arXiv subjects

Ariel Pacetti

Publications and source records attributed to Ariel Pacetti.

At least 37 records · Page 2Linked to original sources

arXiv2018

Anticyclotomic $p$-adic $L$-functions for elliptic curves at some additive reduction primes

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 $L'(E,χ,1)$.

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arXiv2018

On the number of Galois orbits of newforms

Counting the number of Galois orbits of newforms in $S_k(Γ_0(N))$ and giving some arithmetic sense to this number is an interesting open problem. The case $N=1$ corresponds to Maeda's conjecture (still an open problem) and the expected number of orbits in this case is 1, for any $k \ge 16$. In this article we give local invariants of Galois orbits of newforms for general $N$ and count their number. Using an existence result of newforms with prescribed local invariants we prove a lower bound for the number of non-CM Galois orbits of newforms for $Γ_0(N)$ for large enough weight $k$ (under some technical assumptions on $N$). Numerical evidence suggests that in most cases this lower bound is indeed an equality, thus we leave as a Question the possibility that a generalization of Maeda's conjecture could follow from our work. We finish the paper with some natural generalizations of the problem and show some of the implications that a generalization of Maeda's conjecture has.

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arXiv2016

On Heegner Points for primes of additive reduction ramifying in the base field

Let $E$ be a rational elliptic curve, and $K$ be an imaginary quadratic field. In this article we give a method to construct Heegner points when $E$ has a prime bigger than $3$ of additive reduction ramifying in the field $K$. The ideas apply to more general contexts, like constructing Darmon points attached to real quadratic fields which is presented in the appendix.

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arXiv2015

Heegner points on Cartan non-split curves

Let $E$ be an elliptic curve of conductor $N$, and let $K$ be an imaginary quadratic field such that the root number of $E/K$ is $-1$. Let $O$ be an order in $K$ and assume that there exists an odd prime $p$, such that $p^2 \mid\mid N$, and $p$ is inert in $O$. Although there are no Heegner points on $X_0(N)$ attached to $O$, in this article we construct such points on Cartan non-split curves. In order to do that we give a method to compute Fourier expansions for forms in Cartan non-split curves, and prove that the constructed points form a Heegner system as in the classical case.

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arXiv2014

Connectedness of Hecke Algebras and the Rayuela conjecture: a path to functoriality and modularity

Let $ρ_1$ and $ρ_2$ be a pair of residual, odd, absolutely irreducible two-dimensional Galois representations of a totally real number field $F$. In this article we propose a conjecture asserting existence of "safe" chains of compatible systems of Galois representations linking $ρ_1$ to $ρ_2$. Such conjecture implies the generalized Serre's conjecture and is equivalent to Serre's conjecture under a modular version of it. We prove a weak version of the modular variant using the connectedness of certain Hecke algebras, and we comment on possible applications of these results to establish some cases of Langlands functoriality.

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arXiv2014

Theta Lifts of Bianchi Modular Forms and Applications to Paramodularity

We explain how the work of Johnson-Leung and Roberts on lifting Hilbert modular forms for real quadratic fields to Siegel modular forms can be adapted to imaginary quadratic fields. For this we use archimedean results from Harris, Soudry, Taylor and replace the global arguments of Roberts by the non-vanishing result of Takeda. As an application of our lifting result, we exhibit an abelian surface $B$ defined over $\mathbb{Q}$, which is not restriction of scalars of an elliptic curve and satisfies the Brumer-Kramer Paramodularity Conjecture.

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arXiv2014

Shimura correspondence for level $p^2$ and the central values of $L$-series II

Given a Hecke eigenform $f$ of weight $2$ and square-free level $N$, by the work of Kohnen, there is a unique weight $3/2$ modular form of level $4N$ mapping to $f$ under the Shimura correspondence. Furthermore, by the work of Waldspurger the Fourier coefficients of such a form are related to the quadratic twists of the form $f$. Gross gave a construction of the half integral weight form when $N$ is prime, and such construction was later generalized to square-free levels. However, in the non-square free case, the situation is more complicated since the natural construction is vacuous. The problem being that there are too many special points so that there is cancellation while trying to encode the information as a linear combination of theta series. In this paper, we concentrate in the case of level $p^2$, for $p>2$ a prime number, and show how the set of special points can be split into subsets (indexed by bilateral ideals for an order of reduced discriminant $p^2$) which gives two weight $3/2$ modular forms mapping to $f$ under the Shimura correspondence. Moreover, the splitting has a geometric interpretation which allows to prove that the forms are indeed a linear combination of theta series associated to ternary quadratic forms. Once such interpretation is given, we extend the method of Gross-Zagier to the case where the level and the discriminant are not prime to each other to prove a Gross-type formula in this situation.

