Tomasz Jędrzejak

• Ph. D. + Habilitation
• Professor at University of Szczecin

Consider a one-parameter family of hyperelliptic curves Ca : y 2 = x 5 + 3ax 4 − 2a 2 x 3 − 6a 3 x 2 + 3a 4 x + a 5 dened over Q, and their Jacobians Ja where without loss of generality a is a non-zero squarefree integer. Clearly, the curve Ca is a quadratic twist by a of C 1. Note that Ja has complex multiplication by the quartic eld Q −2 + √ 2. F...

This is a new version (Theorem 1 is modified)

This paper is a continuation of our previous one under the same title. In both articles we study the hyperelliptic curves Ca : y 2 = x 5 + ax dened over Q, and their Jacobians Ja (without loss of generality a is a non-zero 8th power free integer). Previously we considered the case when the polynomial x 4 +a is irreducible in Q [x] and obtained (und...

This paper is a continuation of our previous one under the same title. In both articles, we study the hyperelliptic curves [Formula: see text] defined over [Formula: see text], and their Jacobians [Formula: see text] (without loss of generality a is a nonzero 8th power free integer). Previously, we considered the case when the polynomial [Formula:...

Let $K = \mathbb Q(\sqrt{-q})$, where $q$ is a prime congruent to $3$ modulo $4$. Let $A = A(q)$ denote the Gross curve over the Hilbert class field $H$ of $K$. In this note we use Magma to calculate the values $L(E/H, 1)$ for all such $q$'s up to some reasonable ranges for all primes $q$ congruent to $7$ modulo $8$. All these values are non-zero,...

Consider the hyperelliptic curves Ca:y2=x5+ax defined over Q, and their Jacobians Ja. Without loss of generality a is a non-zero 8th power free integer. Our aim is to obtain upper bounds for ra:=rankJa(Q). In particular, we would like to find infinite subfamily of Ja with rank 0. We show that under certain assumptions on the quartic field Q(−a4), r...

This article is a continuation of our previous paper [9] concerning elliptic curves Ep,m : y 2 = x(x − 2 m)(x + p), where p and p + 2 m are primes. There we proved inter alia that E p,1 has at most two non-torsion integral points, and E p,2 has no such points. Now by using completely dierent methods, namely an analysis of local height functions, we...

presentation based on our paper in Journal of Number Theory in 2019

Let $K=\Bbb Q(\sqrt{-q})$, where $q$ is a prime congruent to $3$ modulo $4$. Let $A=A(q)$ denote the Gross curve. Let $E=A^{(-\beta)}$ denote its quadratic twist, with $\beta=\sqrt{-q}$. The curve $E$ is defined over the Hilbert class field $H$ of $K$. We use Magma to calculate the values $L(E/H,1)$ for all such $q$'s up to some reasonable ranges (...

We classify elliptic curves over Q with a rational point of order 2 or ≥4 and good reduction outside two odd primes. We also exhibit some families of elliptic curves with a rational point of order 3, collect some general existence/non-existence results, and present some information concerning upper bounds for the rank.

This is extended version of my talk at IMPANGA seminar (11.01.2019). The subject of this talk is located at the intersection of the number theory and algebraic geometry, it is called arithmetic algebraic geometry or diophantine geometry.

The title equations are connected with Jacobians of hyperelliptic curves C m,a,b : y 2 = x 2m + ax + b dened over Q. More precisely, these equations have a nontrivial solution if and only if the class of the divisor ∞ + − ∞ − is a torsion point in Jacobian Jac C m,a,b , where ∞ + and ∞ − are two points at innity in C m,a,b. We show that if ab = 0 t...

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 th...

Consider two families of hyperelliptic curves (over Q) C^n,a : y^2 = x^n + a and C_n,a : y^2 = x(x^n + a), and their Jacobians J^n,a, J_n,a respectively. We give the 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 present the results of our search for the orders of Tate–Shafarevich groups for the quadratic twists of \(E=X_0(49)\).

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)). To this end we compute the zeta function of C_3,5,a (resp. C_q,p,a) over F_l for primes l≡1,2,4,8,11(mod15) (resp. for prime...

We consider the elliptic curves Eu : y2 = x3 + ux2 − 16x and their quadratic twists Eu n by a squarefree integer n, where u2 + 64 = p1 . . . pl, (pi are primes). When l ≤ 2, n ≡ 1(mod 4) and all prime divisors of n are congruent to 3 modulo 4 we give a complete description of sizes of Selmer groups of Eu n in terms of number of even partitions of s...

We consider the Fermat elliptic curve E2 : x3 + y3 = 2 and prove (using descent methods) that its quadratic twists have rank zero for a positive proportion of squarefree integers with fixed number of prime divisors. We also prove similar result for rank zero cubic twists of this curve. Then we present detailed description of rank zero quadratic and...

Let $f\in\Q[x]$ be a square-free polynomial of degree $\geq 3$ and $m\geq 3$ be an odd positive integer. Based on our earlier investigations we prove that there exists a function $D_{1}\in\Q(u,v,w)$ such that the Jacobians of the curves \begin{equation*} C_{1}:\;D_{1}y^2=f(x),\quad C_{2}:\;y^2=D_{1}x^m+b,\quad C_{3}:\;y^2=D_{1}x^m+c, \end{equation*...

Consider the families of curves Cn,A:y2=x^n+Ax and Cn,A:y2=x^n+A where A is a nonzero rational. Let Jn,A and Jn,A denote their respective Jacobian varieties. The torsion points of C3,A(Q) and C3,A(Q) are well known. We show that for any nonzero rational A the torsion subgroup of J7,A(Q) is a 2-group, and for A<>4a^4,−1728,−1259712 this subgroup is...

It is classical that a natural number n is congruent i� the rank of Q-points on En : y^2 = x^3 − n^2x is positive. In this paper, following Tada [Ta], we consider generalised congruent numbers. We extend this classical criterion to several in nite families of real number elds (Theorems 1 and 11).

In this paper we consider the Diophantine equation (x+y)(x^2+Bxy + y^2) = Dz^p, where B, D are integers (B 6= ±2, D 6= 0) and p is a prime > 5. We give the Kraus type criterions on nonsolvability of this equation (explicitly, for many B and D) in terms of Galois representations and modular forms. We apply these criterions to hundreds of equations (...

In this paper, we prove that the unboundedness of ranks in families of Jacobian varieties of twisted Fermat curves is equivalent to the divergence of certain infinite series.

We characterize torsion subgroup of the Jacobian of the curve C_A : y^2 = x^5 + Ax, where A 6= 0 is 8th power free integer. As an application of our result we show that for any quadruple a1; a2; a3; a4 of pairwise distinct non-zero integers there exists an in�nite set of integers D with the property that the Jacobian of C_aiD is of positive rank fo...

We study the family of curves Fm (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 distribution of root numbers of the L-functions of the hyperelliptic curves associated to the Fm (p). As a corollary we obtain that the jacobians of the curves Fm (5) with even analytic rank and those with odd...

We give bounds for the canonical height of rational and integral points on cubic twists of the Fermat elliptic curve. As a corollary we prove that there is no integral arithmetic progression on certain curves in this family.(Received January 10 2005)