March 15, 2012 CARMA International Number Theory Conference. Number of solutions of Thue equations. Joint papers with Alf

Transcription

1 March 15, 2012 CARMA International Number Theory Conference Abstract So far, a rather small number of families of Thue curves having only trivial integral oints have been exhibited. In a joint work with Claude Levesque, for each number field K of degree at least three and for each finite set S of laces of K containing the infinite laces, we roduce families of curves related to the units of the number field, having only trivial S integral oints. Some families of curves with only trivial S integral oints (joint work with Claude Levesque) Michel Waldschmidt Institut de Mathe matiques de Jussieu (Univ. Paris VI) htt:// miw/ 1 / 65 Joint aers with Alf 2 / 65 Number of solutions of Thue equations Brindza, B. ; Pinte r, A. ; van der Poorten, A. J. ; W.M. (1997) Loxton, John H. ; Mignotte, Maurice ; van der Poorten, Alfred J. ; Waldschmidt, Michel A lower bound for linear forms in the logarithms of algebraic numbers. C. R. Math. Re. Acad. Sci. Canada 9 (1987), no. 2, MR (88j:11041) (Reviewer : P. L. Cijsouw) Let F 2 Z[X, Y ] be an irreducible binary form of degree n 3 and let m be a ositive integer having s distinct rime factors. Then the equation F (x, y ) = m has at most 2n2 (s + 1) + 13n solutions with Brindza, Be la ; Pinte r, A kos ; van der Poorten, Alfred J. ; Waldschmidt, Michel On the distribution of solutions of Thue s equation. Number theory in rogress, Vol. 1 (Zakoane-Kos cielisko, 1997), 35 46, de Gruyter, Berlin, MR (2000c:11048) (Reviewer : Yann Bugeaud) 21n2 M 5 m, max( x, y ) where = 3 / 65 1 n (n 1)2 4 / 65

2 Thue equations Liouville s inequality Axel Thue ( ) Let F 2 Z[X, Y ] be a homogeneous olynomial with rational integer coe cients having at least 3 non roortional linear factors over the field of algebraic numbers. Let m 2 Z, m 6= 0. Then the Diohantine equation F (X, Y )=m has only finitely many solutions (x, y) 2 Z Z. Liouville s inequality. Let be an algebraic number of degree d 2. There exists c > 0 such that, for any /q 2 Q, q c q d Joseh Liouville, /65 6/65 Thue equations and Diohantine aroximation Liouville s estimate for the rational Diohantine aroximation of 3 2 : 3 2 q > 1 9q 3 for su ciently large q. Mike Bennett (1997) : for any /q 2 Q, 3 2 q 1 4 q 2.5 Mike Bennett htt:// bennett/ For any /q 2 Q, 3 2 q 1 4 q 2.5 For any (x, y) 2 Z 2 with x > 0, x 3 2y 3 x. 7/65 8/65

3 Connection between Diohantine aroximation and Diohantine equations Let ale satisfy 0 < ale ale 3. The following conditions are equivalent : (i) There exists c 1 > 0 such that 3 2 q for any /q 2 Q. (ii) There exists c 2 > 0 such that c 1 q ale x 3 2y 3 c 2 x 3 ale for any (x, y) 2 Z 2 having x > 0. Imrovements of Liouville s inequality In the lower bound q > c( ) q d for real algebraic number of degree d 3, the exonent d of q in the denominator of the right hand side was relaced by ale with any ale > (d/2) + 1 by A. Thue (1909), 2 d by C.L. Siegel in 1921, 2d by F. Dyson and A.O. Gel fond in 1947, any ale > 2 by K.F. Roth in /65 10 / 65 Thue Siegel Roth Theorem Families of Thue equations Axel Thue ( ) Carl Ludwig Siegel ( ) Klaus Friedrich Roth (1925 ) The first families of Thue equations having only trivial solutions were introduced by A. Thue himself. (a +1)X n ay n =1. He roved that the only solution in ositive integers x, y is x = y =1for n rime and a su ciently large in terms of n. For n =3this equation has only this solution for a 386. For any real algebraic number, for any > 0, the set of /q 2 Q with /q < q 2 is finite. M. Bennett (2001) roved that this is true for all a and n with n 3 and a / / 65

