arXiv subjects

Matija Kazalicki

Publications and source records attributed to Matija Kazalicki.

At least 19 recordsLinked to original sources

arXiv2026

Gross vectors modulo 2 and elliptic curves of prime conductor

Let p > 3 be a prime, and let S_p denote the geometric isomorphism classes of supersingular elliptic curves in characteristic p whose j-invariants lie in F_p. For each negative fundamental discriminant -D for which p is inert in Q(sqrt(-D)), let m_i(D), i in S_p, be the integral coefficients of the corresponding Gross vector. We prove that the vectors (m_i(D) mod 2)_{i in S_p} span F_2^{S_p}. The key step reduces the parity of the representation numbers of Gross's ternary lattices to representation by rank-two sublattices perpendicular to Frobenius. Using Ibukiyama's explicit maximal orders, the resulting primitive binary forms are identified with those occurring in the Xiao--Zhou--Deng--Qu parametrization of supersingular elliptic curves over F_p. Class field theory and Chebotarev's theorem then allow the individual supersingular coordinates to be isolated. As a consequence, if E/Q has prime conductor p and positive Mordell--Weil rank, then every coefficient of its Brandt eigenvector indexed by S_p is even, proving a conjecture of Kazalicki and Kohen. Thus an odd coefficient at a rational supersingular class is an algebraic certificate of rank 0. Combining this parity theorem with formulas of Mestre and Gross--Kudla, we also prove that the modular degree of every positive-rank elliptic curve of prime conductor and root number +1 is divisible by 4. Consequently, Watkins' conjecture holds for all such curves of rank 2.

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arXiv2026

Jordan rigidity of full matrix algebras

Let $\mathbb{F}$ be a field of characteristic different from $2$, and let $M_n(\mathbb{F})^+$ denote the Jordan algebra of all $n\times n$ matrices over $\mathbb{F}$ with product $X\circ Y:=(XY+YX)/2$. We prove a rigidity theorem for $M_n(\mathbb{F})^+$, $n\ge2$: if $\mathcal{J}$ is any $2$-torsion-free Jordan ring and $ϕ:M_n(\mathbb{F})^+\to\mathcal{J}$ is a Jordan multiplicative (product-preserving) map, then $ϕ(0)$ is an idempotent and $X\mapstoϕ(X)-ϕ(0)$ is either zero or an injective Jordan ring homomorphism. Thus, up to an idempotent constant, preservation of the Jordan product alone forces additivity and the zero-or-injective dichotomy. When specialized to associative codomains, the theorem yields the Jacobson--Rickart decomposition into homomorphic and antihomomorphic parts. In particular, for maps $M_n(\mathbb F)^+\to M_k(\mathbb K)^+$, where $\mathbb K$ is a field of characteristic different from $2$, we also obtain a block normal form governed by finite-dimensional $\mathbb K$-representations of $\mathbb F$, together with a criterion for the existence of nonconstant maps.

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arXiv2026

Quartic Rational Diophantine Quadruples and the Euler Surface

We prove that there exist infinitely many quartic rational Diophantine quadruples, that is, sets of four pairwise distinct nonzero rational numbers whose pairwise products increased by 1 are fourth powers in Q. To the best of our knowledge, no examples of such quadruples were previously known. Our construction is motivated by computer experiments and leads naturally to the classical Euler surface E:X^4+Y^4=Z^4+W^4. We show that every rational point on a suitable Zariski-open subset of E yields a quartic rational Diophantine quadruple, thereby obtaining a rational map from the Euler surface to the parameter space of quartic quadruples. In particular, Euler's classical parametrization produces the first explicit infinite family of quartic rational Diophantine quadruples. We also explain that the same mechanism extends to arbitrary exponents k>1, with the Euler surface replaced by the Fermat--Euler surface E_k:X^k+Y^k=Z^k+W^k. For even k, every rational point on a suitable open subset of E_k gives rise to a kth power rational Diophantine quadruple, while for odd k one obtains such quadruples on the locus where W/Z is a square.

