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Distribution of values of Hecke characters of infinite order

C. S. Rajan (1998)

Acta Arithmetica

We show that the number of primes of a number field K of norm at most x, at which the local component of an idele class character of infinite order is principal, is bounded by O(x exp(-c√(log x))) as x → ∞, for some absolute constant c > 0 depending only on K.

Division-ample sets and the Diophantine problem for rings of integers

Gunther Cornelissen, Thanases Pheidas, Karim Zahidi (2005)

Journal de Théorie des Nombres de Bordeaux

We prove that Hilbert’s Tenth Problem for a ring of integers in a number field K has a negative answer if K satisfies two arithmetical conditions (existence of a so-called division-ample set of integers and of an elliptic curve of rank one over K ). We relate division-ample sets to arithmetic of abelian varieties.

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