Submitted / Accepted

• Efficient computation of non-archimedean theta functions
  Masdeu, M., Xarles, X.; (2024, to appear in Math. Comp.)

  Abstract

  We describe an efficient iterative algorithm for the computation of theta functions of non-archimedean Schottky groups and, more generally, of (non-archimedean) discontinuous groups.

  arxiv.org/abs/2408.14918

Published

20. Plectic p-adic invariants
    Fornea, M., Guitart, X., Masdeu, M.; Advances in Mathematics, Volume 406 (2022)

    Abstract

    For modular elliptic curves over number fields of narrow class number one, and with multiplicative reduction at a collection of p-adic primes, we define new p-adic invariants. Inspired by Nekovar and Scholl's plectic conjectures, we believe these invariants control the Mordell--Weil group of higher rank elliptic curves and we support our expectations with numerical experiments.

    
    https://dx.doi.org/10.1016/j.aim.2022.108484
    ArXiv
    
21. A quaternionic construction of p-adic singular moduli
    Guitart, X., Masdeu, M., Xarles, X.; Research in the Mathematical Sciences, Volume 8, Issue 3 (2021)

    Abstract

    Rigid meromorphic cocycles were introduced by Darmon and Vonk as a conjectural $p$-adic extension of the theory of singular moduli to real quadratic base fields. They are certain cohomology classes of $\textrm{SL}_2(\mathbb{Z}[1/p])$ which can be evaluated at real quadratic irrationalities and the values thus obtained are conjectured to lie in algebraic extensions of the base field. In this article we present a similar construction of cohomology casses in which $\textrm{SL}_2(\mathbb{Z}[1/p])$ is replaced by an order in an indefinite quaternion algebra over a totally real number field $F$. These quaternionic cohomology classes can be evaluated at elements in almost totally complex extensions $K$ of $F$, and we conjecture that the corresponding values lie in algebraic extensions of $K$. We also report on extensive numerical evidence for this algebraicity conjecture.

    
    https://dx.doi.org/10.1007/s40687-021-00274-3
    ArXiv
    
22. Special values of triple-product p-adic L-functions and non-crystalline diagonal classes
    Gatti, F., Guitart, X., Masdeu, M., Rotger, V.; Journal de Théorie des Nombres de Bordeaux, Volume 33, Issue 3.1 (2022)

    Abstract

    The main purpose of this note is to understand the arithmetic encoded in the special value of the p-adic L-function $L^g_p(f,g,h)$ associated to a triple of modular forms $(f,g,h)$ of weights $(2,1,1)$, in the case where the classical L-function $L(f\otimes g\otimes h,s)$ -which typically has sign +1- does not vanish at its central critical point $s=1$. When $f$ corresponds to an elliptic curve $E/\mathbb{Q}$ and the classical L-function vanishes, the Elliptic Stark Conjecture of Darmon-Lauder-Rotger predicts that $L^g_p(f,g,h)(2,1,1)$ is either 0 (when the order of vanishing of the complex L-function is $>2$) or related to logarithms of global points on $E$ and a certain Gross-Stark unit associated to $g$. We complete the picture proposed by the Elliptic Stark Conjecture by providing a formula for the value $L^g_p(f,g,h)(2,1,1)$ in the case where $L(f\otimes g\otimes h,1)\neq 0$.

    
    https://dx.doi.org/10.5802/jtnb.1179
    ArXiv
    
23. An automorphic approach to Darmon points
    Guitart, X., Masdeu, M., Molina, S.; Indiana University Mathematics Journal, Volume 69, Issue 4 (2020)

    Abstract

    We give archimedean and non-archimedean constructions of Darmon points on modular abelian varieties attached to automorphic forms over arbitrary number fields and possibly non-trivial central character. An effort is made to present a unifying point of view, emphasizing the automorphic nature of the construction.

    
    https://dx.doi.org/10.1512/iumj.2020.69.7949
    ArXiv
    
24. Computing p-adic periods of abelian varieties from automorphic forms
    Guitart, X., Masdeu, M.; Contemporary Mathematics, (2019)

