arXiv subjects

Matija Kazalicki

Publications and source records attributed to Matija Kazalicki.

27 records · Page 2Linked to original sources

arXiv2019

Congruences for sporadic sequences and modular forms for non-congruence subgroups

In the course of the proof of the irrationality of zeta(2) R. Apery introduced numbers b_n = \sum_{k=0}^n {n \choose k}^2{n+k \choose k}. Stienstra and Beukers showed that for the prime p > 3 Apery numbers satisfy congruence b((p-1)/2) = 4a^2-2p mod p, if p = a^2+b^2 (where a is odd). Later, Zagier found some generalizations of Apery numbers, so called sporadic sequences, and recently Osburn and Straub proved similar congruences for all but one of the six Zagier's sporadic sequences (three cases were already known to be true) and conjectured the congruence for the sixth sequence. In this paper we prove that remaining congruence by studying Atkin and Swinnerton-Dyer congruences between Fourier coefficients of certain cusp form for non-congurence subgroup.

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arXiv2018

Diophantine m-tuples in finite fields and modular forms

For a prime p, a Diophantine m-tuple in $\mathbb{F}_p$ is a set of m nonzero elements of $\mathbb{F}_p$ with the property that the product of any two of its distinct elements is one less than a square. In this paper, we present formulas for the number $N^{(m)}(p)$ of Diophantine m-tuples in $\mathbb{F}_p$ for m=2,3 and 4. Fourier coefficients of certain modular forms appear in the formula for the number of Diophantine quadruples. We prove that asymptotically $N^{(m)}(p)=\frac{1}{2^{m \choose 2 }}\frac{p^m}{m!} + o(p^m)$, and also show that if $p>2^{2m-2}m^2$, then there is at least one Diophantine m-tuple in $\mathbb{F}_p$.

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arXiv2016

Supersingular zeros of divisor polynomials of elliptic curves of prime conductor

For a prime number $p$ we study the zeros modulo $p$ of divisor polynomials of rational elliptic curves $E$ of conductor $p$. Ono made the observation that these zeros of are often $j$-invariants of supersingular elliptic curves over $\overline{\mathbb{F}_p}$. We show that these supersingular zeros are in bijection with zeros modulo $p$ of an associated quaternionic modular form $v_E$. This allows us to prove that if the root number of $E$ is $-1$ then all supersingular $j$-invariants of elliptic curves defined over $\mathbb{F}_{p}$ are zeros of the corresponding divisor polynomial. If the root number is $1$ we study the discrepancy between rank $0$ and higher rank elliptic curves, as in the latter case the amount of supersingular zeros in $\mathbb{F}_p$ seems to be larger. In order to partially explain this phenomenon, we conjecture that when $E$ has positive rank the values of the coefficients of $v_E$ corresponding to supersingular elliptic curves defined over $\mathbb{F}_p$ are even. We prove this conjecture in the case when the discriminant of $E$ is positive, and obtain several other results that are of independent interest.

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arXiv2016

On a special case of Watkins' conjecture

Watkins' conjecture asserts that for a rational elliptic curve $E$ the degree of the modular parametrization is divisible by $2^r$, where $r$ is the rank of $E$. In this paper we prove that if the modular degree is odd then $E$ has rank $0$. Moreover, we prove that the conjecture holds for all rank two rational elliptic curves of prime conductor and positive discriminant.

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arXiv2016

More on Diophantine sextuples

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 Dujella, Kazalicki, Mikic and Szikszai recently proved that there exist infinitely many rational Diophantine sextuples. In this paper, generalizing the work of Piezas, we describe a method for generating new parametric formulas for rational Diophantine sextuples.

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arXiv2015

There are infinitely many rational Diophantine sextuples

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. In this paper, we prove that there exist infinitely many rational Diophantine sextuples.

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arXiv2013

Modular forms, de Rham cohomology and congruences

In this paper we show that Atkin and Swinnerton-Dyer type of congruences hold for weakly modular forms (modular forms that are permitted to have poles at cusps). Unlike the case of original congruences for cusp forms, these congruences are nontrivial even for congruence subgroups. On the way we provide an explicit interpretation of the de Rham cohomology groups associated to modular forms in terms of "differentials of the second kind". As an example, we consider the space of cusp forms of weight 3 on a certain genus zero quotient of Fermat curve X^N+Y^N=Z^N. We show that the Galois representation associated to this space is Grossencharacter of a cyclotomic field $\Q(ζ_N)$. Moreover, for N=5 the space does not admit a "$p$-adic Hecke eigenbasis" for (non-ordinary) primes $p\equiv 2,3 \pmod{5}$, which provides a counterexample for original Atkin and Swinnerton-Dyer speculaction (see [2], [7], [8]).

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arXiv2013

Modular forms, hypergeometric functions and congruences

Using the theory of Stienstra and Beukers, we prove various elementary congruences for the numbers \sum \binom{2i_1}{i_1}^2\binom{2i_2}{i_2}^2...\binom{2i_k}{i_k}^2, where k,n \in N, and the summation is over the integers i_1, i_2, ...i_k >= 0 such that i_1+i_2+...+i_k=n. To obtain that, we study the arithmetic properties of Fourier coefficients of certain (weakly holomorphic) modular forms.

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arXiv2012

Modular parametrizations of certain elliptic curves

Kaneko and Sakai recently observed that certain elliptic curves whose associated newforms (by the modularity theorem) are given by the eta-quotients can be characterized by a particular differential equation involving modular forms and Ramanujan-Serre differential operator. In this paper, we study certain properties of modular parametrizations associated to the elliptic curves over Q, and as a consequence we generalize and explain some of their findings.

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