Search ResultsShowing 1-20 of 61
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arXiv:2505.04149 (Published 2025-05-07)
Torsion of Rational Elliptic Curves over the $\mathbb{Z}_p$-Extensions of Quadratic Fields
Comments: 12 pagesCategories: math.NTLet $E$ be an elliptic curve defined over $\mathbb{Q}$. For a quadratic number field $K$ and an odd prime number $p$, let $L$ be a $\mathbb{Z}_p$-extension of $K$. We prove that $E(L)_{\text{tors}}=E(K)_{\text{tors}}$ when $p>5$. It enables us to classify the groups that can be realized as the torsion subgroup $E(L)_{\text{tors}}$, by using the classification of torsion subgroups over the quadratic fields.
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arXiv:2505.09637 (Published 2025-05-05)
Explicit quadratic large sieve inequality
In this article, we obtain an explicit version of Heath-Brown's large sieve inequality for quadratic characters and discuss its applications to $L$-functions and quadratic fields.
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arXiv:2501.17602 (Published 2025-01-29)
Local-global principle for isogenies of elliptic curves over quadratic fields
Comments: 9 pagesCategories: math.NTIn this paper, we prove that the local-global principle of $11$-isogenies for elliptic curves over quadratic fields does not fail. This gives a positive answer to a conjecture by Banwait and Cremona. The proof is based on the determination of the set of quadratic points on the modular curve $X_{D_{10}}(11)$.
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arXiv:2412.20080 (Published 2024-12-28)
Parameterized families of quadratic fields with $n$-rank at least 2
Comments: 7 pages. Published in `Class Groups of Number Fields and Related Topics', Springer Proceedings in Mathematics & Statistics, vol 470, Springer, SingaporeJournal: Class Groups of Number Fields and Related Topics, Springer Proceedings in Mathematics & Statistics, vol 470, 2024, Springer, SingaporeCategories: math.NTTags: journal articleWe construct parameterized families of imaginary (resp. real) quadratic fields whose class groups have $n$-rank at least $2$.
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arXiv:2410.18389 (Published 2024-10-24)
Heavenly elliptic curves over quadratic fields
Comments: 30 pages, 1 figure, 2 tablesCategories: math.NTMotivated by a long-standing question of Ihara, we investigate heavenly abelian varieties -- abelian varieties defined over a number field $K$ that exhibit constrained $\ell$-adic Galois representations at some rational prime $\ell$. We demonstrate a finiteness result for heavenly elliptic curves where $\ell$ is fixed and the field of definition varies. Introducing the notion of a balanced heavenly abelian variety, characterized by the structure of its $\ell$-torsion as a group scheme, we show that all heavenly abelian varieties are balanced for sufficiently large $\ell$, with this result holding uniformly once the degree of the number field and the dimension of the abelian variety are fixed. We study the Frobenius traces on balanced heavenly elliptic curves, showing that they satisfy certain congruences modulo $\ell$ akin to those of elliptic curves with complex multiplication. Conjecturally, we propose that balanced elliptic curves over quadratic number fields must possess complex multiplication. Finally, we produce an explicit list of elliptic curves with irrational $j$-invariants, which contains all heavenly elliptic curves with complex multiplication defined over quadratic fields, supported by computational evidence that every curve on the list is heavenly.
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arXiv:2409.07941 (Published 2024-09-12)
No proper generalized quadratic forms are universal over quadratic fields
Ondřej Chwiedziuk et al.Categories: math.NTWe consider generalized quadratic forms over real quadratic number fields and prove, under a natural positive-definiteness condition, that a generalized quadratic form can only be universal if it contains a quadratic subform that is universal. We also construct an example illustrating that the positive-definiteness condition is necessary.
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arXiv:2404.06858 (Published 2024-04-10)
Number Theory in OSCAR
Comments: Submitted as chapter for the upcoming book on the computer algebra system OSCARCategories: math.NTKeywords: computational algebraic number theory, galois module structure, brief introduction, quadratic fields, cohen-lenstra heuristicTags: book chapterWe give a brief introduction to computational algebraic number theory in OSCAR. Our main focus is on number fields, rings of integers and their invariants. After recalling some classical results and their constructive counterparts, we showcase the functionality in two examples related to the investigation of the Cohen-Lenstra heuristic for quadratic fields and the Galois module structure of rings of integers.
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arXiv:2403.13359 (Published 2024-03-20)
$6$-torsion and integral points on quartic surfaces
Categories: math.NTWe prove matching upper and lower bounds for the average of the 6-torsion of class groups of quadratic fields. Furthermore, we count the number of integer solutions on an affine quartic surface.
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arXiv:2401.08828 (Published 2024-01-16)
2-Selmer Groups over Multiquadratic Extensions
Comments: 31 pages, comments welcome!Categories: math.NTLet K be a multiquadratic number field. We investigate the average dimension of 2-Selmer groups over K for the family of all elliptic curves over the rational numbers (ordered by height). We give upper and lower bounds for this average. In the special case of quadratic fields, these bounds are arbitrarily close for a positive proportion of K. Our bounds are achieved by studying the genus theory invariant for 2-Selmer groups over such fields, whose average we similarly bound and, in many cases, determine. We make use of a variant of the Ekedahl sieve for local sums, which we present in appropriate generality for further applications.