    Abstract

    We give an overview of the paper "Periods of modular GL2-type abelian varieties and p-adic integration", in which we exploit Darmon’s p-adic L-invariants to compute p-adic uniformizations of certain motives attached to modular forms. We illustrate our methods with new examples.

    
    https://dx.doi.org/10.1090/conm/732/14787
    PDF
25. On Heegner points for primes of additive reduction ramifying in the base field
    Kohen, D., Pacetti, A.; Transactions of the American Mathematical Society, Volume 370, Issue 2 (2017)

    Abstract

    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.

    
    https://dx.doi.org/10.1090/tran/6990
    ArXiv
    
26. Periods of modular GL2-type abelian varieties and p-adic integration
    Guitart, X., Masdeu, M.; Experimental Mathematics, Volume 27, Issue 3 (2017)

    Abstract

    Let $F$ be a number field and $\mathfrak{N}\subset \mathcal{O}_F$ an integral ideal. Let $f$ be a modular newform over $F$ of level $\Gamma_0(\mathfrak{N})$ with rational Fourier coefficients. Under certain additional conditions, Guitart-Masdeu-Sengun constructed a $p$-adic lattice which is conjectured to be the Tate lattice of an elliptic curve $E_f$ whose $L$-function equals that of $f$. The aim of this note is to generalize this construction when the Hecke eigenvalues of $f$ generate a number field of degree $d\geq 1$, in which case the geometric object associated to $f$ is expected to be, in general, an abelian variety $A_f$ of dimension $d$. We also provide numerical evidence supporting the conjectural construction in the case of abelian surfaces.

    
    https://dx.doi.org/10.1080/10586458.2017.1284624
    ArXiv
    
27. Poincaré duality isomorphisms in tensor categories
    Masdeu, M., Seveso, M.; Journal of Pure and Applied Algebra, Volume 222, Issue 10 (2018)

    Abstract

    If for a vector space V of dimension g over a characteristic zero field we denote by $\wedge^iV$ its alternating powers, and by $V^\vee$ its linear dual, then there are natural Poincar\'e isomorphisms: $\wedge^i V^\vee \cong \wedge^{g-i} V$. We describe an analogous result for objects in rigid pseudo-abelian $\mathbb{Q}$-linear ACU tensor categories.

    
    https://dx.doi.org/10.1016/j.jpaa.2017.11.015
    ArXiv
    
28. Dirac operators in tensor categories and the motive of quaternionic modular forms
    Masdeu, M., Seveso, M.; Advances in Mathematics, Volume 313 (2017)

    Abstract

    We define a motive whose realizations afford modular forms (of arbitrary weight) on an indefinite division quaternion algebra. This generalizes work of Iovita--Spiess to odd weights in the spirit of Jordan--Livné. It also generalizes a construction of Scholl to indefinite division quaternion algebras.

    
    https://dx.doi.org/10.1016/j.aim.2017.03.034
    ArXiv
    
29. Uniformization of modular elliptic curves via p-adic periods
    Guitart, X., Masdeu, M., Şengün, M.; Journal of Algebra, Volume 445 (2016)

    Abstract

    The Langlands Programme predicts that a weight 2 newform $f$ over a number field $K$ with integer Hecke eigenvalues generally should have an associated elliptic curve $E_f$ over $K$. In our previous paper, we associated, building on works of Darmon and Greenberg, a $p$-adic lattice to $f$, under certain hypothesis, and implicitly conjectured that this lattice is commensurable with the $p$-adic Tate lattice of $E_f$ . In this paper, we present this conjecture in detail and discuss how it can be used to compute, directly from $f$, an explicit Weierstrass equation for the conjectural $E_f$ . We develop algorithms to this end and implement them in order to carry out extensive systematic computations in which we compute Weierstrass equations of hundreds of elliptic curves, some with huge heights, over dozens of number fields. The data we obtain provide overwhelming amount of support for the conjecture and furthermore demonstrate that the conjecture provides an efficient tool to building databases of elliptic curves over number fields.

