Solvability of Diophantine Equations

Attila Bérczes (University of Debrecen): On arithmetic properties of solutions of norm form equations. Abstract. Let α be an algebraic number of degree n and K := Q(α). Consider the norm form equation NK/Q(x0 + x1α+ x2α + . . .+ xn−1α) = b in x0, . . . , xn−1 ∈ Z. (1) Let H denote the solution set of (1). Arranging the elements of H in an |H| × n array H, one may ask at least two natural questions about arithmetical progressions appearing in H. The “horizontal” one: do there exist infinitely many rows of H, which form arithmetic progressions; and the “vertical” one: do there exist arbitrary long arithmetic progressions in some column of H? In the near past both questions were studied thoroughly by Bazsó, Dujella, Hajdu, Pethő, Tadic, Ziegler and the speaker. In the talk the emphasis will be on explicit results obtained by the speaker and his co-authors (Pethő and Ziegler). More precisely, in the case when α is a root of the irreducible polynomial x − a with 1 < a ≤ 100 the authors explicitly determined all solutions of the norm form equations (1) with b = 1 in which the coordinates x0, . . . , xn−1 form an arithmetic progression. Similarly, when α is the root of the Thomas polynomial fa := X − (a − 1)X − (a + 2)X − 1 (a ∈ Z) we determined all solutions of the inequality |NK/Q(x0 + x1α+ x2α)| ≤ |2a+ 1|, whose coordinates from an arithmetic progression. The problems were solved using Baker’s method, and combining it with local methods, the modular method and built-in Thue-solvers of PARI and MAGMA. Let α be an algebraic number of degree n and K := Q(α). Consider the norm form equation NK/Q(x0 + x1α+ x2α + . . .+ xn−1α) = b in x0, . . . , xn−1 ∈ Z. (1) Let H denote the solution set of (1). Arranging the elements of H in an |H| × n array H, one may ask at least two natural questions about arithmetical progressions appearing in H. The “horizontal” one: do there exist infinitely many rows of H, which form arithmetic progressions; and the “vertical” one: do there exist arbitrary long arithmetic progressions in some column of H? In the near past both questions were studied thoroughly by Bazsó, Dujella, Hajdu, Pethő, Tadic, Ziegler and the speaker. In the talk the emphasis will be on explicit results obtained by the speaker and his co-authors (Pethő and Ziegler). More precisely, in the case when α is a root of the irreducible polynomial x − a with 1 < a ≤ 100 the authors explicitly determined all solutions of the norm form equations (1) with b = 1 in which the coordinates x0, . . . , xn−1 form an arithmetic progression. Similarly, when α is the root of the Thomas polynomial fa := X − (a − 1)X − (a + 2)X − 1 (a ∈ Z) we determined all solutions of the inequality |NK/Q(x0 + x1α+ x2α)| ≤ |2a+ 1|, whose coordinates from an arithmetic progression. The problems were solved using Baker’s method, and combining it with local methods, the modular method and built-in Thue-solvers of PARI and MAGMA. Imin Chen (Simon Fraser University, Burnaby): Applications of Q-curves to Diophantine equations. Abstract. The use of elliptic curves over Q and their modularity has shown itself to be a powerful tool for tackling certain types of diophantine questions. The use of elliptic curves over Q and their modularity has shown itself to be a powerful tool for tackling certain types of diophantine questions.