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arXiv:2305.19969 (Published 2023-05-31)
On the $p$-isogenies of elliptic curves with multiplicative reduction over quadratic fields
Comments: 13 pages, comments are welcome!Categories: math.NTLet $q > 5$ be a prime and $K$ a quadratic number field. In this article we extend a previous result of Najman and the author and prove that if $E/K$ is an elliptic curve with potentially multiplicative reduction at all primes $\mathfrak q \mid q$, then $E$ does not have prime isogenies of degree greater than $71$ and different from $q$. As an application to our main result, we present a variant of the asymptotic version of Fermat's Last Theorem over quadratic imaginary fields of class number one.
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arXiv:2212.14139 (Published 2022-12-29)
The matrix equation $aX^m+bY^n=cI$ over $M_2(\mathbb{Z})$
Categories: math.NTLet $\mathbb{N}$ be the set of all positive integers and let $a,\, b,\, c$ be nonzero integers such that $\gcd\left(a,\, b,\, c\right)=1$. In this paper, we prove the following three results: (1) the solvability of the matrix equation $aX^m+bY^n=cI,\,X,\,Y\in M_2(\mathbb{Z}),\, m,\, n\in\mathbb{N}$ can be reduced to the solvability of the corresponding Diophantine equation if $XY\neq YX$ and the solvability of the equation $ax^m+by^n=c,\, m,\, n\in\mathbb{N}$ in quadratic fields if $XY=YX$; (2) we determine all non-commutative solutions of the matrix equation $X^n+Y^n=c^nI,\,X,\,Y\in M_2(\mathbb{Z}),\,n\in\mathbb{N},\,n\geq3$, and the solvability of this matrix equation can be reduced to the solvability of the equation $x^n+y^n=c^n,\, n\in\mathbb{N},\,n\geq3$ in quadratic fields if $XY=YX$; (3) we determine all solutions of the matrix equation $aX^2+bY^2=cI,\,X,\,Y\in M_2(\mathbb{Z})$.
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arXiv:2212.11724 (Published 2022-12-22)
ECm And The Elliott-Halberstam Conjecture For Quadratic Fields
The complexity of the elliptic curve method of factorization (ECM) is proven under the celebrated conjecture of existence of smooth numbers in short intervals. In this work we tackle a different version of ECM which is actually much more studied and implemented, especially because it allows us to use ECM-friendly curves. In the case of curves with complex multiplication (CM) we replace the heuristics by rigorous results conditional to the Elliott-Halberstam (EH) conjecture. The proven results mirror recent theorems concerning the number of primes p such thar p -- 1 is smooth. To each CM elliptic curve we associate a value which measures how ECM-friendly it is. In the general case we explore consequences of a statement which translated EH in the case of elliptic curves.
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arXiv:2207.14053 (Published 2022-07-28)
Distribution of primes represented by polynomials and Multiple Dedekind zeta functions
Comments: 18 pagesCategories: math.NTn this paper, we state several conjectures regarding distribution of primes and of pairs of primes represented by irreducible homogeneous polynomial in two variables $f(a,b)$. We formulate conjectures with respect to the slope $t=b/a$ for any irreducible polynomial $f$. Here, we formulate a conjecture for all irreducible polynomials. We also consider conjectures for distribution of pairs of primes. It show unexpected relation to multiple Dedekind zeta function - at $s=2$ for one prime and at $(s_1,s_2)=(2,2)$ for pairs of primes. We tested the conjecture for pairs of primes for several quadratic fields. The conjecture for pairs of primes and multiple Dedekind zeta function over the Gaussian integers provide error less than a tenth of a percent. We also tested conjectures that compare sets of primes in a pair of different quadratic fields. Numerically, such quotients can be expressed in terms of regulators and class numbers. Some of the data, together with the code, is available on GitHub, (see \cite{Zouberou}).
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arXiv:2206.13931 (Published 2022-06-28)
Unlimited lists of fundamental units of quadratic fields -- Applications
Comments: 26 pages. Elementary subject about real quadratic fields with new algorithms using given PARI programsCategories: math.NTWe use the polynomials $m_s(t) = t^2 - 4 s$, $s \in \{-1, 1\}$, in an elementary process giving arbitrary large lists of {\it fundamental units} of quadratic fields of discriminants listed in ascending order. More precisely, let $\mathbf{B} \gg 0$; then as $t$ grows from $1$ to $\mathbf{B}$, for each {\it first occurrence} of a square-free integer $M \geq 2$, in the factorization $m_s(t) =: M r^2$, the unit $\frac{1}{2} \big(t + r \sqrt{M}\big)$ is the fundamental unit of norm $s$ of $\mathbb{Q}(\sqrt M)$, even if $r >1$ (Theorem 4.1). Using $m_{s\nu}(t) = t^2 - 4 s \nu$, $\nu \geq 2$, the algorithm gives arbitrary large lists of {\it fundamental solutions} to $u^2 - M v^2= 4s\nu$ (Theorem 4.11). We deduce, for $p>2$ prime, arbitrary large lists of {\it non $p$-rational} quadratic fields (Theorems 6.3, 6.4, 6.5) and of degree $p-1$ imaginary fields with non-trivial $p$-class group (Theorems 7.1,7.2). PARI programs are given to be copied and pasted.