    
    https://dx.doi.org/10.1016/j.jalgebra.2015.06.021
    ArXiv
    MathSciNet
    
30. A $p$-adic construction of ATR points on $\mathbb{Q}$-curves
    Guitart, X., Masdeu, M.; Publicacions Matemàtiques, Volume 59 (2015)

    Abstract

    In this note we consider certain elliptic curves defined over real quadratic fields isogenous to their Galois conjugate. We give a construction of algebraic points on these curves defined over almost totally real number fields. The main ingredient is the system of Heegner points arising from Shimura curve uniformizations. In addition, we provide an explicit p-adic analytic formula which allows for the effective, algorithmic calculation of such points.

    
    https://dx.doi.org/10.5565/PUBLMAT_59215_09
    ArXiv
    MathSciNet
    
31. Darmon points on elliptic curves over number fields of arbitrary signature
    Guitart, X., Masdeu, M., Şengün, M.; Proceedings of the London Mathematical Society, Volume 111, Issue 2 (2015)

    Abstract

    We present new constructions of complex and p-adic Darmon points on elliptic curves over base fields of arbitrary signature. We conjecture that these points are global and present numerical evidence to support our conjecture.

    
    https://dx.doi.org/10.1112/plms/pdv033
    ArXiv
    MathSciNet
    
32. Elementary Matrix Decomposition and The Computation of Darmon Points with Higher Conductor
    Guitart, X., Masdeu, M.; Mathematics of Computation, Volume 84, Issue 292 (2014)

    Abstract

    We extend the algorithm of Darmon-Green and Darmon-Pollack for computing $p$-adic Darmon points on elliptic curves to the case of composite conductor. We also extend the algorithm of Darmon-Logan for computing ATR Darmon points to treat curves of nontrivial conductor. Both cases involve an algorithmic decomposition into elementary matrices in congruence subgroups $\Gamma(N)$ for ideals $N$ in certain rings of $S$-integers. We use these extensions to provide additional evidence in support of the conjectures on the rationality of Darmon points.

    
    https://dx.doi.org/10.1090/S0025-5718-2014-02853-6
    ArXiv
    MathSciNet
    
33. Computing fundamental domains for the Bruhat-Tits tree for GL_2(Qp), p-adic automorphic forms, and the canonical embedding of Shimura curves
    Franc, C., Masdeu, M.; LMS Journal of Computation and Mathematics, Volume 17, Issue 1 (2014)

    Abstract

    In this paper we describe an algorithm for computing certain quaternionic quotients of the Bruhat-Tits tree for GL2(Qp). These quotients are of arithmetic interest as they describe bad fibers of integral models of Shimura curves.

    
    https://dx.doi.org/10.1112/S1461157013000235
    ArXiv
    MathSciNet
    
34. Overconvergent cohomology and quaternionic Darmon points
    Guitart, X., Masdeu, M.; Journal of the London Mathematical Society, Volume 90, Issue 2 (2014)

    Abstract

    We develop the (co)homological tools that make effective the construction of the quaternionic Darmon points introduced by Matthew Greenberg. In addition, we use the overconvergent cohomology techniques of Pollack--Pollack to allow for the efficient calculation of such points. Finally, we provide the first numerical evidence supporting the conjectures on their rationality.

    
    https://dx.doi.org/10.1112/jlms/jdu036
    ArXiv
    MathSciNet
    
35. Computation of ATR Darmon Points on Nongeometrically Modular Elliptic Curves
    Guitart, X., Masdeu, M.; Experimental Mathematics, Volume 22, Issue 1 (2013)

    Abstract

    ATR points were introduced by Darmon as a conjectural construction of algebraic points on certain elliptic curves for which in general the Heegner point method is not available. So far the only numerical evidence, provided by Darmon-Logan and Gärtner, concerned curves arising as quotients of Shimura curves. In those special cases the ATR points can be obtained from the already existing Heegner points, thanks to results of Zhang and Darmon-Rotger-Zhao.

