80 citations · 169 across the 21 of their papers we have counts for
23 papers
Defining using unit groups
Barry Mazur, Karl Rubin, Alexandra Shlapentokh
We consider first-order definability and decidability questions over rings of integers of algebraic extensions of $\Q$, paying attention to the uniformity of definitions. The unifo…
Existential definability and diophantine stability
Barry Mazur, Karl Rubin, Alexandra Shlapentokh
Let be a number field, let be an algebraic (possibly infinite degree) extension of , and let be their rings of integers. Suppose is an abelian…
Big fields that are not large
Barry Mazur, Karl Rubin
A subfield of is if every smooth curve over with a rational point has infinitely many rational points. A subfield of i…
Algebraic maps constant on isomorphism classes of unpolarized abelian varieties are constant
Eric Rains, Karl Rubin, Travis Scholl +2
We show that if a rational map is constant on each isomorphism class of unpolarized abelian varieties of a given dimension, then it is a constant map. Our results are motivated by…
Arithmetic conjectures suggested by the statistical behavior of modular symbols
Barry Mazur, Karl Rubin
Suppose is an elliptic curve over and is a Dirichlet character. We use statistical properties of modular symbols to estimate heuristically the probability that…
Diophantine stability
Barry Mazur, Karl Rubin, Michael Larsen
If is an irreducible algebraic variety over a number field , and is a field containing , we say that is diophantine-stable for if . We prove th…