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Papers by Manisha Kulkarni
In [Bil99] Bilu classied the pairs of polynomials f; g over a eld of charac-teristic 0 such that ... more In [Bil99] Bilu classied the pairs of polynomials f; g over a eld of charac-teristic 0 such that f(X) g(Y) has an irreducible factor of degree 2. This note extends his results to arbitrary characteristic. Our method is completely dierent from Bilu's. The main bulk of the work handles the case of positive
Cornell University - arXiv, Mar 22, 2014
We show that a two dimensional ℓ-adic representation of the absolute Galois group of a number fie... more We show that a two dimensional ℓ-adic representation of the absolute Galois group of a number field which is locally potentially equivalent to a GLp2q-ℓadic representation ρ at a set of places of K of positive upper density is potentially equivalent to ρ. For an elliptic curver E defined over a number field K and for a place v of K of good reduction for E, let F pE; vq denote the Frobenius field of E at v, given by the splitting field of the characteristic polynomial of the Frobenius automorphism at v acting on the Tate module of E. As an application, suppose E 1 and E 2 defined over a number field K, with at least one of them without complex multiplication. We prove that the set of places v of K of good reduction such that the corresponding Frobenius fields are equal has positive upper density if and only if E 1 and E 2 are isogenous over some extension of K. For an elliptic curve E defined over a number field K, we show that the set of finite places of K such that the Frobenius field F pE, vq at v equals a fixed imaginary quadratic field F has positive upper density if and only if E has complex multiplication by F .
This note extends the characteristic 0 results in [Bil99] to arbitrary characteristic. The method... more This note extends the characteristic 0 results in [Bil99] to arbitrary characteristic. The method is completely different from Bilu’s. The main bulk of the work handles the case of positive characteristic. Indeed, if one skips all
INDIAN JOURNAL OF PURE AND APPLIED …, 1997
... Page 2. 1 108 MAN ISHA V KULKARNI When L: O, no general result is known regarding the structu... more ... Page 2. 1 108 MAN ISHA V KULKARNI When L: O, no general result is known regarding the structure of OM as a AM/L module, however when M is a tame abelian extension, of L, we have MJ Taylor's theorem in Taylor? which ...
Let K be any quadratic field with O-K its ring of integers. We study the solutions of cubic equat... more Let K be any quadratic field with O-K its ring of integers. We study the solutions of cubic equations, which represent elliptic curves defined over Q, in quadratic fields and prove some interesting results regarding the solutions by using elementary tools. As an application we consider the Diophantine equation r + s + t = rst = 1 in O-K. This Diophantine equation gives an elliptic curve defined over Q with finite Mordell-Weil group. Using our study of the solutions of cubic equations in quadratic fields we present a simple proof of the fact that except for the ring of integers of Q(i) and Q(root 2), this Diophantine equation is not solvable in the ring of integers of any other quadratic fields, which is already proved in [4].
If f, g ∈ K[X] where K has odd characteristic, and f(X)−g(Y ) has an irreducible quadratic factor... more If f, g ∈ K[X] where K has odd characteristic, and f(X)−g(Y ) has an irreducible quadratic factor in K[X, Y ], we show that f and g must have the same degree and can be expressed in terms of Dickson polynomials. This generalizes a theorem of Bilu in characteristic zero.
Journal of Number Theory, 2016
We show that a two dimensional ℓ-adic representation of the absolute Galois group of a number fie... more We show that a two dimensional ℓ-adic representation of the absolute Galois group of a number field which is locally potentially equivalent to a GL(2)-ℓadic representation ρ at a set of places of K of positive upper density is potentially equivalent to ρ.
Indagationes Mathematicae
In this paper we consider the general Diophantine equation of the form
Let E 1 and E 2 be two elliptic curves over a number field K. For a place v of K of good reductio... more Let E 1 and E 2 be two elliptic curves over a number field K. For a place v of K of good reduction for E 1 and for E 2 , let F (E 1 , v) and F (E 2 , v) denote the splitting fields of the characteristic polynomials of the Frobenius automorphism at v acting on the Tate modules of E 1 and E 2 respectively. F (E 1 , v) and F (E 2 , v) are called the Frobenius fields of E 1 and E 2 at v. Assume that at least one of the two elliptic curves is without complex multiplication. Then, we show that the set of places v of K of good reduction such that F (E 1 , v) = F (E 2 , v) has positive upper density if and only if E 1 and E 2 are isogenous over some extension of K. We use this result to prove that, for an elliptic curve E over a number field K, the set of finite places v of K such that F (E, v) equals a fixed imaginary quadratic field F has positive upper density iff E has complex multiplication by F .
