 11 · 31 4.4190 ... Nitaj such that a + b = c we have There exists an absolute constant k such that, if g(7)=143 Dickson 1936 5 2 9 15 2 7 11 positive integers such that gcd(x, y, z) = 1 and • The abc conjecture for a cyclotomic number field K implies Greenberg’s conjecture for in- 2 · 11 · 19 + 5 · 37 · 47 = 3 · 7 · 743 4.2680 ... Nitaj x and y are positive integers satisfying 19 11 11 3 2 1/3 3 2 · 13 · 103 + 7 = 3 · 5 · 11 4.2678 ... de Weger. eκR (log R) ≥ c, finitely many primes p: the Iwasawa invariants λp(K) and µp(K) vanish. (Ichimura using a 1 1 1 lemma of Sumida 1998) + + < 1. A sufficient condition p q r Rad(x + i) = Rad(y + i) where R = Rad(abc) and κ is an absolute Dickson, Pillai (1936) • Effective abc implies Dressler’s conjecture: between two positive integers having the same constant. prime factors, there is always a prime. (Cochrane and Dressler 1999) Beal’s Prize (1M$), supported by the AMS, will be given for a proof or a disproof of the state- for i = 0, 1, . . . , k − 1, then x = y. ment that there is no solution to the Fermat-Catalan equation with coprime integers (x, y, z) and k • The uniform abc conjecture for number fields implies a lower bound for the class number of p, q, r all ≥ 3. The 10 known solutions are If is such that The ABC conjecture for polynomials 2k{(3/2)k} + b(3/2)kc ≤ 2k − 2 then imaginary quadratic fields (Granville and Stark 2000), and Mahler has shown that this implies The abc conjecture implies Erdos-Woods˝ Bretschneider’s conjecture holds for k. that the associated L-function has no Siegel zeros. Hurwitz, Stothers and Mason( ≈1900, 1981, 1984) Pillai’s conjecture (1945) 1 + 23 = 32, 25 + 72 = 34, 73 + 132 = 29, 27 + 173 = 712, 5 4 2 8 2 3 Langevin (1996) 3 + 11 = 122 , 33 + 1 549 034 = 15 613 , Let K be an algebraically closed field of 3 2 7 3 2 7 1414 + 2213459 = 65 , 9262 + 15312 283 = 113 , characteristic zero. For any Let k be a positive integer. The equation 7 3 2 8 3 2 17 + 76271 = 21063928 , 43 + 96222 = 30042907 . Already in 1975, Langevin studied the radical Q ai P = γ i(x − αi) call the radical of P the of n(n + k) (with gcd(n, k) = 1) using lower polynomial Rad(P ) = Q (x − α ). Then for p q Is the abc conjecture optimal? i i x − y = k, bounds for linear forms in logarithms of Mahler’s theorem (1957) any three relatively prime polynomials A, B, C The abc conjecture implies asymptotic Fermat- algebraic numbers (Baker’s method). Theorems by Stewart and Tijdeman and later by van such that A + B = C we have where the unknowns x, y, p and q take integer values, all ≥ 2, has only finitely many solutions Catalan conjecture The condition of Dickson and Pillai is true for Frankenhuijsen (1986, 2012) max(deg(A), deg(B), deg(C)) (x, y, p, q). all but a finite set of integers k. ≤ deg(Rad(ABC)) − 1. Tijdeman (1988) , An elementary proof The case k = 1 An elementary study shows that the condition Stewart-Tijdeman (1986): For any δ > 0 there on (p, q, r) actually implies Cassels, Tijdeman, Langevin, Mignotte are infinitely many triples (a, b, c) with • Since A + B = C we have A0 + B0 = C0 and then W (A, B) = W (C,B) = W (A, C), where gcd(a, b) = 1 and c = a + b for which W (A, B) = AB0 − A0B. 1 1 1 41 Kubina and Wunderlich (1990) created a fast algorithm to test the conjecture up to large values of k. + + ≤ · 0 0 p q r 42  (log R)1/2  • Since A, B, C are relatively prime W (A, B) 6= 0. Indeed AB = A B would imply that A Dirichlet’s approximation theorem (≈1830) c > R exp (4 − δ) , divides A0. log log R 1 The abc conjecture applied to ε = and 0 0 0 84 Effective bound assuming abc (2011) • Clearly each of GA := gcd(A, A ), GB := gcd(B,B ) and GC := gcd(C,C ) divides (a, b, c) = (xp, yq, zr) implies For any irrational α there exist infinitely many where R = Rad(abc). W (A, B). Since A, B and C are relatively prime, GAGBGC divides W (A, B). Then relatively prime pairs (p, q) such that A discussion between David and Waldschmidt In 2012 van Frankenhuijsen showed that 4 − δ deg(GA) + deg(GB) + deg(GC ) ≤ deg(W (A, B)) zr(1−2ε) > xyz ≥ Rad(xpyqzr) > κ(ε)zr(1−ε). lead to a proof of Mahler’s result as a can be replaced in the inequality above by by ≤ deg(A) + deg(B) − 1. consequence of abc. Laishram proved that 6.008. p 1 • Since for all P , Rad(P ) = P/ gcd(P,P 0) the theorem follows. α − < · Bretschneider’s conjecture follows from the q q2 The case of fixed (p, q, r) explicit version of abc due to Baker. The same The equation |xp − yq| = 1 has no integer Darmon and Granville (1995) author proved a series of explicit results in a Heuristic: solution (x, y, p, q) with p, q > 1 and This theorem implies that the Pell-Fermat joint work with Shorey. max(xp, yq) > exp exp exp exp(730). equation Rad(a), Rad(b) and Rad(a + b) are independent For each triple (p, q, r) with 1 + 1 + 1 < 1 The abc conjecture for meromorphic function fields p q r Robert, Stewart and Tenenbaum (2014) there exist only finitely many coprime solutions x2 − dy2 = 1 The Catalan-Mihailescu˘ theorem (1844, 2002) (x, y, z) to the Fermat-Catalan equation. The value distribution theory was introduced For any δ > 0 there exists κ(δ) > 0 such that by Nevanlinna. The abc conjecture was for any abc triple with R = Rad(abc) > 8, The only solution to the equation has non-trivial solutions for any squarefree Baker’s explicit version of the abc conjecture (2004) extended to this context by Pei-Chu and d > 1, a result which was previously proved by Chung-Chun and later by Vojta. ! Lagrange (1766) and extended in a work on √  log R 1/2 p q Let (a, b, c) be three integers such that x − y = 1 quadratic forms by Gauss (1801). c < κ(δ)R exp (4 3 + δ) . The cases (p, p, 2) and (p, p, 3) gcd(a, b) = 1 and c = a + b. Then log log R 2 3 with x, y > 0 and p, q > 1 is 3 − 2 = 1. Darmon and Merel (1997) 6 (log R)ω Further, there exist infinitely many triples The Thue-Siegel-Roth’s theorem (1909, 1921, 1955) c ≤ R 5 ω! (a, b, c) such that gcd(a, b) = 1 and c = a + b for which The Fermat-Catalan equation has no solution in For any irrational algebraic number α and any with R = Rad(abc) the radical of abc and The Lang-Waldschmidt conjecture (1978) relatively prime positive integers for p = q ≥ 4 Mochizuki announces a proof (2012) positive ε the set of relatively prime integers ω = ω(abc) the number of distinct prime ! and r = 2 and also for p = q ≥ 3 and r = 3. √  log R 1/2 p, q such that factors of abc. c > R exp (4 3 − δ) . Let ε > 0. There exists a constant c(ε) > 0 log log R Inter-universal Teichmuler¨ IV: log-volume p q with the following property. If x 6= y , then p 1 computations and set theoretic foundations. α − < q q2+ε is finite. The submitted proof is more than 500 pages long and is currently (November 2017) in the |xp − yq| ≥ c(ε) max{xp, yq}κ−ε Siegel’s theorem (1929) Mordell-Faltings theorem (1922, 1983) process of being peer-reviewed. Szpiro’s conjecture (1983) The number fields abc conjecture implies a refinement Let g be the genus of a smooth algebraic curve Bombieri (1994) Let g be the genus of an equation P (x, y) = 0 in a given coordinate system, with coefficients of coefficients in Q. If g ≥ 2 then the equation Given any ε > 0, there exists a constant in a number field K. If g ≥ 1, then the curve κ = 1 − 1 − 1 C(ε) > 0 such that, for any over has only finitely many solutions with with p q . The abc conjecture implies that in the has only finitely many integer points. References (x, y) ∈ Q2. Q with minimal discriminant ∆ and inequality conductor N, • An extensive litterature on the subject is available on the abc home page created and maintained by Abderrahmane Nitaj: https://nitaj.users.lmno.cnrs.fr/abc.html. The abc conjecture implies Lang-Waldschmidt and p 1 6+ε α − < |∆| < C(ε)N . q q2+ε • This poster is a summary of the article “The abc conjecture and some of its consequences”. therefore Pillai’s conjecture Michel Waldschmidt. 2015, available online at https://webusers.imj-prg.fr/ The effective abc conjecture implies effective Siegel The effective abc implies effective Mordell ˜michel.waldschmidt/articles/pdf/abcLahoreProceedings.pdf. The abc conjecture implies Szpiro’s conjecture of the Thue-Siegel-Roth theorem one can Hall’s conjecture (1971) replace the exponent ε with Surroca (2004) Elkies (1991) Oesterle´ (1988) Authors The case p = 3, q = 2: If x3 6= y2, then −1/2 −1 κ(log q) (log log q) , In the proof she uses a theorem of Bely¨ı. The effective version of Mordell’s conjectures amounts to giving bounds on the heights of Razvan Barbulescu and Michel Waldschmidt, Imj-prg (CNRS, Univ. Paris 6, Univ Paris 7). 3 2 3 2 1/6 Conversely, Szpiro’s conjecture implies a weak |x − y | ≥ c max{x , y } . form of the abc conjecture, with 1 − ε replaced where κ depends only on α. rational points. by 5/6 − ε. To deduce Hall’s conjecture from the abc Under the (effective) abc conjecture for a conjecture, one would need to take ε = 0, number field K, the (effective) conjecture of which is not allowed. Mordell holds for the same K.