    In this paper we compute for the first time an algebraic ATR point on a curve which is not uniformizable by any Shimura curve, thus providing the first piece of numerical evidence that Darmon's construction works beyond geometric modularity. To this purpose we improve the method proposed by Darmon and Logan by removing the requirement that the real quadratic field be norm-euclidean, and accelerating the numerical integration of Hilbert modular forms.

    
    https://dx.doi.org/10.1080/10586458.2013.738564
    ArXiv
    MathSciNet
    
36. Continued fractions in 2-stage euclidean quadratic fields
    Guitart, X., Masdeu, M.; Mathematics of Computation, Volume 82, Issue 282 (2012)

    Abstract

    We discuss continued fractions on real quadratic number fields of class number $1$. If the field has the property of being $2$-stage euclidean, a generalization of the euclidean algorithm can be used to compute these continued fractions. Although it is conjectured that all real quadratic fields of class number $1$ are $2$-stage euclidean, this property has been proven for only a few of them. The main result of this paper is an algorithm that, given a real quadratic field of class number $1$, verifies this conjecture, and produces as byproduct enough data to efficiently compute continued fraction expansions. If the field was not $2$-stage euclidean, then the algorithm would not terminate. As an application, we enlarge the list of known $2$-stage euclidean fields, by proving that all real quadratic fields of class number $1$ and discriminant less than $8000$ are $2$-stage euclidean.

    
    https://dx.doi.org/10.1090/S0025-5718-2012-02620-2
    ArXiv
    MathSciNet
    
37. CM cycles on Shimura curves, and $p$-adic L-functions
    Masdeu, M.; Compositio Mathematica, Volume 148, Issue 4 (2012)

    Abstract

    Let $f$ be a modular form of weight $k\geq 2$ and level $N$, let $K$ be a quadratic imaginary field, and assume that there is a prime $p$ exactly dividing $N$. Under certain arithmetic conditions on the level and the field $K$, one can attach to this data a $p$-adic L-function $L_p(f,K,s)$, as done by Bertolini-Darmon-Iovita-Spiess. In the case of $p$ being inert in $K$, this analytic function of a $p$-adic variable $s$ vanishes in the critical range $s=1,\ldots,k-1$, and therefore one is interested in the values of its derivative in this range. We construct, for $k\geq 4$, a Chow motive endowed with a distinguished collection of algebraic cycles which encode these values, via the $p$-adic Abel-Jacobi map.

    Our main result generalizes the result obtained by Iovita-Spiess, which gives a similar formula for the central value $s=k/2$. Even in this case our construction is different from the one found by Iovita-Spiess.

    
    https://dx.doi.org/10.1112/S0010437X12000206
    ArXiv
    MathSciNet
    
38. Bezout Domains and Elliptic Curves
    Goldbring, I., Masdeu, M.; Communications in Algebra, Volume 36, Issue 12 (2008)

    Abstract

    Let $k$ be a fixed algebraic closure of $\mathbb{Q}$ and $k(t)^\textrm{ac}$ a fixed algebraic closure of $k(t)$. Let $S \subset k[t]\setminus \{0\}$ be a multiplicative set. Let $A = S^{-1}(k[t])$ and $\widetilde{A}$ be the integral closure of $A$ in $k(t)^\textrm{ac}$. We use elliptic curves to develop a necessary condition on $S$ for $\widetilde{A}$ to be a Bezout domain. We give some examples of $S$ which fail to satisfy this condition. As a consequence, we eliminate some candidates for a good Rumely domain of characteristic $0$ with algebraic subring $k$.

    
    https://dx.doi.org/10.1080/00927870802182408
    MathSciNet
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39. A note on dihedral polynomials of prime degree
    Lario, Joan-C., Masdeu, Marc; Indian J. Pure Appl. Math., (2005)

    Abstract

    We present an algorithm to determine all roots of a prime degree $p$ polynomial with dihedral Galois group $D_{2p}$ as rational functions of any two of them. This must be seen as an effective version of a more general result of Galois valid for prime degree equations with solvable group.

    https://dx.doi.org/
    MathSciNet
    PDF

Thesis