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arXiv2014

Congruences between modular forms modulo prime powers

Given a prime $p \ge 5$ and an abstract odd representation $ρ_n$ with coefficients modulo $p^n$ (for some $n \ge 1$) and big image, we prove the existence of a lift of $ρ_n$ to characteristic $0$ whenever local lifts exist (under some technical conditions). Moreover, we can chose the inertial type of our lift at all primes but finitely many (where the lift is of Steinberg type). We apply this result to the realm of modular forms, proving a level lowering theorem modulo prime powers and providing examples of level raising. In particular, our method shows that given a modular eigenform $f$ without Complex Multiplication or inner twists, for all primes $p$ but finitely many, and for all positive integers $n$, there exists another eigenform $g\neq f$, which is congruent to $f$ modulo $p^n$.

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arXiv2013

Hecke and Sturm bounds for Hilbert modular forms over real quadratic fields

In this article we give an analogue of Hecke and Sturm bounds for Hilbert modular forms over real quadratic fields. Let $K$ be a real quadratic field and $\Om_K$ its ring of integers. Let $Γ$ be a congruence subgroup of $\SL_2(\Om_K)$ and $M_{(k_1,k_2)}(Γ)$ the space of Hilbert modular forms of weight $(k_1,k_2)$ for $Γ$. The first main result is an algorithm to construct a finite set $S$, depending on $K$, $Γ$ and $(k_1,k_2)$, such that if the Fourier expansion coefficients of a form $G \in M_{(k_1,k_2)}(Γ)$ vanish on the set $S$, then $G$ is the zero form. The second result corresponds to the same statement in the Sturm case, i.e. suppose that all the Fourier coefficients of the form $G$ lie in a finite extension of $\Q$, and let $\id{p}$ be a prime ideal in such extension, whose norm is unramified in $K$; suppose furthermore that the Fourier expansion coefficients of $G$ lie in the ideal $\id{p}$ for all the elements in $S$, then they all lie in the ideal $\id{p}$.

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arXiv2012

Modularity of the Consani-Scholten quintic

We prove that the Consani-Scholten quintic, a Calabi-Yau threefold over QQ, is Hilbert modular. For this, we refine several techniques known from the context of modular forms. Most notably, we extend the Faltings-Serre-Livne method to induced four-dimensional Galois representations over QQ. We also need a Sturm bound for Hilbert modular forms; this is developed in an appendix by Jose Burgos Gil and the second author.

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arXiv2012

Computing ideal classes representatives in quaternion algebras

Let $K$ be a totally real number field and let $B$ be a totally definite quaternion algebra over $K$. In this article, given a set of representatives for ideal classes for a maximal order in $B$, we show how to construct in an efficient way a set of representatives of ideal classes for any Bass order in $B$. The algorithm does not require any knowledge of class numbers, and improves the equivalence checking process by using a simple calculation with global units. As an application, we compute ideal classes representatives for an order of level 30 in an algebra over the real quadratic field $\Q[\sqrt{5}]$.

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arXiv2010

On the change of root numbers under twisting and applications

The purpose of this article is to show how the root number of a modular form changes by twisting in terms of the local Weil-Deligne representation at each prime ideal. As an application, we show how one can for each odd prime $p$, determine whether a modular form (or a Hilbert modular form) with trivial nebentypus is Steinberg, Principal Series or Supercuspidal at $p$ by analyzing the change of sign under a suitable twist. We also explain the case $p=2$, where twisting is not enough in general.

