@article{12312, abstract = {Let $\ell$ be a prime number. We classify the subgroups $G$ of $\operatorname{Sp}_4(\mathbb{F}_\ell)$ and $\operatorname{GSp}_4(\mathbb{F}_\ell)$ that act irreducibly on $\mathbb{F}_\ell^4$, but such that every element of $G$ fixes an $\mathbb{F}_\ell$-vector subspace of dimension 1. We use this classification to prove that the local-global principle for isogenies of degree $\ell$ between abelian surfaces over number fields holds in many cases -- in particular, whenever the abelian surface has non-trivial endomorphisms and $\ell$ is large enough with respect to the field of definition. Finally, we prove that there exist arbitrarily large primes $\ell$ for which some abelian surface $A/\mathbb{Q}$ fails the local-global principle for isogenies of degree $\ell$.}, author = {Lombardo, Davide and Verzobio, Matteo}, issn = {1420-9020}, journal = {Selecta Mathematica}, number = {2}, publisher = {Springer Nature}, title = {{On the local-global principle for isogenies of abelian surfaces}}, doi = {10.1007/s00029-023-00908-0}, volume = {30}, year = {2024}, } @article{12313, abstract = {Let P be a nontorsion point on an elliptic curve defined over a number field K and consider the sequence {Bn}n∈N of the denominators of x(nP). We prove that every term of the sequence of the Bn has a primitive divisor for n greater than an effectively computable constant that we will explicitly compute. This constant will depend only on the model defining the curve.}, author = {Verzobio, Matteo}, issn = {0030-8730}, journal = {Pacific Journal of Mathematics}, number = {2}, pages = {331--351}, publisher = {Mathematical Sciences Publishers}, title = {{Some effectivity results for primitive divisors of elliptic divisibility sequences}}, doi = {10.2140/pjm.2023.325.331}, volume = {325}, year = {2023}, } @unpublished{12311, abstract = {In this note we prove a formula for the cancellation exponent $k_{v,n}$ between division polynomials $\psi_n$ and $\phi_n$ associated with a sequence $\{nP\}_{n\in\mathbb{N}}$ of points on an elliptic curve $E$ defined over a discrete valuation field $K$. The formula is identical with the result of Yabuta-Voutier for the case of finite extension of $\mathbb{Q}_{p}$ and generalizes to the case of non-standard Kodaira types for non-perfect residue fields.}, author = {Naskręcki, Bartosz and Verzobio, Matteo}, booktitle = {arXiv}, title = {{Common valuations of division polynomials}}, doi = {10.48550/arXiv.2203.02015}, year = {2022}, } @article{12308, abstract = {Let P and Q be two points on an elliptic curve defined over a number field K. For α∈End(E), define Bα to be the OK-integral ideal generated by the denominator of x(α(P)+Q). Let O be a subring of End(E), that is a Dedekind domain. We will study the sequence {Bα}α∈O. We will show that, for all but finitely many α∈O, the ideal Bα has a primitive divisor when P is a non-torsion point and there exist two endomorphisms g≠0 and f so that f(P)=g(Q). This is a generalization of previous results on elliptic divisibility sequences.}, author = {Verzobio, Matteo}, issn = {2522-0160}, journal = {Research in Number Theory}, keywords = {Algebra and Number Theory}, number = {2}, publisher = {Springer Nature}, title = {{Primitive divisors of sequences associated to elliptic curves with complex multiplication}}, doi = {10.1007/s40993-021-00267-9}, volume = {7}, year = {2021}, } @article{12309, abstract = {Take a rational elliptic curve defined by the equation y2=x3+ax in minimal form and consider the sequence Bn of the denominators of the abscissas of the iterate of a non-torsion point. We show that B5m has a primitive divisor for every m. Then, we show how to generalize this method to the terms of the form Bmp with p a prime congruent to 1 modulo 4.}, author = {Verzobio, Matteo}, issn = {0065-1036}, journal = {Acta Arithmetica}, keywords = {Algebra and Number Theory}, number = {2}, pages = {129--168}, publisher = {Institute of Mathematics, Polish Academy of Sciences}, title = {{Primitive divisors of elliptic divisibility sequences for elliptic curves with j=1728}}, doi = {10.4064/aa191016-30-7}, volume = {198}, year = {2021}, } @unpublished{12314, abstract = {In literature, there are two different definitions of elliptic divisibility sequences. The first one says that a sequence of integers $\{h_n\}_{n\geq 0}$ is an elliptic divisibility sequence if it verifies the recurrence relation $h_{m+n}h_{m-n}h_{r}^2=h_{m+r}h_{m-r}h_{n}^2-h_{n+r}h_{n-r}h_{m}^2$ for every natural number $m\geq n\geq r$. The second definition says that a sequence of integers $\{\beta_n\}_{n\geq 0}$ is an elliptic divisibility sequence if it is the sequence of the square roots (chosen with an appropriate sign) of the denominators of the abscissas of the iterates of a point on a rational elliptic curve. It is well-known that the two sequences are not equivalent. Hence, given a sequence of the denominators $\{\beta_n\}_{n\geq 0}$, in general does not hold $\beta_{m+n}\beta_{m-n}\beta_{r}^2=\beta_{m+r}\beta_{m-r}\beta_{n}^2-\beta_{n+r}\beta_{n-r}\beta_{m}^2$ for $m\geq n\geq r$. We will prove that the recurrence relation above holds for $\{\beta_n\}_{n\geq 0}$ under some conditions on the indexes $m$, $n$, and $r$.}, author = {Verzobio, Matteo}, booktitle = {arXiv}, title = {{A recurrence relation for elliptic divisibility sequences}}, doi = {10.48550/arXiv.2102.07573}, year = {2021}, } @article{12310, abstract = {Let be a sequence of points on an elliptic curve defined over a number field K. In this paper, we study the denominators of the x-coordinates of this sequence. We prove that, if Q is a torsion point of prime order, then for n large enough there always exists a primitive divisor. Later on, we show the link between the study of the primitive divisors and a Lang-Trotter conjecture. Indeed, given two points P and Q on the elliptic curve, we prove a lower bound for the number of primes p such that P is in the orbit of Q modulo p.}, author = {Verzobio, Matteo}, issn = {0022-314X}, journal = {Journal of Number Theory}, keywords = {Algebra and Number Theory}, number = {4}, pages = {378--390}, publisher = {Elsevier}, title = {{Primitive divisors of sequences associated to elliptic curves}}, doi = {10.1016/j.jnt.2019.09.003}, volume = {209}, year = {2020}, }