4 Families of Thue equations (continued) E. Thomas in 1990 studied the families of equations F a (X, Y )=1associated with D. Shanks simlest cubic fields (cf. John Friedlander s lecture), viz. F a (X, Y )=X 3 (a 1)X 2 Y (a +2)XY 2 Y 3. According to E. Thomas (1990) and M. Mignotte (1993), for a 4 the only solutions are (0, 1), (1, 0) and ( 1, +1), while for the cases a =0, 1, 3, there exist some nontrivial solutions, too, which are given exlicitly by Thomas. For the same form F a (X, Y ), all solutions of the Thue inequality F a (X, Y ) ale 2a +1have been found by M. Mignote A. Pethő and F. Lemmermeyer (1996). The family of Thue s equations attached to some quintic fields by E. Lehmer do not seem to have been investigated from this oint of view so far. Families of Thue equations (continued) 13 / 65 Families of Thue equations (continued) E. Lee and M. Mignotte with N. Tzanakis studied in 1991 and 1992 the family of cubic Thue equations X 3 ax 2 Y (a +1)XY 2 Y 3 =1. The left hand side is X (X + Y )(X (a +1)Y ) Y 3. For a , there are only the solutions (1, 0), (0, 1), (1, 1), ( a 1, 1), (1, a). In 2000, M. Mignotte could rove the same result for all a 3. Families of Thue equations (continued) 14 / 65 I. Wakabayashi roved in 2003 that for a , the equation X 3 a 2 XY 2 + Y 3 =1 has exactly the five solutions (0, 1), (1, 0), (1, a 2 ), (±a, 1). I. Wakabayashi in 2002 used Padé aroximation for solving the Diohantine inequality X 3 + axy 2 + by 3 ale a + b +1 A. Togbé considered the family of equations X 3 (n 3 2n 2 +3n 3)X 2 Y n 2 XY 2 Y 3 = ±1 in For n 1, the only solutions are (±1, 0) and (0, ±1). for arbitrary b and a a b 4 as well as for b 2 {1, 2} and 15 / / 65

5 Families of Thue equations (continued) Families of Thue equations (continued) E. Thomas considered some families of Diohantine equations X 3 bx 2 Y + cxy 2 Y 3 =1 for restricted values of b and c. Family of quartic equations : Further work on equations of degrees u to 8 by J.H. Chen, I. Gaál, C. Heuberger, B. Jadrijević, G. Lettl, C. Levesque, M. Mignotte, A. Pethő, R. Roth, R. Tichy, E. Thomas, A. Togbé, P. Voutier, I. Wakabayashi, P. Yuan, V. Ziegler... X 4 ax 3 Y X 2 Y 2 + axy 3 + Y 4 = ±1 (A. Pethő 1991, M. Mignotte, A. Pethő and R. Roth, 1996). The left hand side is X (X Y )(X + Y )(X ay )+Y / / 65 Families of Thue equations (continued) Slit families of E. Thomas (1993) : ny (X i (a)y ) Y n = ±1, i=1 where 1,..., n are olynomials in Z[a]. Surveys by I. Wakabayashi (2002) and C. Heuberger (2005). New families of Thue equations Let K be a number field. For each " 2 Z K, let f "(X ) 2 Z[X ] be the irreducible olynomial of " over Q. Denote by d =[Q(") :Q] its degree. Set F " (X, Y )=Y d f " (X /Y ). Hence F " (X, Y ) 2 Z[X, Y ] is an irreducible binary form with integer coe cients. A corollary of our main result is the following : Corollary Let K be a number field and let m 2 K, m 6= 0. Then the set (x, y, ") 2 Z 2 Z K xy 6= 0, [Q(") :Q] 3, F "(x, y) =m is finite. 19 / / 65

6 Thue Mahler equations Let F 2 Z[X, Y ] be a homogeneous olynomial with rational integer coe cients having at least 3 non roortional linear factors over the field of algebraic numbers. Let m 2 Z, m 6= 0. Let 1,..., s be rime numbers. Then the Diohantine equation F (X, Y )=m Z Zs s has only finitely many solutions (x, y, z 1,...,z s ) 2 Z 2+s with z j 0 for j =1,...,s, xy 6= 0 and gcd(xy, 1 s )=1. S integers, S units Let K be a number field and S be a finite set of laces of K containing the infinite laces. The ring O S of S-integers of K is defined by O S = {x 2 K x v ale 1foreachv 62 S}. The grou O S of S-units of K is the grou of units of O S, namely O S = {x 2 K x v =1 foreach v 62 S}. 21 / / 65 Two secial cases For S the set of infinite laces of K, O S is the ring Z K of integers of K and O S is the grou Z K of units of K. For K = Q, S = {1, 1,..., s }, with s 0 O S = {a/b 2 Q b = z 1 1 zs s with z 1,...,z s in Z, z j 0 } and Hence O S = {t 1 1 ts s with t 1,...,t s in Z}. O S = {a/b 2 Q a 2 Z, b 2 Z \ O S } Thue Mahler equations over a number field We will consider the Thue Mahler equations F (X, Y )=E, where the two unknowns X, Y take resectively values x, y in the ring of S integers of K while the unknown E takes its values " in the grou of S units of K. If (x, y, ") is a solution, namely F (x, y) =", and if d denotes the degree of F, then, for all 2 O S, the trile ( x, y, d ") is also a solution : F ( x, y) = d ". 23 / / 65

