arXiv subjects
Publications and source records attributed to Ariel Pacetti.
Hypergeometric motives are family of motives associated to hypergeometric local systems. Their special features, in particular their rigidity, makes them more tractable than general motives. In the present article we prove most of the properties that they are expected to satisfy in the rank $2$ case.
In this article we study solutions to the generalized Fermat equation $x^q+y^p+z^r=0 $ using hypergeometric motives within the framework of the modular method. In doing so, we give an explicit description of the ramification behavior at primes dividing $2qr$ and analyze the contribution of trivial solutions. We identify a general obstruction to the modular method that accounts for its failure in many instances. As an application, assuming a standard large image conjecture, we prove that the previous equation admits no nontrivial primitive solutions $(a,b,c)$ with $3 \nmid c$, when $q=5,$ $r=3$ and $p$ is a prime sufficiently large.
Over the last two hundred years different transformation formulas for Gauss' hypergeometric function ${}_2F_1$ were discovered. The goal of the present article is to study their arithmetic analogue for the underlying hypergeometric motive. As an application, we show how these transformation properties can be used in the study of some Diophantine equations.
In the beautiful article [11] Darmon proposed a program to study integral solutions of the generalized Fermat equation $Ax^p+By^q=Cz^r$. In the aforementioned article, Darmon proved many steps of the program, by exhibiting models of hyperelliptic/superelliptic curves lifting what he called ''Frey representations'', Galois representations over a finite field of characteristic $p$. The goal of the present article is to show how hypergeometric motives are more natural objects to obtain the global representations constructed by Darmon, allowing to prove most steps of his program without the need of algebraic models.
In a remarkable article Ribet showed how to attach rational $2$-dimensional representations to elliptic ${\mathbb Q}$-curves. An abelian variety $A$ is a (weak) $K$-variety if it is isogenous to all of its $\text{Gal}_K$-conjugates. In this article we study the problem of attaching an absolutely irreducible $\ell$-adic representation of $\text{Gal}_K$ to an abelian $K$-variety, which sometimes has smaller dimension than expected. When possible, we also construct a Galois-equivariant pairing, which restricts the image of this representation. As an application of our construction, we prove modularity of abelian surfaces over ${\mathbb Q}$ with potential quaternionic multiplication.
Let $\mathcal{C}$ be a hyperelliptic curve $y^2 = p(x)$ defined over a number field $K$ with $p(x)$ integral of odd degree. The purpose of the present article is to prove lower and upper bounds for the $2$-Selmer group of the Jacobian of $\mathcal{C}$ in terms of the class group of the $K$-algebra $K[x]/(p(x))$. Our main result is a formula relating these two quantities under some mild hypothesis. We provide some examples that prove that our lower and upper bounds are as sharp as possible. As a first application, we study the rank distribution of the $2$-Selmer group in families of quadratic twists. Under some extra hypothesis we prove that among prime quadratic twists, a positive proportion has fixed $2$-Selmer group. As a second application, we study the family of octic twists of the genus $2$ curve $y^2 = x^5 + x$.
We present a method to compute finite index subgroups of $PSL_2(\mathbb{Z})$. Our strategy follows Kulkarni's ideas, the main contribution being a recursive method to compute bivalent trees and their automorphism group. As a concrete application, we compute all subgroups of index up to 20. We then use this database to produce tables with several arithmetical properties.
Let $f$ and $g$ be two different newforms without complex multiplication having the same coefficient field. The main result of the present article proves that a congruence between the Galois representations attached to $f$ and to $g$ for a large prime $p$ implies an isomorphism between the endomorphism algebras of the abelian varieties $A_f$ and $A_g$ attached to $f$ and $g$ by the Eichler-Shimura construction. This implies important relations between their building blocks. A non-trivial application of our result is that for all prime numbers $d$ congruent to $3$ modulo $8$ satisfying that the class number of $\mathbb{Q}(\sqrt{-d})$ is prime to $3$, the equation $x^4+dy^2 =z^p$ has no non-trivial primitive solutions when $p$ is large enough. We prove a similar result for the equation $x^2+dy^6=z^p$.
We work out the exact relationship between algebraic modular forms for a two-by-two general unitary group over a definite quaternion algebra, and those arising from genera of positive-definite quinary lattices, relating stabilisers of local lattices with specific open compact subgroups, paramodular at split places, and with Atkin-Lehner operators. Combining this with the recent work of R\"osner and Weissauer, proving conjectures of Ibukiyama on Jacquet-Langlands type correspondences (mildly generalised here), provides an effective tool for computing Hecke eigenvalues for Siegel modular forms of degree two and paramodular level. It also enables us to prove examples of congruences of Hecke eigenvalues connecting Siegel modular forms of degrees two and one. These include some of a type conjectured by Harder at level one, supported by computations of Fretwell at higher levels, and a subtly different congruence discovered experimentally by Buzzard and Golyshev.