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arXiv2026

The Jordan multiplication semigroup of matrix algebras is the full endomorphism semigroup

Let $\mathbb{K}$ be a field of characteristic different from $2$, and let $M_n(\mathbb{K})$ be the algebra of all $n\times n$ matrices over $\mathbb{K}$. We consider the corresponding special Jordan algebra $\mathcal{A}:=M_n(\mathbb{K})^+$ with symmetrized product $A\circ B:=(AB+BA)/2$, and write $\mathcal{A}_{\mathrm v}:=M_n(\mathbb{K})$ for the underlying $\mathbb{K}$-vector space of $\mathcal{A}$. For $A\in\mathcal{A}$, let $\mathrm{L}_A(X):=A\circ X$ be the multiplication operator. We consider the Jordan multiplication semigroup generated by all multiplication operators, \[ \mathrm{JMS}(\mathcal{A}):=\langle \mathrm{L}_A:A\in\mathcal{A}\rangle\subseteq \mathrm{End}_{\mathbb{K}}(\mathcal{A}_{\mathrm v}). \] We prove that $\mathrm{JMS}(\mathcal{A})=\mathrm{End}_{\mathbb{K}}(\mathcal{A}_{\mathrm v})$. Equivalently, every $\mathbb{K}$-linear endomorphism of $\mathcal{A}_{\mathrm v}$ is a composition of multiplication operators. The proof is primarily linear-algebraic. The main step is to show that $\mathrm{SL}(\mathcal{A}_{\mathrm v})\subseteq \mathrm{JMS}(\mathcal{A})$ by constructing elementary transvections inside the semigroup. We then prove determinant surjectivity on the unit group of $\mathrm{JMS}(\mathcal{A})$ and combine it with the existence of a singular element of rank $n^2-1$ to obtain the full endomorphism semigroup. In the finite-field case, the determinant-surjectivity step is established via Jacobi-sum estimates.

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arXiv2026

Seven squares from three numbers

We study triples {a,b,c} of distinct nonzero rational numbers such that a+1,b+1,c+1,ab+1,ac+1,bc+1 and abc+1 are all perfect squares. We prove that there exist infinitely many such triples. In contrast, we show that no triple of positive integers has this property.

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arXiv2025

Equidistribution of Diophantine pairs among the equivalence classes of quadratic forms

For a fixed integer n, a pair of nonzero integers {a, c} is called a D(n)-pair if the product ac plus n is a perfect square. In this short note we prove that D(n)-pairs are asymptotically equidistributed (via their associated quadratic forms) among proper SL_2(Z)-equivalence classes of binary quadratic forms of discriminant 4n with fixed content. As a consequence, we obtain a more streamlined and simpler proof of Badesa's asymptotic formula for the number of D(n)-pairs.

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arXiv2025

Improving elliptic curve rank classification using multi-value and learned Mestre-Nagao sums

Determining the rank of an elliptic curve E/Q remains a central challenge in number theory. Heuristics such as Mestre--Nagao sums are widely used to estimate ranks, but there is considerable room for improving their predictive power. This paper introduces two novel methods for enhancing rank classification using Mestre--Nagao sums. First, we propose a ``multi-value'' approach that simultaneously uses two distinct sums, S_0 and S_5, evaluated over multiple ranges. This multi-sum perspective significantly improves classification accuracy over traditional single-sum heuristics. Second, we employ machine learning -- specifically deep neural networks -- to learn optimal, potentially conductor-dependent weightings for Mestre--Nagao sums directly from data. Our results indicate that adaptively weighted sums offer a slight edge in rank classification over traditional methods.

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arXiv2025

Second moments and the bias conjecture for the family of cubic pencils

For a 1-parametric family $E_k$ of elliptic curves over $\mathbb{Q}$ and a prime $p$, consider the second moment sum $M_{2,p}(E_k)=\sum_{k \in \mathbb{F}_p} a_{k,p}^2$, where $a_{k,p}=p+1-\#E_k(\mathbb{F}_p)$. Inspired by Rosen and Silverman's proof of Nagao conjecture which relates the first moment of a rational elliptic surface to the rank of Mordell-Weil group of the corresponding elliptic curve, S. J. Miller initiated the study of the asymptotic expansion of $M_{2,p}(E_k)=p^2+O(p^{3/2})$ (which by the work of Deligne and Michel has cohomological interpretation). He conjectured that, similar to the first moment case, the largest lower-order term that does not average to 0 has a negative bias. In this paper, we provide an explicit formula for the second moment $M_{2,p}(\mathcal{E}_{U})$ of $$ \mathcal{E}_{U}:y^2=P(x)U+Q(x), $$ where $\textrm{deg } P(x), \textrm{deg } Q(x)\leq 3$. For a generic choice of polynomials $P(x)$ and $Q(x)$ this formula is expressed in terms of the point count of a certain genus two curve. As an application, we prove that the Bias conjecture holds for the pencil of the cubics $\mathcal{E}_U$.