7 Equivalence classes Definition. Two solutions (x, y, ") and (x 0, y 0, " 0 ) in O 2 S O S of the equation F (X, Y )=E are said to be equivalent modulo O S if the oints of P1 (K) with rojective coordinates (x : y) and (x 0 : y 0 ) are the same. In other terms, two solutions (x, y, ") and (x 0, y 0, " 0 ) are equivalent if there exists 2 O S such that where d is the degree of F. x 0 = x, y 0 = y, " 0 = d " Thue Mahler equations (continued) For any finite set S of laces of K containing all the archimedean laces, for every m 2 K and for any binary homogeneous form F (X, Y ) with the roerty that the olynomial F (X, 1) 2 K[X ] has at least three linear factors involving three distinct roots in K, the Thue-Mahler equation F (X, Y )=me has but a finite number of classes of solutions (x, y, ") 2 OS 2 O S (namely : the set of solutions (x, y, ") 2 OS 2 O S can be written as the union of a finite number of equivalence classes modulo O S ). 25 / / 65 A secial case For any finite set S of laces of K containing all the archimedean laces, the Thue-Mahler equation XY (X Y )=E has but a finite number of classes of solutions (x, y, ") 2 O 2 S O S. Fact : this secial case is equivalent to the general case! Siegel S unit equation For any finite set S of laces of K containing all the archimedean laces, the S unit equation E 1 + E 2 =1 has but a finite number of solutions (" 1, " 2 ) in O S O S. Fact : this statement is also equivalent to the finiteness of the number of classes of solutions of the Thue Mahler equation XY (X Y )=E. X = E 0, Y = E 2, X Y = E 1, E 1 + E 2 = E 0, E 0 E 1 E 2 = E. 27 / / 65

8 Integral oints on P 1 minus three oints Families of Thue Mahler equations A further equivalent statement is the following one : For any finite set S of laces of K containing all the archimedean laces, every set of S integral oints of P 1 (K) minus three oints is finite. A more general corollary of our main result is the following : Corollary Further, let 1,..., s be finitely many rimes. Then the set of (x, y, z 1,...,z s, ") 2 Z 2+s Z K with z j 0 for j =1,...,s, xy 6= 0and gcd(xy, 1 s )=1such that [Q(") :Q] 3 and F " (x, y) =m z 1 1 zs s is finite. 29 / / 65 The general equation Let K be a number field, S afinitesetoflacesofk containing the infinite laces, µ, 1, 2, 3 nonzero elements in K. Consider the equation (X 1 E 1 Y )(X 2 E 2 Y )(X 3 E 3 Y )Z = µe, where the variables take for values (x, y, z, " 1, " 2, " 3, ") 2 O 3 S (O S )4. Trivial solutions are solutions with xy = 0. Two nontrivial solutions (x, y, z, " 1, " 2, " 3, ") and (x 0, y 0, z 0, " 0 1, " 0 2, " 0 3, " 0 ) are called S 3 deendent if there exist S units 1, 2 and 3 in O S such that x 0 = x 1, y 0 = y 1 1 3, z 0 = z 2, " 0 i = " i 3, " 0 = " The main result Theorem The set of classes of S 3 deendence of the nontrivial solutions of the equation (x, y, z, " 1, " 2, " 3, ") 2 O 3 S (O S )4 (X 1 E 1 Y )(X 2 E 2 Y )(X 3 E 3 Y )Z = µe satisfying Card{ 1 " 1, 2 " 2, 3 " 3 } =3is finite. The number of these classes is bounded by an exlicit constant deending only on K, µ, 1, 2, 3 and the rank s of the grou O S. 31 / / 65