The purpose of the present article is to show how the modular method together with different techniques can be used to prove non-existence of primitive non-trivial solutions of the equation $x^2+dy^6=z^p$ for square-free values $1 \le d \le 20$ following the approach of [PT]. The main innovation is to make use of the symplectic argument over ramified extensions to discard solutions, together with a multi-Frey approach to deduce large image of Galois representations.
In the article [PV] a general procedure to study solutions of the equations $x^4-dy^2=z^p$ was presented for negative values of $d$. The purpose of the resent article is to extend our previous results to positive values of $d$. On doing so, we give a description of the extension $\mathbb{Q}(\sqrt{d},\sqrt{\epsilon})/\mathbb{Q}(\sqrt{d})$ (where $\epsilon$ is a fundamental unit) needed to prove the existence of a Hecke character over $\mathbb{Q}(\sqrt{d})$ with prescribed local conditions. We also extend some "large image" results due to Ellenberg regarding images of Galois representations coming from $\mathbb{Q}$-curves from imaginary to real quadratic fields.
A superelliptic curve over a DVR ${\mathcal O}$ of residual characteristic $p$ is a curve given by an equation $C:y^n=f(x)$. The purpose of the present article is to describe the Galois representation attached to such a curve under the hypothesis that $f(x)$ has all its roots in the fraction field of ${\mathcal O}$ and that $p \nmid n$. Our results are inspired on the algorithm given in [BW17] but our description is given in terms of a cluster picture as defined in [DDMM18].
In this article we study the equations $x^4+dy^2=z^p$ and $x^2+dy^6=z^p$ for positive square-free values of $d$. A Frey curve over $\mathbb{Q}(\sqrt{-d})$ is attached to each primitive solution, which happens to be a $\mathbb{Q}$-curve. Our main result is the construction of a Hecke character $\chi$ satisfying that the Frey elliptic curve representation twisted by $\chi$ extends to $\text{Gal}_\mathbb{Q}$, therefore (by Serre's conjectures) corresponds to a newform in $S_2(n,\varepsilon)$ for explicit values of $n$ and $\varepsilon$. Following some well known results and elimination techniques (together with some improvements) it provides a systematic procedure to study solutions of the above equations and allows us to prove non-existence of non-trivial primitive solutions for large values of $p$ of both equations for new values of $d$.
Let $K$ be a number field and $E/K$ be an elliptic curve with no $2$-torsion points. In the present article we give lower and upper bounds for the $2$-Selmer rank of $E$ in terms of the $2$-torsion of a narrow class group of a certain cubic extension of $K$ attached to $E$. As an application, we prove (under mild hypotheses) that a positive proportion of prime conductor quadratic twists of $E$ have the same $2$-Selmer group.
We study the rational Bianchi newforms (weight 2, trivial character, with rational Hecke eigenvalues) in the LMFDB that are not associated to elliptic curves, but instead to abelian surfaces with quaternionic multiplication. Two of these examples exhibit a rather special kind of behaviour: we show they arise from twisted base change of a classical newform with nebentypus character of order 4 and eight inner twists.
The main result of this paper is to extend from $\Q$ to each of the nine imaginary quadratic fields of class number one a result of Serre (1987) and Mestre-Oesterlé (1989), namely that if $E$ is an elliptic curve of prime conductor then either $E$ or a $2$-, $3$- or $5$-isogenous curve has prime discriminant. For four of the nine fields, the theorem holds with no change, while for the remaining five fields the discriminant of a curve with prime conductor is either prime or the square of a prime. The proof is conditional in two ways: first that the curves are modular, so are associated to suitable Bianchi newforms; and second that a certain level-lowering conjecture holds for Bianchi newforms. We also classify all elliptic curves of prime power conductor and non-trivial torsion over each of the nine fields: in the case of $2$-torsion, we find that such curves either have CM or with a small finite number of exceptions arise from a family analogous to the Setzer-Neumann family over $\Q$.
Let $E$ be a rational elliptic curve and let $p$ be an odd prime of additive reduction. Let $K$ be an imaginary quadratic field and fix a positive integer $c$ prime to the conductor of $E$. The main goal of the present article is to define an anticyclotomic $p$-adic $L$-function $Ł$ attached to $E/K$ when $E/\QQ_p$ attains semistable reduction over an abelian extension. We prove that $Ł$ satisfies the expected interpolation properties; namely, we show that if $χ$ is an anticyclotomic character of conductor $cp^n$ then $χ(Ł)$ is equal (up to explicit constants) to $L(E,χ,1)$ or $L'(E,χ,1)$.
Generalizing the method of Faltings-Serre, we rigorously verify that certain abelian surfaces without extra endomorphisms are paramodular. To compute the required Hecke eigenvalues, we develop a method of specialization of Siegel paramodular forms to modular curves. In the appendix, Serre proves a result extending his work on the reduction of G-invariant bilinear forms modulo primes to the case of G-covariant forms.