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arXiv:2203.10672 (Published 2022-03-20)
Isogenies over quadratic fields of elliptic curves with rational $j$-invariant
We determine the possible degrees of cyclic isogenies defined over quadratic fields for non-CM elliptic curves with rational $j$-invariant.
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arXiv:2201.04291 (Published 2022-01-12)
Congruences for odd class numbers of quadratic fields with odd discriminant
Comments: 17 pagesCategories: math.NTFor any distinct two primes $p_1\equiv p_2\equiv 3$ $(\text{mod }4)$, let $h(-p_1)$, $h(-p_2)$ and $h(p_1p_2)$ be the class numbers of the quadratic fields $\mathbb{Q}(\sqrt{-p_1})$, $\mathbb{Q}(\sqrt{-p_2})$ and $\mathbb{Q}(\sqrt{p_1p_2})$, respectively. Let $\omega_{p_1p_2}:=\frac{1+\sqrt{p_1p_2}}{2}$ and let $\Psi(\omega_{p_1p_2})$ be the Hirzebruch sum of $\omega_{p_1p_2}$. We show that $h(-p_1)h(-p_2)\equiv h(p_1p_2)\Psi(\omega_{p_1p_2})/n$ $(\text{mod }8)$, where $n=6$ (respectively, $n=2$) if $\min\{p_1,p_2\}>3$ (respectively, otherwise). We also consider the real quadratic order with conductor $2$ in $\mathbb{Q}(\sqrt{p_1p_2})$.
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arXiv:2103.14861 (Published 2021-03-27)
Continued Fractions, Quadratic Fields, and Factoring: Some Computational Aspects
Comments: 8 pages, to appear on Collectio Cyphrarum. arXiv admin note: text overlap with arXiv:1905.10704Categories: math.NTLegendre discovered that the continued fraction expansion of $\sqrt N$ having odd period leads directly to an explicit representation of $N$ as the sum of two squares. In this vein, it was recently observed that the continued fraction expansion of $\sqrt N$ having even period directly produces a factor of composite $N$. It is proved here that these apparently fortuitous occurrences allow us to propose and apply a variation of Shanks' infrastructural method which significantly reduces the asymptotic computational burden with respect to currently used factoring techniques.
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arXiv:2011.14528 (Published 2020-11-30)
Powers of Gauss sums in quadratic fields
Comments: 15 pages; 2 tablesIn the past two decades, many researchers have studied {\it index $2$} Gauss sums, where the group generated by the characteristic $p$ of the underling finite field is of index $2$ in the unit group of ${\mathbb Z}/m{\mathbb Z}$ for the order $m$ of the associated multiplicative character of the filed. A complete solution to the problem of evaluating index $2$ Gauss sums was given by Yang and Xia~(2010). In particular, it is known that some nonzero integal powers of the Gauss sums in this case are in quadratic fields over the field of rational numbers. On the other hand, McEliece (1974), Evans (1981) and Aoki (2004, 2012) studied {\it pure} Gauss sums, some nonzero integral powers of which are in the field of rational numbers. In this paper, we study Gauss sums, some integral powers of which are in quadratic fields over the field of rational numbers. This class of Gauss sums is a generalization of index $2$ Gauss sums and an extension of pure Gauss sums to quadratic fields.
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arXiv:2007.13147 (Published 2020-07-26)
Pair arithmetical equivalence for quadratic fields
Comments: 34 pagesGiven two distinct number fields $K$ and $M$, and finite order Hecke characters $\chi$ of $K$ and $\eta$ of $M$ respectively, we say that the pairs $(\chi, K)$ and $(\eta, M)$ are arithmetically equivalent if the associated L-functions coincide: $$L(s, \chi, K) = L(s, \eta, M) .$$ When the characters are trivial, this reduces to the question of fields with the same Dedekind zeta function, investigated by Gassman in 1926, who found such fields of degree 180, and by Perlis (1977) and others, who showed that there are no nonisomorphic fields of degree less than $7$. We construct infinitely many such pairs where the fields are quadratic. This gives dihedral automorphic forms induced from characters of different quadratic fields. We also give a classification of such characters of order 2 for the quadratic fields of our examples, all with odd class number.
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arXiv:2006.07987 (Published 2020-06-14)
High $\ell$-torsion rank for class groups over function fields
Comments: 5 pages, comments welcome!We prove that in the function field setting, $\ell$-torsion in the class groups of quadratic fields can be arbitrarily large. In fact, we explicitly produce a family whose $\ell$-rank growth matches the growth in the setting of genus theory, which might be best possible. We do this by specifically focusing on the Artin-Schreir curves $y^2=x^q-x$.