9 A secial case It turns out that the secial case of the equation (X Y )(X E 1 Y )(X E 2 Y )=E is equivalent to the general case. Two solutions (x, y, " 1, " 2, ") and (x 0, y 0, " 0 1, " 0 2, " 0 ) in OS 2 (O S )3 of this equation are called S deendent if there exists 2 O S such that x 0 = x, y 0 = y, " 0 1 = " 1, " 0 2 = " 2, " 0 = " 3. Theorem The number of classes of S deendence of the solutions of the equation (X Y )(X E 1 Y )(X E 2 Y )=E is finite. Connection with a result of P. Vojta Let D be a divisor of P n with at least n +2distinct comonents. Then any set of D integral oints on P n is degenerate (namely : is contained in a roer Zarisky closed set). With n =4, with rojective coordinates (X : Y : Z : E 1 : E 2 ) and with the divisor ZE 1 E 2 (X Y )(XZ E 1 Y )(XZ E 2 Y )=0 on P 4, one deduces that the set of solutions of the equation is degenerate. (X Y )(X E 1 Y )(X E 2 Y )=E 33 / / 65 Generalized S unit equation Integral oints on P n minus n +2hyerlanes Let n 1 be an integer and let S afinitesetoflacesofk including the archimedean laces. Then the equation E E n =0 has but finitely many classes modulo O S of solutions (" 0,...," n ) 2 (O S )n+1 for which no roer subsum P i2i " i vanishes, with I being a subset of {0,...,n}, with at least two elements and at most n. Let n 1 be an integer and let S afinitesetoflacesofk including the archimedean laces. Then for any set of n +2 distinct hyerlanes H 0,...,H n+1 in P n (K), the set of S integral oints of P n (K) \ (H 0 [ [ H n+1 ) is contained in afiniteunionofhyerlanesofp n (K). Work of P. Vojta. 35 / / 65

10 Generalized Siegel unit equation and integral oints The finiteness of non degenerate solutions of the generalized S unit equation is equivalent to the statement on integral oints on P n minus n +2hyerlanes, and both statements deend on Schmidt s Subsace Theorem. The statement on the generalized S unit equation is our main tool for the roof of our finiteness results on families of Thue Mahler Diohantine equations. Schmidt s Subsace Theorem (1970) For m 2 let L 0,...,L m 1 be m indeendent linear forms in m variables with algebraic coe cients. Let > 0. Then the set {x =(x 0,...,x m 1 ) 2 Z m ; L 0 (x) L m 1 (x) ale x } is contained in the union of finitely many roer subsaces of Q m. Wolfgang M. Schmidt 37 / / 65 Schmidt s Subsace Theorem For x =(x 0,...,x m 1 ) 2 Z m,define x =max{ x 0,..., x m 1 }. W.M. Schmidt (1970) : For m 2 let L 0,...,L m 1 be m indeendent linear forms in m variables with algebraic coe cients. Let > 0. Then the set {x =(x 0,...,x m 1 ) 2 Z m ; L 0 (x) L m 1 (x) ale x } is contained in the union of finitely many roer subsaces of Q m. Examle : m =2, L 0 (x 0, x 1 )=x 0, L 1 (x 0, x 1 )= x 0 x 1. Roth s Theorem : for any real algebraic irrational number, for any > 0, the set of /q 2 Q with /q < q 2 is finite. Schmidt s subsace Theorem Several laces Let m 2 be a ositive integer, S afinitesetoflacesofq containing the infinite lace. For each v 2 S let L 0,v,...,L m 1,v be m indeendent linear forms in m variables with algebraic coe cients in the comletion of Q at v. Let > 0. Then the set of x =(x 0,...,x m 1 ) 2 Z m such that Y L 0,v (x) L m 1,v (x) v ale x v2s is contained in the union of finitely many roer subsaces of Q m. 39 / / 65