If f, g ∈ K[X] where K has odd characteristic, and f (X)−g(Y ) has an irreducible quadratic facto... more If f, g ∈ K[X] where K has odd characteristic, and f (X)−g(Y ) has an irreducible quadratic factor in K[X, Y ], we show that f and g must have the same degree and can be expressed in terms of Dickson polynomials. This generalizes a theorem of Bilu in characteristic zero.
Proceedings Mathematical Sciences
Let K/F be a cyclic extension of prime degree l over a number field F containing l-th roots of un... more Let K/F be a cyclic extension of prime degree l over a number field F containing l-th roots of unity. If F has class number 1, we study the structure of the l-Sylow subgroup of K and obtain its rank using genus theory. Frank Gerth III studied the case of the prime 3. In this article, we generalize his approach and obtain some results for general l. Following that, we obtain more complete results for l = 5 and F = Q(zeta_5). The rank of the 5-class group of K is expressed in terms of power residue symbols. We compare our results with tables obtained using SAGE (under GRH).
Proceedings - Mathematical Sciences, 2011
proved that a product of consecutive integers can never be a perfect power. That is, the equation... more proved that a product of consecutive integers can never be a perfect power. That is, the equation x(x + 1)(x + 2)...(x + (m − 1)) = y n has no solutions in positive integers x, m, n where m, n > 1 and y ∈ Q. We consider the equation where 0 ≤ a 1 < a 2 < · · · < a k are integers and, with r ∈ Q, n ≥ 3 and we prove a finiteness theorem for the number of solutions x in Z, y in Q. Following that, we show that, more interestingly, for every nonzero integer n > 2 and for any nonzero integer r which is not a perfect n-th power for which the equation admits solutions, k is bounded by an effective bound.
Indagationes Mathematicae, 2007
Indagationes Mathematicae, 2005
Let a, b be nonzero rational numbers and C(y) a polynomial with rational coefficients. We study t... more Let a, b be nonzero rational numbers and C(y) a polynomial with rational coefficients. We study the Diophantine equations and aBm(x)=bfn(y)+C(y)
In [Bil99] Bilu classied the pairs of polynomials f; g over a eld of charac-teristic 0 such that ... more In [Bil99] Bilu classied the pairs of polynomials f; g over a eld of charac-teristic 0 such that f(X) g(Y) has an irreducible factor of degree 2. This note extends his results to arbitrary characteristic. Our method is completely dierent from Bilu's. The main bulk of the work handles the case of positive
Cornell University - arXiv, Mar 22, 2014
We show that a two dimensional ℓ-adic representation of the absolute Galois group of a number fie... more We show that a two dimensional ℓ-adic representation of the absolute Galois group of a number field which is locally potentially equivalent to a GLp2q-ℓadic representation ρ at a set of places of K of positive upper density is potentially equivalent to ρ. For an elliptic curver E defined over a number field K and for a place v of K of good reduction for E, let F pE; vq denote the Frobenius field of E at v, given by the splitting field of the characteristic polynomial of the Frobenius automorphism at v acting on the Tate module of E. As an application, suppose E 1 and E 2 defined over a number field K, with at least one of them without complex multiplication. We prove that the set of places v of K of good reduction such that the corresponding Frobenius fields are equal has positive upper density if and only if E 1 and E 2 are isogenous over some extension of K. For an elliptic curve E defined over a number field K, we show that the set of finite places of K such that the Frobenius field F pE, vq at v equals a fixed imaginary quadratic field F has positive upper density if and only if E has complex multiplication by F .
This note extends the characteristic 0 results in [Bil99] to arbitrary characteristic. The method... more This note extends the characteristic 0 results in [Bil99] to arbitrary characteristic. The method is completely different from Bilu’s. The main bulk of the work handles the case of positive characteristic. Indeed, if one skips all
INDIAN JOURNAL OF PURE AND APPLIED …, 1997
... Page 2. 1 108 MAN ISHA V KULKARNI When L: O, no general result is known regarding the structu... more ... Page 2. 1 108 MAN ISHA V KULKARNI When L: O, no general result is known regarding the structure of OM as a AM/L module, however when M is a tame abelian extension, of L, we have MJ Taylor's theorem in Taylor? which ...