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arXiv2024

Murmurations of Mestre-Nagao sums

This paper investigates the detection of the rank of elliptic curves with ranks 0 and 1, employing a heuristic known as the Mestre-Nagao sum \[ S(B) = \frac{1}{\log{B}} \sum_{\substack{p<B \\ \textrm{good reduction}}} \frac{a_p(E)\log{p}}{p}, \] where $a_p(E)$ is defined as $p + 1 - \#E(\mathbb{F}_p)$ for an elliptic curve $E/\mathbb{Q}$ with good reduction at prime $p$. This approach is inspired by the Birch and Swinnerton-Dyer conjecture. Our observations reveal an oscillatory behavior in the sums, closely associated with the recently discovered phenomena of murmurations of elliptic curves. Surprisingly, this suggests that in some cases, opting for a smaller value of $B$ yields a more accurate classification than choosing a larger one. For instance, when considering elliptic curves with conductors within the range of $[40\,000,45\,000]$, the rank classification based on $a_p$'s with $p < B = 3\,200$ produces better results compared to using $B = 50\,000$. This phenomenon finds partial explanation in the recent work of Zubrilina.

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arXiv2024

Rational Diophantine sextuples with strong pair

A set of $m$ distinct nonzero rationals $\{a_1, a_2,\ldots, a_m\}$ such that $a_i a_j+1$ is a perfect square for all $1\le i <j \le m$, is called a rational Diophantine $m$-tuple. If in addition, $a_i^2+1$ is a perfect square for $1\le i\le m$, then we say the $m$-tuple is strong. In this paper, we construct infinite families of rational Diophantine sextuples containing a strong Diophantine pair.

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arXiv2022

Ranks of elliptic curves and deep neural networks

Determining the rank of an elliptic curve E/Q is a hard problem, and in some applications (e.g. when searching for curves of high rank) one has to rely on heuristics aimed at estimating the analytic rank (which is equal to the rank under the Birch and Swinnerton-Dyer conjecture). In this paper, we develop rank classification heuristics modeled by deep convolutional neural networks (CNN). Similarly to widely used Mestre-Nagao sums, it takes as an input the conductor of E and a sequence of normalized a_p-s (where a_p=p+1-#E(F_p) if p is a prime of good reduction) in some range (p<10^k for k=3,4,5), and tries to predict rank (or detect curves of ``high'' rank). The model has been trained and tested on two datasets: the LMFDB and a custom dataset consisting of elliptic curves with trivial torsion, conductor up to 10^30, and rank up to 10. For comparison, eight simple neural network models of Mestre-Nagao sums have also been developed. Experiments showed that CNN performed better than Mestre-Nagao sums on the LMFDB dataset (interestingly neural network that took as an input all Mestre-Nagao sums performed much better than each sum individually), while they were roughly equal on custom made dataset.

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arXiv2022

Quadratic twists of genus one curves and Diophantine quintuples

Motivated by the theory of Diophantine $m$-tuples, we study rational points on quadratic twists $H^d:d y^2=(x^2+6x-18)(-x^2+2x+2)$, where $|d|$ is a prime. If we denote by $S(X)=\{ d \in \mathbb{Z}: H^d(\mathbb{Q})\ne \emptyset, |d| \textrm{ is a prime}\textrm{ and } |d| < X\},$ then, by assuming some standard conjectures about the ranks of elliptic curves in the family of quadratic twists, we prove that as $X \rightarrow \infty$ $$\frac{43}{256}+o(1)\le \frac{\#S(X)}{2π(X)}\le \frac{46}{256}+o(1).$$