11 Sketch of roof of the main theorem Let 1, 2, 3, µ be nonzero elements of the number field K. Consider a solution (x, y, z, " 1, " 2, " 3, ") in O 3 S (O S )4 of the equation (X 1 E 1 Y )(X 2 E 2 Y )(X 3 E 3 Y )Z = µe satisfying xy 6= 0and Card{ 1 " 1, 2 " 2, 3 " 3 } =3: (x 1 " 1 y)(x 2 " 2 y)(x 3 " 3 y)z = µ". Sketch of roof of the main theorem (continued) Set j = x j " j y ( j =1, 2, 3), so that 1 2 3z = µ". À la Siegel, eliminate x and y among the three equations We deduce where 1 = x 1 " 1 y, 2 = x 2 " 2 y, 3 = x 3 " 3 y. u 12 u 13 + u 23 u 21 + u 31 u 32 =0, u ij = i i j, (i, j =1, 2, 3, i 6= j). This is a generalized S unit equation with six terms. But nontrivial subsums may vanish / / 65 Thue s equations and aroximation Let f 2 Z[X ] be an irreducible olynomial of degree d and let F (X, Y )=Y d f (X /Y ) be the associated homogeneous binary form of degree d. Then the following two assertions are equivalent : (i) For any integer k 6= 0, the set of (x, y) 2 Z 2 verifying F (x, y) =k is finite. (ii) For any real number ale > 0 and for any root 2 C of f, the set of rational numbers /q verifying q ale ale q d AvariantofLiouville sinequality Claude Levesque and M.W., Aroximation of an algebraic number by roducts of rational numbers and units, Journal of the Australian Mathematical Society, Secial Issue dedicated to Alf van der Poorten, to aear. Let 2 C be an algebraic number of degree d. There exists a constant c 1 such that, for any /q 2 Q and for any unit " of Q( ) such that " 6= /q, we have " q c 1 q d " d 1 is finite. 43 / / 65

12 Quadratic case Refinement in degree 3 Let 0 be the fundamental unit > 1 of the real quadratic field Q( ). For any n 0 with at most one excetion, there exists aconstantc 2 and infinitely many rational numbers /q such that n 0 q ale c 2 q 2 n 0 and infinitely many rational numbers /q such that n 0 q ale c 2 q 2 n 0 Consequence of the finiteness result of S integral oints on Thue s curves : Let be an algebraic number of degree d. For any constant ale > 0, the set of airs (/q, ") 2 Q Z K such that [Q(" ) :Q] 3 and is finite. " q ale ale q d " d 1 45 / / 65 Corvaja Zannier Denote by k k the distance to the nearest integer : for x 2 R, kxk := min n2z x Let Q denote the field of comlex numbers which are algebraic over Q. Following P. Corvaja and U. Zannier (2004), call a (comlex) algebraic number a seudo Pisot number if (i) > 1 and all its conjugates have (comlex) absolute value strictly less than 1 ; (ii) has integral trace : Tr Q( )/Q ( ) 2 Z. n. Corvaja Zannier The main Theorem of Corvaja and Zannier, whose roof also rests on Schmidt s Subsace Theorem, can be stated as follows. Let Q be a finitely generated multilicative grou of algebraic numbers, let 2 Q be a non zero algebraic number and let > 0 be fixed. Then there are only finitely many airs (q, ") 2 Z with =[Q(") :Q] such that q" > 1, q" is not a seudo Pisot number and 0 < k q"k < 1 e h(") q + 47 / / 65

13 E ectivity (work in rogess) An e ective refinement of Liouville s estimate Exlicit uer bounds for the number of solutions or for the number of classes of solutions are obtained by means of quantitative versions of the Subsace Theorem of W.M. Schmidt, but e ective bounds for the solutions or for the heights of the solutions are not available in general. Let K be a number field and let 2 K. There exists an e ectively comutable constant c 3 > 0 such that, for any unit " 2 Z K and any rational number /q with " 6= /q, " q log( " +2) c 3 log max{, q, 2}. In a few secial cases we are able to roduce e ective results. 49 / / 65 On the Brahmaguta Fermat Pell equations The equation x 2 dy 2 = ±1, where the unknowns x and y are ositive integers while d is a fixed ositive integer which is not a square, has been mistakenly called with the name of Pell by Euler. It was investigated by Indian mathematicians since Brahmaguta (628) who solved the case d =92, next by Bhaskara II (1150) for d =61and Narayana (during the 14-th Century) for d =103. Brahmaguta ( ) Brahmashutasiddhanta : Solve in integers the equation x 2 92y 2 =1 The smallest solution is x =1151, y =120. Comosition method : samasa Brahmagutaidentity (a 2 db 2 )(x 2 dy 2 )=(ax + dby) 2 d(ay + bx) 2. htt://mathworld.wolfram.com/brahmagutasproblem.html htt://www-history.mcs.st-andrews.ac.uk/histtoics/pell.html 51 / / 65

14 Bhaskara II or Bhaskaracharya ( ) Narayana Pandit Lilavati Ujjain (India) (Bijaganita, 1150) x 2 61y 2 =1 x 2 103y 2 =1 x = , y = Cyclic method (Chakravala) :roduceasolutiontopell s equation x 2 dy 2 =1starting from a solution to a 2 db 2 = k with a small k. htt://www-history.mcs.st-andrews.ac.uk/histtoics/pell.html x =227528, y = =1. 53 / / 65 References to Indian mathematics Brahmaguta Fermat Pell equations André Weil Number theory. An aroach through history. From Hammurai to Legendre. Birkhäuser Boston, Inc., Boston, Mass., (1984) 375. MR 85c:01004 Connection with Continued fractions Linear recurrence sequences (cf. the lecture by Hugh Williams) 55 / / 65