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arXiv2008

Proving modularity for a given elliptic curve over an imaginary quadratic field

We present an algorithm to determine if the $L$-series associated to an automorphic representation and the one associated to an elliptic curve over an imaginary quadratic field agree. By the work of Harris-Soudry-Taylor, Taylor and Berger-Harcos (cf. \cite{harris-taylor}, \cite{taylorII} and \cite{berger-harcos}) we can associate to an automorphic representation a family of compatible $p$-adic representations. Our algorithm is based on Faltings-Serre's method to prove that $p$-adic Galois representations are isomorphic.

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arXiv2006

Computing central values of twisted L-series: the case of composite levels

Let f be a newform of weight two and composite level N. We show how to compute weight 3/2 modular forms "associated" to f whose Fourier coefficients are related to the central values of quadratic twists (real and imaginary) of f. We will focus on examples for levels N=27, N=15, and N=75, which exhibit most aspects of our methods.

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arXiv2006

Shimura correspondence for level p^2 and the central values of L-series

Given a weight 2 and level p^2 modular form f, we construct two weight 3/2 modular forms (possibly zero) of level 4p^2 and non trivial character mapping to f via the Shimura correspondence. Then we relate the coefficients of the constructed forms to the central value of the L-series of certain imaginary quadratic twists of f. Furthermore, we give a general framework for our construction that applies to any order in definite quaternion algebras, with which one can, in principle, construct weight 3/2 modular forms of any level, provided one knows how to compute ideal classes representatives.

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arXiv2005

Newton-Hensel Interpolation Lifting

The main result of this paper is a new version of Newton-Hensel lifting that relates to interpolation questions. It allows one to lift polynomials in $Z[x]$ from information modulo a prime number $p\ne 2$ to a power $p^k$ for any $k$, and its originality is that it is a mixed version that not only lifts the coefficients of the polynomial but also its exponents. We show that this result corresponds exactly to a Newton-Hensel lifting of a system of $2t$ generalized equations in $2t$ unknowns in the ring of $p$-adic integers $\Z_p$. Finally we apply our results to sparse polynomial interpolation in $\Z[x]$

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arXiv2005

On the embedding problem for $2^+S_4$ representations

Let $2^+S_4$ denote the double cover of $S_4$ corresponding to the element in $H^2(S_4,\Z/2\Z)$ where transpositions lift to elements of order 2 and the product of two disjoint transpositions to elements of order 4 (denoted $\tilde S_4$ in \cite{Serre}). Given an elliptic curve $E$, let $E[2]$ denote its 2-torsion points. Under some conditions on $E$ (as in \cite{Bayer}) elements in $H^1(\Gal_\Q,E[2])\backslash \{0 \}$ correspond to Galois extensions $N$ of $\Q$ with Galois group (isomorphic to) $S_4$. On this work we give an interpretation of the addition law on such fields, and prove that the obstruction for $N$ having a Galois extension $\tilde N$ with $\Gal(\tilde N/ \Q) \simeq 2^+S_4$ gives an homomorphism $s_4^+:H^1(\Gal_\Q,E[2]) \to H^2(\Gal_\Q,\Z/2\Z)$. As a Corollary we can prove (if $E$ has conductor divisible by few primes and high rank) the existence of 1$-dimensional representations attached to $E$ and use them in some examples to construct 3/2 modular forms mapping via the Shimura map to (the modular form attached to) $E$.

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arXiv2004

A formula for the central value of certain Hecke L-functions

Let N = 1 mod 4 be the negative of a prime, K=Q(sqrt{N}) and O_K its ring of integers. Let D be a prime ideal in O_K of prime norm congruent to 3 modulo 4. Under these assumptions, there exists Hecke characters $ψ_{\D}$ of K with conductor $(\D)$ and infinite type $(1,0)$. Their L-series L(ψ_\D,s)$ are associated to a CM elliptic curve E(N,\D) defined over the Hilbert class field of $K$. We will prove a Waldspurger-type formula for L(ψ_\D,s) of the form L(ψ_\D,1) = Ω\sum_{[\A],I} r(\D,[\A],I) m_{[\A],I}([\D]) where the sum is over class ideal representatives I of a maximal order in the quaternion algebra ramified at |N| and infinity and [\A] are class group representatives of $K$. An application of this formula for the case N=-7 will allow us to prove the non-vanishing of a family of L-series of level $7|D|$ over $K$.

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