8 papers
On the fibres of an elliptic surface where the rank does not jump
Jerson Caro, Hector Pasten
For a non-constant elliptic surface over defined over , it is a result of Silverman that the Mordell--Weil rank of the fibres is at least the rank of the…
Non-Diophantine sets in rings of functions
Natalia Garcia-Fritz, Hector Pasten, Thanases Pheidas
Except for a limited number of cases, a complete classification of the Diophantine sets of polynomial rings and fields of rational functions seems out of reach at present. We contr…
A Diophantine definition of the constants in
Natalia Garcia-Fritz, Hector Pasten
It is an old problem in the area of Diophantine definability to determine whether is Diophantine in . We provide a positive answer conditional on two st…
A Chabauty-Coleman bound for surfaces
Jerson Caro, Hector Pasten
Building on work by Chabauty from 1941, Coleman proved in 1985 an explicit bound for the number of rational points of a curve of genus defined over a number field ,…
Elliptic curves with long arithmetic progressions have large rank
Natalia Garcia-Fritz, Hector Pasten
For any family of elliptic curves over the rational numbers with fixed -invariant, we prove that the existence of a long sequence of rational points whose -coordinates form a…
Towards Hilbert's Tenth Problem for rings of integers through Iwasawa theory and Heegner points
Natalia Garcia-Fritz, Hector Pasten
For a positive proportion of primes and , we prove that is Diophantine in the ring of integers of . This provides a new and e…