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arXiv2021

Diophantine triples and K3 surfaces

A Diophantine $m$-tuple with elements in the field $K$ is a set of $m$ non-zero (distinct) elements of $K$ with the property that the product of any two distinct elements is one less than a square in $K$. Let $X: (x^2-1)(y^2-1)(z^2-1)=k^2,$ be a threefold. Its $K$-rational points parametrize Diophantine triples over $K$ such that the product of the elements of the triple that corresponds to the point $(x,y,z,k)\in X(K)$ is equal to $k$. We denote by $\overline{X}$ the projective closure of $X$ and for a fixed $k$ by $X_k$ a variety defined by the same equation as $X$. We prove that the variety $\overline{X}$ is birational to $\mathbb{P}^3$ which leads us to a new rational parametrization of the set of Diophantine triples. Next, specializing to finite fields, we find a correspondence between a K3 surface $X_k$ for a given $k\in\mathbb{F}_{p}^{\times}$ in the prime field $\mathbb{F}_{p}$ of odd characteristic and an abelian surface which is a product of two elliptic curves $E_k\times E_k$ where $E_k: y^2=x(k^2(1 + k^2)^3 + 2(1 + k^2)^2 x + x^2)$. We derive a formula for $N(p,k)$, the number of Diophantine triples over $\mathbb{F}_{p}$ with the product of elements equal to $k$. We show that the variety $\overline{X}$ admits a fibration by rational elliptic surfaces and from it we derive the formula for the number of points on $\overline{X}$ over an arbitrary finite field $\mathbb{F}_{q}$. We reprove the formula for the number of Diophantine triples over $\mathbb{F}_{q}$ from Dujella-Kazalicki(2021). We derive the formula for the second moment of the elliptic surface $E_k$ (and thus confirming Steven J. Miller's Bias conjecture in this particular case) which we describe in terms of Fourier coefficients of a rational newform generating $S_4(Γ_{0}(8))$. Finally, in the Appendix, Luka Lasić defines circular Diophantine $m$-tuples, and describes the parametrization of these sets.

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arXiv2021

Elliptic curves with torsion groups $\mathbb{Z}/8\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z}$

In this paper, we present details of seven elliptic curves over $\mathbb{Q}(u)$ with rank $2$ and torsion group $\mathbb{Z}/ 8\mathbb{Z}$ and five curves over $\mathbb{Q}(u)$ with rank $2$ and torsion group $\mathbb{Z}/ 2\mathbb{Z} \times \mathbb{Z}/ 6\mathbb{Z}$. We also exhibit some particular examples of curves with high rank over $\mathbb{Q}$ by specialization of the parameter. We present several sets of infinitely many elliptic curves in both torsion groups and rank at least $3$ parametrized by elliptic curves having positive rank. In some of these sets we have performed calculations about the distribution of the root number. This has relation with recent heuristics concerning the rank bound for elliptic curves by Park, Poonen, Voight and Wood.

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arXiv2021

Rational $D(q)$-quadruples

For a rational number $q$, a rational $D(q)$-$n$-tuple is a set of $n$ distinct nonzero rationals $\{a_1, a_2, \dots, a_n\}$ such that $a_ia_j+q$ is a rational square for all $1 \leqslant i < j \leqslant n$. For every $q$ we find all rational $m$ such that there exists a $D(q)$-quadruple with product $abcd=m$. We describe all such quadruples using points on a specific elliptic curve depending on $(q,m).$

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arXiv2020

D(n)-quintuples with square elements

For an integer n, a set of m distinct nonzero integers {a_1,a_2,...,a_m} such that a_i a_j+n is a perfect square for all 0<i<j<m+1, is called a D(n)-m-tuple. In this paper, we show that there are infinitely many essentially different D(n)-quintuples with square elements. We obtained this result by constructing genus one curves on a certain double cover of A^2 branched along four curves.

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arXiv2019

Rational Diophantine sextuples containing two regular quadruples and one regular quintuple

A set of $m$ distinct nonzero rationals $\{a_1,a_2,\ldots,a_m\}$ 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. It is proved recently that there are infinitely many rational Diophantine sextuples. In this paper, we construct infinite families of rational Diophantine sextuples with special structure, namely the sextuples containing quadruples and quintuples of certain type.

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arXiv2019

There are infinitely many rational Diophantine sextuples with square denominators

A rational Diophantine m-tuple is a set of m nonzero rationals such that the product of any two of them increased by 1 is a perfect square. The first rational Diophantine quadruple was found by Diophantus, while Euler proved that there are infinitely many rational Diophantine quintuples. In 1999, Gibbs found the first example of a rational Diophantine sextuple, and in 2016 Dujella, Kazalicki, Mikić and Szikszai proved that there are infinitely many of them. In this paper, we prove that there exist infinitely many rational Diophantine sextuples such that the denominators of all the elements in the sextuples are perfect squares.

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