For a ring $R$, Hilbert's Tenth Problem $HTP(R)$ is the set of polynomial equations over $R$, in ... more For a ring $R$, Hilbert's Tenth Problem $HTP(R)$ is the set of polynomial equations over $R$, in several variables, with solutions in $R$. We view $HTP$ as an operator, mapping each set $W$ of prime numbers to $HTP(\mathbb Z[W^{-1}])$, which is naturally viewed as a set of polynomials in $\mathbb Z[X_1,X_2,\ldots]$. For $W=\emptyset$, it is a famous result of Matiyasevich, Davis, Putnam, and Robinson that the jump $\emptyset~\!'$ is Turing-equivalent to $HTP(\mathbb Z)$. More generally, $HTP(\mathbb Z[W^{-1}])$ is always Turing-reducible to $W'$, but not necessarily equivalent. We show here that the situation with $W=\emptyset$ is anomalous: for almost all $W$, the jump $W'$ is not diophantine in $\mathbb Z[W^{-1}]$. We also show that the $HTP$ operator does not preserve Turing equivalence: even for complementary sets $U$ and $\overline{U}$, $HTP(\mathbb Z[U^{-1}])$ and $HTP(\mathbb Z[\overline{U}^{-1}])$ can differ by a full jump. Strikingly, reversals are also possible, with $V<_T W$ but $HTP(\mathbb Z[W^{-1}]) <_T HTP(\mathbb Z[V^{-1}])$.
If A is an abelian variety defined over Q, its Tate module Tℓ(A) affords an ℓ-adic representation... more If A is an abelian variety defined over Q, its Tate module Tℓ(A) affords an ℓ-adic representation ρ of the absolute Galois group GQ. But under suitable constraints on ramification, such a representation cannot exist. Thus, Fontaine [Fo] proved there do not exist abelian varieties over Z, i.e. with everywhere good reduction. Fontaine
... The conductor of an abelian variety Armand Brumer Kenneth Kramer 1 Mathematics Dept., Fordham... more ... The conductor of an abelian variety Armand Brumer Kenneth Kramer 1 Mathematics Dept., Fordham University, Bronx, NY 10458; 2Mathematics Dept., Queens College (CUNY ... Here 03A6(G) denotes the Frattini subgroup of G, generated by p th powers and commutators. ...
For a ring $R$, Hilbert's Tenth Problem $HTP(R)$ is the set of polynomial equations over $R$, in ... more For a ring $R$, Hilbert's Tenth Problem $HTP(R)$ is the set of polynomial equations over $R$, in several variables, with solutions in $R$. We view $HTP$ as an operator, mapping each set $W$ of prime numbers to $HTP(\mathbb Z[W^{-1}])$, which is naturally viewed as a set of polynomials in $\mathbb Z[X_1,X_2,\ldots]$. For $W=\emptyset$, it is a famous result of Matiyasevich, Davis, Putnam, and Robinson that the jump $\emptyset~\!'$ is Turing-equivalent to $HTP(\mathbb Z)$. More generally, $HTP(\mathbb Z[W^{-1}])$ is always Turing-reducible to $W'$, but not necessarily equivalent. We show here that the situation with $W=\emptyset$ is anomalous: for almost all $W$, the jump $W'$ is not diophantine in $\mathbb Z[W^{-1}]$. We also show that the $HTP$ operator does not preserve Turing equivalence: even for complementary sets $U$ and $\overline{U}$, $HTP(\mathbb Z[U^{-1}])$ and $HTP(\mathbb Z[\overline{U}^{-1}])$ can differ by a full jump. Strikingly, reversals are also possible, with $V<_T W$ but $HTP(\mathbb Z[W^{-1}]) <_T HTP(\mathbb Z[V^{-1}])$.
If A is an abelian variety defined over Q, its Tate module Tℓ(A) affords an ℓ-adic representation... more If A is an abelian variety defined over Q, its Tate module Tℓ(A) affords an ℓ-adic representation ρ of the absolute Galois group GQ. But under suitable constraints on ramification, such a representation cannot exist. Thus, Fontaine [Fo] proved there do not exist abelian varieties over Z, i.e. with everywhere good reduction. Fontaine
... The conductor of an abelian variety Armand Brumer Kenneth Kramer 1 Mathematics Dept., Fordham... more ... The conductor of an abelian variety Armand Brumer Kenneth Kramer 1 Mathematics Dept., Fordham University, Bronx, NY 10458; 2Mathematics Dept., Queens College (CUNY ... Here 03A6(G) denotes the Frattini subgroup of G, generated by p th powers and commutators. ...
Uploads
Papers by Kenneth Kramer