Conveners
Number Theory
- Borka Jadrijević
Number Theory
- Zrinka Franušić (Department of Mathematics. Faculty of Science. University of Zagreb)
Number Theory
- Petar Orlić (University of Zagreb)
Number Theory
- Ivan Soldo (School of Applied Mathematics and Informatics, Josip Juraj Strossmayer University of Osijek)
We find the number of homogeneous polynomials of degree $d$ such that they vanish on cuspidal modular forms of even weight $m\geq 2$ that form a basis for $S_m(\Gamma_0(N))$. We use these cuspidal forms to embedd $X_0(N)$ to projective space and we find the Hilbert polynomial of the graded ideal of the projective curve that is the image of this embedding.
Let $ \Gamma $ be a congruence subgroup of $ \mathrm{Sp}_{2n}(\mathbb Z) $. Using Poincaré series of $ K $-finite matrix coefficients of integrable discrete series representations of $ \mathrm{Sp}_{2n}(\mathbb R) $, we construct a spanning set for the space $ S_\rho(\Gamma) $ of Siegel cusp forms for $ \Gamma $ of weight $ \rho $, where $ \rho $ is an irreducible polynomial representation of $...
In this talk we give an explicit characterization of all bases of $\varepsilon-$canonical number systems ($\varepsilon-$CNS) with finiteness property in quadratic number fields for all values $\varepsilon\in\lbrack0,1)$. This result is a consequence of the recent result of Jadrijević and Miletić on the characterization of quadratic $\varepsilon-$CNS polynomials. Our result includes the...
Since their introduction the Bent functions had an important role in designing S-boxes for block and stream chipers. While the theory behind Bent functions has substantially been developed, their count is still unknown for number of variables $n>8,$ and is probably the most important question left. Recently, some improvements on the upper bound of number of the Bent functions are presented...
Let $\mathcal R$ be a commutative ring with unity and $n\in \mathcal R$, $n\not=0$. A $D(n)$-quadruple in $\mathcal R$ is a set of four elements in $\mathcal{R}\backslash\{0\}$ with the property that the product of any two of its distinct elements increased by $n$ is a square in $\mathcal{R}$. It is interesting that in some rings $D(n)$-quadruples are related to the representations of...
For a non-zero element $n\in \mathbb{Z}[X]$, a set of $m$ distinct non-zero elements from $\mathbb{Z}[X]$, such that the product of any two of them increased by $n$ is a square of an element of $\mathbb{Z}[X]$, is called a Diophantine $m$-tuple with the property $D(n)$ or simply a $D(n)$-$m$-tuple in $\mathbb{Z}[X]$. We prove that there does not exist a $D(2X+1)$-quadruple in $\mathbb{Z}[X]$,...
A set $\{a, b, c, d\}$ of four distinct non-zero polynomials in $\mathbb{R}[X]$, which are not all constants, is called a polynomial $D(4)$-quadruple in $\mathbb{R}[X]$ if the product of any two of its distinct elements, increased by 4, is a square of a polynomial in $\mathbb{R}[X]$.
We prove some properties of these sets, and to tackle the problem of regularity of polynomial...
For a given irreducible and monic polynomial $f(x) \in \mathbb{Z}[x]$ of degree $4$, we consider the quadratic twists by square-free integers $q$ of the genus one quartic ${H\, :\, y^2=f(x)}$
$H_q \, :\, qy^2=f(x)$.
Let $L$ denotes the set of positive square-free integers $q$ for which $H_q$ is everywhere locally solvable. For a real number $x$, let ${L(x)= \#\{q\in L:\, q \leq x\}}$ be the...
We determine all possible degrees of cyclic isogenies of elliptic curves with rational $j$-invariant defined over degree $d$ extensions of $\mathbb Q$ for $d=3,5,7$.
The same question for $d=1$ has been answered by Mazur and Kenku in 1978-1982, and Vukorepa answered the question for $d=2$. All possible prime degrees of isogenies were previously found by Najman.
We use well known results...
Gonality of an algebraic curve is defined as the minimal degree of a nonconstant morphism from that curve to the projective space $\mathbb{P}^1$. In this talk, I will present the methods used to determine the $\mathbb{Q}$-gonality of the modular curve $X_0(N)$ and its quotients.
A set of $m$ distinct nonzero rationals $\{a_1, a_2, ... , a_m\}$ such that $a_ia_j + 1$ is a perfect square for all $1 \leq i < j \leq m$, is called a rational Diophantine $m$-tuple. If, in addition, $a_i^2 + 1$ is a perfect square for $1 \leq i \leq m$, then we say the $m$-tuple is strong. In this talk, we will describe a construction of infinite families of rational Diophantine sextuples...
A set of $m$ non-zero elements of a commutative ring $R$ with unity $1$ is called a $D(−1)$-$m$-tuple if the product of any two of its distinct elements decreased by $1$ is a square in $R$.
We investigate $D(−1)$-tuples in rings $\mathbb{Z}[\sqrt{-k}]$, k ≥ 2. We prove that, under certain technical conditions, there does not exist a $D(−1)$-quadruple of the form $\{a, 2^i
p^j,c,d\}$ in...
The set $\{a_1, a_2, \ldots , a_m\}$ in a commutative ring $R$ such that $a_i\ne 0$, $i=1,\ldots,m$ and $a_ia_j+n$ is a square in $R$ for all $1\le i< j\le m$ is called a Diophantine $D(n)$-$m$-tuple in the ring $R$.
Let $N$ be a positive integer such that $4N^2+1=q^j$, $q$ is a prime and $j$ is a positive integer. In this talk, we will discuss the extendibility of the Diophantine...
Let $1< a < b < c$ be multiplicatively dependent integers (i.e., there exist nontrivial integer exponents $x, y, z$, such that $a^x b^y c^z = 1$). Is it possible that $a+1, b+1, c+1$ are multiplicatively dependent as well? It turns out that this is easy to answer. We will discuss related, more difficult questions, which will lead to Diophantine equations. We will solve some of them using lower...
