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Filip Najman

Publications and source records attributed to Filip Najman.

At least 37 records · Page 2Linked to original sources
arXiv2022

Hyperelliptic modular curves $X_0(n)$ and isogenies of elliptic curves over quadratic fields

Let $n$ be an integer such that the modular curve $X_0(n)$ is hyperelliptic of genus $\ge2$ and such that the Jacobian of $X_0(n)$ has rank $0$ over $\mathbb Q$. We determine all points of $X_0(n)$ defined over quadratic fields, and we give a moduli interpretation of these points. As a consequence, we show that up to $\overline{\mathbb Q}$-isomorphism, all but finitely many elliptic curves with $n$-isogenies over quadratic fields are in fact $\mathbb Q$-curves, and we list all exceptions. We also show that, again with finitely many exceptions up to $\overline{\mathbb Q}$-isomorphism, every $\mathbb Q$-curve $E$ over a quadratic field $K$ admitting an $n$-isogeny is $d$-isogenous, for some $d\mid n$, to the twist of its Galois conjugate by some quadratic extension $L$ of $K$; we determine $d$ and $L$ explicitly.

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arXiv2022

Splitting of primes in number fields generated by points on some modular curves

We study the splitting of primes in number fields generated by points on modular curves. Momose was the first to notice that quadratic points on $X_1(n)$ generate quadratic fields over which certain primes split in a particular way and his results were later expanded upon by Krumm. We prove results about the splitting behaviour of primes in quadratic fields generated by points on the modular curves $X_0(n)$ which are hyperelliptic (except for $n=37$) and in cubic fields generated by points on $X_1(2, 14)$.

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arXiv2021

$\mathbb Q$-curves over odd degree number fields

By reformulating and extending results of Elkies, we prove some results on $\mathbb Q$-curves over number fields of odd degree. We show that, over such fields, the only prime isogeny degrees~$\ell$ which an elliptic curve without CM may have are those degrees which are already possible over~$\mathbb Q$ itself (in particular, $\ell\le37$), and we show the existence of a bound on the degrees of cyclic isogenies between $\mathbb Q$-curves depending only on the degree of the field. We also prove that the only possible torsion groups of $\mathbb Q$-curves over number fields of degree not divisible by a prime $\ell\leq 7$ are the $15$ groups that appear as torsion groups of elliptic curves over $\mathbb Q$. Complementing these theoretical results we give an algorithm for establishing whether any given elliptic curve $E$ is a $\mathbb Q$-curve, which involves working only over $\mathbb Q(j(E))$.

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arXiv2021

Sporadic points of odd degree on $X_1(N)$ coming from $\mathbb{Q}$-curves

We say a closed point $x$ on a curve $C$ is sporadic if there are only finitely many points on $C$ of degree at most deg$(x)$. In the case where $C$ is the modular curve $X_1(N)$, most known examples of sporadic points come from elliptic curves with complex multiplication (CM). We seek to understand all sporadic points on $X_1(N)$ corresponding to $\mathbb{Q}$-curves, which are elliptic curves isogenous to their Galois conjugates. This class contains not only all CM elliptic curves, but also any elliptic curve $\overline{\mathbb{Q}}$-isogenous to one with a rational $j$-invariant, among others. In this paper, we show that all non-CM $\mathbb{Q}$-curves giving rise to a sporadic point of odd degree lie in the $\overline{\mathbb{Q}}$-isogeny class of the elliptic curve with $j$-invariant $-140625/8$. In addition, we show that a stronger version of this finiteness result would imply Serre's Uniformity Conjecture.

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arXiv2021

Irreducibility of mod p Galois representations of elliptic curves with multiplicative reduction over number fields

In this note we prove that for every integer $d \geq 1$, there exists an explicit constant $B_d$ such that the following holds. Let $K$ be a number field of degree $d$, let $q > \max\{d-1,5\}$ be any rational prime that is totally inert in $K$ and $E$ any elliptic curve defined over $K$ such that $E$ has potentially multiplicative reduction at the prime $\mathfrak q$ above $q$. Then for every rational prime $p> B_d$, $E$ has an irreducible mod $p$ Galois representation. This result has Diophantine applications within the "modular method". We present one such application in the form of an Asymptotic version of Fermat's Last Theorem that has not been covered in the existing literature.

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arXiv2020

Fields of definition of elliptic curves with prescribed torsion

We prove that all elliptic curves over quadratic fields with a subgroup isomorphic to $C_{16}$, as well as all elliptic curves over cubic fields with a subgroup isomorphic to $C_2\times C_{14}$, are base changes of elliptic curves defined over $\Q$. We obtain these results by studying geometric properties of modular curves and maps between modular curves, and then obtaining a modular description of these curves and maps.

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arXiv2020

An algorithm for determining torsion growth of elliptic curves

We present a fast algorithm that takes as input an elliptic curve defined over $\mathbb Q$ and an integer $d$ and returns all the number fields $K$ of degree $d'$ dividing $d$ such that $E(K)_{tors}$ contains $E(F)_{tors}$ as a proper subgroup, for all $F \varsubsetneq K$. We ran this algorithm on all elliptic curves of conductor less than 400.000 (a total of 2.483.649 curves) and all $d \leq 23$ and collected various interesting data. In particular, we find a degree 6 sporadic point on $X_1(4,12)$, which is so far the lowest known degree a sporadic point on $X_1(m,n)$, for $m\geq 2$.

