Consider an irreducible, admissible representation $π$ of GL(2,$F$) whose restriction to GL(2,$F)^+$ breaks up as a sum of two irreducible representations $π_+ + π_-$. If $π=r_θ$, the Weil representation of GL(2,$F$) attached to a character $θ$ of $K^*$ which does not factor through the norm map from $K$ to $F$, then $χ\in \hat{K^*}$ with $(χ>. θ^{-1})|_{F^{*}}=ω_{K/F}$ occurs in ${r_θ}_+$ if and only if $ε(θχ^{-1},ψ_0)=ε(\bar θχ^{-1},ψ_0)=1$ and in ${r_θ}_-$ if and only if both the epsilon factors are -1. But given a conductor $n$, can we say precisely how many such $χ$ will appear in $π$? We calculate the number of such characters at each given conductor $n$ in this work.
We compute the Fourier expansion of vector valued Eisenstein series for the Weil representation associated to an even lattice. To this end, we define certain twists by Dirichlet characters of the usual Eisenstein series associated to isotropic elements in the discriminant form of the underlying lattice. These twisted functions still form a generating system for the space of Eisenstein series but have better multiplicative properties than the individual Eisenstein series. We adapt a method of Bruinier and Kuss to obtain algebraic formulas for the Fourier coefficients of the twisted Eisenstein s…
We show that in Weil's converse theorem the functional equations of multiplicative twists for at least the first $\sqrt{\frac{p-24}{3}}$ moduli are needed in order to prove the modularity for $Γ_0(p)$.
We present a proof of the Chevalley-Weil Theorem that is somewhat different from the proofs appearing in the literature and with somewhat weaker hypotheses, of purely topological type. We also provide a discussion of the assumptions, and an application to solutions of generalized Fermat equations, where our statement allows to simplify the original argument of Darmon and Granville.
We show that a Frobenius-semisimple Weil representation over a local field K is determined by its Euler factors over the extensions of K. The construction is explicit, and we illustrate it for l-adic representations attached to elliptic and genus 2 curves. As an application, we construct an absolutely simple 2-dimensional abelian variety over Q all of whose quadratic twists must have positive rank, according to the Birch-Swinnerton-Dyer conjecture.
In this paper, we consider a family of twists of a superelliptic curve over a global field, and obtain results on the distribution of the Mordell-Weil rank of these twists. Our results have applications to the distribution of the number of rational points.
Watkins conjectured that for an elliptic curve $E$ over $\mathbb{Q}$ of Mordell-Weil rank $r$, the modular degree of $E$ is divisible by $2^r$. If $E$ has non-trivial rational $2$-torsion, we prove the conjecture for all the quadratic twists of $E$ by squarefree integers with sufficiently many prime factors.
Suppose that $M$ is an even lattice with dual $M^{*}$ and level $N$. Then the group $Mp_{2}(\mathbb{Z})$, which is the unique non-trivial double cover of $SL_{2}(\mathbb{Z})$, admits a representation $ρ_{M}$, called the Weil representation, on the space $\mathbb{C}[M^{*}/M]$. The main aim of this paper is to show how the formulae for the $ρ_{M}$-action of a general element of $Mp_{2}(\mathbb{Z})$ can be obtained by a direct evaluation which does not depend on ``external objects'' such as theta functions. We decompose the Weil representation $ρ_{M}$ into $p$-parts, in which each $p$-part can be…
Given two elliptic curves defined over a number field K, not both with j-invariant zero, we show that there are infinitely many $D\in K^\times$ with pairwise distinct image in $ K^\times/{K^\times}^2 $, such that the quadratic twist of both curves by D have positive Mordell-Weil rank. The proof depends on relating the values of pairs of cubic polynomials to rational points on another elliptic curve, and on a fiber product construction.
We study the arithmetic of the twist of the Fermat quartic defined by $X^4 + Y^4 + Z^4 = 0$ which has no $\mathbb{Q}$-rational point. We calculate the Mordell--Weil group of the Jacobian variety explicilty. We show that the degree $0$ part of the Picard group is a free $\mathbb{Z}/2\mathbb{Z}$-module of rank $2$, whereas the Mordell--Weil group is a free $\mathbb{Z}/2\mathbb{Z}$-module of rank $3$. Thus the relative Brauer group is non-trivial. We also show that this quartic violates the local-global property for linear determinantal representations.
It is well known(cf. Weil, Gérardin's works) that there are two different Weil representations of a symplectic group over an odd finite field. By a twisted action, we show that one can reorganize them as a representation of a related projective symplectic similitude group. We also discuss the even field case by following Genestier-Lysenko and Gurevich-Hadani's works on geometric Weil representations in characteristic two. As a result, we approach some of their results from the lattice model, which is inspired by MVW, Prasad and Takeda's works.
14H52 : Elliptic curves Let E be the elliptic curve given by a Mordell equation y^2=x^3-A where A is an integer. For certain A, we use Stoll's formula to compute a lower bound for the proportion of square-free integers D up to X such that the Mordell-Weil rank of the quadratic twist by D is less than 2k, for given non-negative k. We also compute an upper bound for a certain average rank of quadratic twists of E.
We characterize quadratic twists of $y^2=x(x-a^2)(x+b^2)$ with Mordell-Weil groups and $2$-primary part of Shafarevich-Tate groups being isomorphic to $(\mathb Z/2\mathbb Z)^2$ under certain conditions. We also obtain the distribution result of these elliptic curves.
Let $E$ be a separable quadratic extension of a locally compact field $F$ of positive characteristic. Asai γ-factors are defined for smooth irreducible representations πof ${\rm GL}_n(E)$. If σis the Weil-Deligne representation of $\mathcal{W}_E$ corresponding to πunder the local Langlands correspondence, we show that the Asai γ-factor is the same as the Deligne-Langlands γ-factor of the Weil-Deligne representation of $\mathcal{W}_F$ obtained from σunder tensor induction. This is achieved by proving that Asai γ-factors are characterized by their local properties together with their role in glo…
Let C be a smooth projective curve defined over a number field and let C' be a twist of C. In this article we relate the l-adic representations attached to the l-adic Tate modules of the Jacobians of C and C' through an Artin representation. This representation induces global relations between the local factors of the respective Hasse-Weil L-functions. We make these relations explicit in a particularly illustrating situation. For every Qbar-isomorphism class of genus 2 curves defined over Q with automorphism group isomorphic to D_8 or D_{12}, except for a finite number, we choose a representat…
Given two pure representations of the absolute Galois group of an $\ell$-adic number field with coefficients in $\overline{\mathbb{Q}}_p$ (with $\ell\neq p$), we show that the Frobenius-semisimplifications of the associated Weil--Deligne representations are twists of each other by an integral power of certain unramified character if they have equal normalized traces. This is an analogue of a recent result of Patankar and Rajan in the context of local Galois representations.
Let $F$ be a non-Archimedean local field. Let $\mathcal{A}_n(F)$ be the set of equivalence classes of irreducible admissible representations of $\textrm{GL}_n(F)$, and $\mathcal{G}_n(F)$ be the set of equivalence classes of n-dimensional Frobenius semisimple Weil-Deligne representations of $W'_F$. The local Langlands correspondence(LLC) establishes the reciprocity maps $\textrm{Rec}_{n,F}: \mathcal{A}_n(F)\longrightarrow \mathcal{G}_n(F)$ , satisfying some nice properties. An important invariant under this correspondence is the L- and $ε$-factors. This is also expected to be true under paralle…