15 Simultaneous Brahmaguta Fermat Pell equations AresultduetoM.Bennett Let a and b be two nonzero distinct rational integers. M.A. Bennett, M. Ciu, M. Mignotte and R. Okazaki (2006) : the system of two equations X 2 az 2 =1, Y 2 bz 2 =1, where the unknowns (X, Y, Z) take ositive integer values, has at most two solutions. An infinite family of coules (a, b) for which this system has exactly two solutions is known exlicitly. Let a and b be two rational integers which are not square. Let u and v be nonzero rational integers with av 6= bu. M. Bennett (1998) : the system of two equations X 2 az 2 = u, Y 2 bz 2 = v, where the unknowns (X, Y, Z) take ositive integer values has at most c 2 min{!(u),!(v)} log( u + v ) solutions, with an absolute ositive constant c, where!(n) is the number of distinct rime factors of n. 57 / / 65 D.W. Masser and J.H. Rickert (1996) For any N, there exist two rational integers u and v such that the system of two equations X 2 2Z 2 = u, Y 2 3Z 2 = v has at least N solutions (x, y, z) in ositive integers. Bugeaud Levesque W. Équations de Fermat-Pell-Mahler simultanées, Publicationes Mathematicae Debrecen, (2011), Let a and b be two rational integers which are not square and such that ab is not a square. Let { 1,..., s } be a finite set of rime numbers. Then the system of two simultaneous equations : ( X 2 az 2 = ± m 1 1 s ms, Y 2 bz 2 = ± n 1 1 ns s, has only finitely many solutions in integers (x, y, z, m 1,...,m s, n 1,...,n s ), with x, y, z > 0 and gcd(x, y, z, 1 s )=1. 59 / / 65

16 Brahmaguta Fermat Pell Mahler equations Equivalence classes Let b 1, b 2 be rational integers, a 1, a 2, c 1, c 2 be nonzero rational integers, S = { 1,..., s } afinitesetofrimenumbers. Set 1 = b 2 1 4a 1 c 1, 2 = b 2 2 4a 2 c 2 and assume that the roduct 1 2 is not a square. Consider the equation (a 1 X 2 + b 1 XZ + c 1 Z 2 )(a 2 Y 2 + b 2 YZ + c 2 Z 2 )=W, Two solutions (x, y, z, w) and (x 0, y 0, z 0, w 0 ) of the equation (a 1 X 2 + b 1 XZ + c 1 Z 2 )(a 2 Y 2 + b 2 YZ + c 2 Z 2 )=W, are called S equivalent if there exists a S-unit u such that x 0 = ux, y 0 = uy, z 0 = uz, w 0 = u 4 w. where the unknowns (X, Y, Z, W ) take their values (x, y, z, w) in Z 3 S Z S with xyz 6= / / 65 Bugeaud Levesque W. Équations de Fermat-Pell-Mahler simultanées, Publicationes Mathematicae Debrecen, (2011), The set of S equivalence classes of solutions (x, y, z, w) 2 Z 3 S Z S of the equation (a 1 X 2 + b 1 XZ + c 1 Z 2 )(a 2 Y 2 + b 2 YZ + c 2 Z 2 )=W, with xyz 6= 0, is finite, and this set has at most ale 1 elements, where ale 1 = t with t =4(!(a 1 a 2 1 s )+1). Consequence Let 1,..., s be distinct rime numbers. The set of (2s +3) tules of rational integers (x, y, z, m 1,...,m s, n 1,...,n s ), with x, y, z > 0 and gcd(x, y, z, 1 s )=1, satisfying ( a1 X 2 + b 1 XZ + c 1 Z 2 = ± m 1 1 s ms, a 2 Y 2 + b 2 YZ + c 2 Z 2 = ± n 1 1 ns s, is finite, and this set has at most ale 2 elements, where ale 2 = (!(a 1a 2 1 s)+1). 63 / / 65

17 March 15, 2012 CARMA International Number Theory Conference Some families of curves with only trivial S integral oints (joint work with Claude Levesque) Michel Waldschmidt Institut de Mathématiques de Jussieu (Univ. Paris VI) htt:// miw/ 65 / 65