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arXiv2019

Growth of torsion groups of elliptic curves upon base change

We study how the torsion of elliptic curves over number fields grows upon base change, and in particular prove various necessary conditions for torsion growth. For a number field $F$, we show that for a large set of number fields $L$, whose Galois group of their normal closure over $F$ has certain properties, it will hold that $E(L)_{tors}=E(F)_{tors}$ for all elliptic curves $E$ defined over $F$. Our methods turn out to be particularly useful in studying the possible torsion groups $E(K)_{tors}$, where $K$ is a number field and $E$ is a base change of an elliptic curve defined over $\mathbb Q$. Suppose that $E$ is a base change of an elliptic curve over $\mathbb Q$ for the remainder of the abstract. We prove that $E(K)_{tors}=E(\mathbb Q)_{tors}$ for all elliptic curves $E$ defined over $\mathbb Q$ and all number fields $K$ of degree $d$, where $d$ is not divisible by a prime $\leq 7$. Using this fact, we determine all the possible torsion groups $E(K)_{tors}$ over number fields $K$ of prime degree $p\geq 7$. We determine all the possible degrees of $[\mathbb Q(P):\mathbb Q]$, where $P$ is a point of prime order $p$ for all $p$ such that $p\not\equiv 8 \pmod 9$ or $\left( \frac{-D}{p}\right)=1$ for any $D\in \{1,2,7,11,19,43,67,163\}$; this is true for a set of density $\frac{1535}{1536}$ of all primes and in particular for all $p<3167$. Using this result, we determine all the possible prime orders of a point $P\in E(K)_{tors}$, where $[K:\mathbb Q]=d$, for all $d\leq 3342296$. Finally, we determine all the possible groups $E(K)_{tors}$, where $K$ is a quartic number field and $E$ is an elliptic curve defined over $\mathbb Q$ and show that no quartic sporadic point on a modular curves $X_1(m,n)$ comes from an elliptic curve defined over $\mathbb Q$.

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arXiv2019

Elliptic Curves over Totally Real Cubic Fields are Modular

We prove that all elliptic curves defined over totally real cubic fields are modular. This builds on previous work of Freitas, Le Hung and Siksek, who proved modularity of elliptic curves over real quadratic fields, as well as recent breakthroughs due to Thorne and to Kalyanswamy.

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arXiv2018

Torsion subgroups of rational elliptic curves over the compositum of all cubic fields

Let $E/\mathbb{Q}$ be an elliptic curve and let $\mathbb{Q}(3^\infty)$ be the compositum of all cubic extensions of $\mathbb{Q}$. In this article we show that the torsion subgroup of $E(\mathbb{Q}(3^\infty))$ is finite and determine 20 possibilities for its structure, along with a complete description of the $\overline{\mathbb{Q}}$-isomorphism classes of elliptic curves that fall into each case. We provide rational parameterizations for each of the 16 torsion structures that occur for infinitely many $\overline{\mathbb{Q}}$-isomorphism classes of elliptic curves, and a complete list of $j$-invariants for each of the 4 that do not.

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arXiv2016

Tamagawa Numbers of elliptic curves with $C_{13}$ torsion over quadratic fields

Let $E$ be an elliptic curve over a number field $K$, $c_v$ the Tamagawa number of $E$ at $v$, and let $c_E=\prod_{v}c_v$. Lorenzini proved that $v_{13}(c_E)$ is postive for all elliptic curves over quadratic fields with a point of order $13$. Krumm conjectured, based on extensive computation, that the $13$-adic valuation of $c_E$ is even for all such elliptic curves. In this note we prove this conjecture and furhtermore prove that there is an unique such curve satisfying $v_{13}(c_E)=2$.

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arXiv2015

Isogenies of non-CM elliptic curves with rational $j$-invariants over number fields

We unconditionally determine $I_\Q(d)$, the set of possible prime degrees of cyclic $K$-isogneies of elliptic curves with $\Q$-rational $j$-invariants and without complex multiplication over number fields $K$ of degree $\leq d$, for $d\leq 7$, and give an upper bound for $I_\Q(d)$ for $d>7$. Assuming Serre's uniformity conjecture, we determine $I_\Q(d)$ exactly for all positive integers $d$.

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arXiv2015

A criterion to rule out torsion groups for elliptic curves over number fields

We present a criterion for proving that certain groups of the form $\mathbb Z/m\mathbb Z\oplus\mathbb Z/n\mathbb Z$ do not occur as the torsion subgroup of any elliptic curve over suitable (families of) number fields. We apply this criterion to eliminate certain groups as torsion groups of elliptic curves over cubic and quartic fields. We also use this criterion to give the list of all torsion groups of elliptic curves occurring over a specific cubic field and over a specific quartic field.

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arXiv2015

Torsion of rational elliptic curves over cubic fields

Let E be an elliptic curve defined over Q. We study the relationship between the torsion subgroup E(Q)_tors and the torsion subgroup E(K)_tors, where K is a cubic number field. In particular, We study the number of cubic number fields K such that E(Q)_tors\neq E(K)_tors.

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arXiv2014

The number of twists with large torsion of an elliptic curve

For an elliptic curve $E/\Q$, we determine the maximum number of twists $E^d/\Q$ it can have such that $E^d(\Q)_{tors}\supsetneq E(\Q)[2]$. We use these results to determine the number of distinct quadratic fields $K$ such that $E(K)_{tors}\supsetneq E(\Q)_{tors}$. The answer depends on $E(\Q)_{tors}$ and we give the best possible bound for all the possible cases.

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