Let K be any quadratic field with O-K its ring of integers. We study the solutions of cubic equat... more Let K be any quadratic field with O-K its ring of integers. We study the solutions of cubic equations, which represent elliptic curves defined over Q, in quadratic fields and prove some interesting results regarding the solutions by using elementary tools. As an application we consider the Diophantine equation r + s + t = rst = 1 in O-K. This Diophantine equation gives an elliptic curve defined over Q with finite Mordell-Weil group. Using our study of the solutions of cubic equations in quadratic fields we present a simple proof of the fact that except for the ring of integers of Q(i) and Q(root 2), this Diophantine equation is not solvable in the ring of integers of any other quadratic fields, which is already proved in [4].
If f, g ∈ K[X] where K has odd characteristic, and f(X)−g(Y ) has an irreducible quadratic factor... more If f, g ∈ K[X] where K has odd characteristic, and f(X)−g(Y ) has an irreducible quadratic factor in K[X, Y ], we show that f and g must have the same degree and can be expressed in terms of Dickson polynomials. This generalizes a theorem of Bilu in characteristic zero.
Journal of Number Theory, 2016
We show that a two dimensional ℓ-adic representation of the absolute Galois group of a number fie... more We show that a two dimensional ℓ-adic representation of the absolute Galois group of a number field which is locally potentially equivalent to a GL(2)-ℓadic representation ρ at a set of places of K of positive upper density is potentially equivalent to ρ.
Indagationes Mathematicae
In this paper we consider the general Diophantine equation of the form
Let E 1 and E 2 be two elliptic curves over a number field K. For a place v of K of good reductio... more Let E 1 and E 2 be two elliptic curves over a number field K. For a place v of K of good reduction for E 1 and for E 2 , let F (E 1 , v) and F (E 2 , v) denote the splitting fields of the characteristic polynomials of the Frobenius automorphism at v acting on the Tate modules of E 1 and E 2 respectively. F (E 1 , v) and F (E 2 , v) are called the Frobenius fields of E 1 and E 2 at v. Assume that at least one of the two elliptic curves is without complex multiplication. Then, we show that the set of places v of K of good reduction such that F (E 1 , v) = F (E 2 , v) has positive upper density if and only if E 1 and E 2 are isogenous over some extension of K. We use this result to prove that, for an elliptic curve E over a number field K, the set of finite places v of K such that F (E, v) equals a fixed imaginary quadratic field F has positive upper density iff E has complex multiplication by F .
If f, g ∈ K[X] where K has odd characteristic, and f (X)−g(Y ) has an irreducible quadratic facto... more If f, g ∈ K[X] where K has odd characteristic, and f (X)−g(Y ) has an irreducible quadratic factor in K[X, Y ], we show that f and g must have the same degree and can be expressed in terms of Dickson polynomials. This generalizes a theorem of Bilu in characteristic zero.
Proceedings Mathematical Sciences
Let K/F be a cyclic extension of prime degree l over a number field F containing l-th roots of un... more Let K/F be a cyclic extension of prime degree l over a number field F containing l-th roots of unity. If F has class number 1, we study the structure of the l-Sylow subgroup of K and obtain its rank using genus theory. Frank Gerth III studied the case of the prime 3. In this article, we generalize his approach and obtain some results for general l. Following that, we obtain more complete results for l = 5 and F = Q(zeta_5). The rank of the 5-class group of K is expressed in terms of power residue symbols. We compare our results with tables obtained using SAGE (under GRH).
Proceedings - Mathematical Sciences, 2011
proved that a product of consecutive integers can never be a perfect power. That is, the equation... more proved that a product of consecutive integers can never be a perfect power. That is, the equation x(x + 1)(x + 2)...(x + (m − 1)) = y n has no solutions in positive integers x, m, n where m, n > 1 and y ∈ Q. We consider the equation where 0 ≤ a 1 < a 2 < · · · < a k are integers and, with r ∈ Q, n ≥ 3 and we prove a finiteness theorem for the number of solutions x in Z, y in Q. Following that, we show that, more interestingly, for every nonzero integer n > 2 and for any nonzero integer r which is not a perfect n-th power for which the equation admits solutions, k is bounded by an effective bound.
Indagationes Mathematicae, 2007
Indagationes Mathematicae, 2005
Let a, b be nonzero rational numbers and C(y) a polynomial with rational coefficients. We study t... more Let a, b be nonzero rational numbers and C(y) a polynomial with rational coefficients. We study the Diophantine equations and aBm(x)=bfn(y)+C(y)