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Congruent numbers over real number fields

100%
Jędrzejak T.
Colloquium Mathematicum
|
2012
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tom 128
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nr 2
179-186
EN
It is classical that a natural number n is congruent iff the rank of ℚ -points on Eₙ: y² = x³-n²x is positive. In this paper, following Tada (2001), we consider generalised congruent numbers. We extend the above classical criterion to several infinite families of real number fields.
2
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On the torsion of the Jacobians of the hyperelliptic curves y² = xⁿ + a and y² = x(xⁿ+a)

100%
Jędrzejak T.
Acta Arithmetica
|
2016
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tom 174
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nr 2
99-120
EN
Consider two families of hyperelliptic curves (over ℚ), $C^{n,a}: y² = xⁿ+a$ and $C_{n,a}: y² = x(xⁿ+a)$, and their respective Jacobians $J^{n,a}$, $J_{n,a}$. We give a partial characterization of the torsion part of $J^{n,a}(ℚ) $ and $J_{n,a}(ℚ)$. More precisely, we show that the only prime factors of the orders of such groups are 2 and prime divisors of n (we also give upper bounds for the exponents). Moreover, we give a complete description of the torsion part of $J_{8,a}(ℚ)$. Namely, we show that $J_{8,a}(ℚ)_{tors} = J_{8,a}(ℚ)[2]$. In addition, we characterize the torsion parts of $J_{p,a}(ℚ)$, where p is an odd prime, and of $J^{n,a}(ℚ)$, where n = 4,6,8. The main ingredients in the proofs are explicit computations of zeta functions of the relevant curves, and applications of the Chebotarev Density Theorem.
3
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The Analytic Rank of a Family of Jacobians of Fermat Curves

100%
Jędrzejak T.
Bulletin of the Polish Academy of Sciences. Mathematics
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2008
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tom 56
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nr 3
199-206
EN
We study the family of curves $F_{m}(p): x^{p} + y^{p} = m$, where p is an odd prime and m is a pth power free integer. We prove some results about the distribution of root numbers of the L-functions of the hyperelliptic curves associated to the curves $F_{m}(p)$. As a corollary we conclude that the jacobians of the curves $F_{m}(5)$ with even analytic rank and those with odd analytic rank are equally distributed.
4
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A note on the torsion of the Jacobians of superelliptic curves $y^{q} = x^{p} + a$

100%
Jędrzejak T.
Banach Center Publications
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2016
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tom 108
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nr 1
143-149
EN
This article is a short version of the paper published in J. Number Theory 145 (2014) but we add new results and a brief discussion about the Torsion Conjecture. Consider the family of superelliptic curves (over ℚ) $C_{q,p,a}: y^{q} = x^{p} + a$, and its Jacobians $J_{q,p,a}$, where 2 < q < p are primes. We give the full (resp. partial) characterization of the torsion part of $J_{3,5,a}(ℚ)$ (resp. $J_{q,p,a}(ℚ)$). The main tools are computations of the zeta function of $C_{3,5,a}$ (resp. $C_{q,p,a}$) over $𝔽_{l}$ for primes l ≡ 1,2,4,8,11 (mod 15) (resp. for primes l ≡ -1 (mod qp)) and applications of the Chebotarev Density Theorem.
5
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Characterization of the torsion of the Jacobians of two families of hyperelliptic curves

100%
Jędrzejak T.
Acta Arithmetica
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2013
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tom 161
|
nr 3
201-218
EN
Consider the families of curves $C^{n,A} : y² = xⁿ + Ax$ and $C_{n,A} : y² = xⁿ + A$ where A is a nonzero rational. Let $J^{n,A}$ and $J_{n,A}$ denote their respective Jacobian varieties. The torsion points of $C^{3,A}(ℚ)$ and $C_{3,A}(ℚ)$ are well known. We show that for any nonzero rational A the torsion subgroup of $J^{7,A}(ℚ)$ is a 2-group, and for A ≠ 4a⁴,-1728,-1259712 this subgroup is equal to $J^{7,A}(ℚ)[2]$ (for a excluded values of A, with the possible exception of A = -1728, this group has a point of order 4). This is a variant of the corresponding results for $J^{3,A}$ (A ≠ 4) and $J^{5,A}$. We also almost completely determine the ℚ-rational torsion of $J_{p,A}$ for all odd primes p, and all A ∈ ℚ∖{0}. We discuss the excluded case (i.e. $A ∈ (-1)^{(p-1)/2}pℕ²$).
6
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Characterization of the torsion of the Jacobian of y² = x⁵+Ax and some applications

64%
Jędrzejak T., Ulas M.
Acta Arithmetica
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2010
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tom 144
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nr 2
183-191
7
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Cubic forms, powers of primes and the Kraus method

51%
Dąbrowski A., Jędrzejak T., Krawciów K.
Colloquium Mathematicum
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2012
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tom 128
|
nr 1
35-48
EN
We consider the Diophantine equation $(x+y)(x²+Bxy+y²) = Dz^{p}$, where B, D are integers (B ≠ ±2, D ≠ 0) and p is a prime >5. We give Kraus type criteria of nonsolvability for this equation (explicitly, for many B and D) in terms of Galois representations and modular forms. We apply these criteria to numerous equations (with B = 0, 1, 3, 4, 5, 6, specific D's, and p ∈ (10,10⁶)). In the last section we discuss reductions of the above Diophantine equations to those of signature (p,p,2).
8
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Tuples of hyperelliptic curves y² = xⁿ+a

51%
Jędrzejak T., Top J., Ulas M.
Acta Arithmetica
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2011
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tom 150
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nr 2
105-113
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