Atkin-Lehner Involutions Research Articles

Equations of hyperelliptic Shimura curves

By constructing suitable Borcherds forms on Shimura curves and using Schofer’s formula for norms of values of Borcherds forms at CM points, we determine all of the equations of hyperelliptic Shimura curves $X_{0}^{D}(N)$. As a byproduct, we also address the problem of whether a modular form on Shimura curves $X_{0}^{D}(N)/W_{D,N}$ with a divisor supported on CM divisors can be realized as a Borcherds form, where $X_{0}^{D}(N)/W_{D,N}$ denotes the quotient of $X_{0}^{D}(N)$ by all of the Atkin–Lehner involutions. The construction of Borcherds forms is done by solving certain integer programming problems.

Notes on Weierstrass Points of Modular Curves $X_0(N)$

We give conditions when the fixed points by the partial Atkin-Lehner involutions on $X_0(N)$ are Weierstrass points as an extension of the result by Lehner and Newman \cite{LN}. Furthermore, we complete their result by determining whether the fixed points by the full Atkin-Lehner involutions on $X_0(N)$ are Weierstrass points or not.

Products of vector valued Eisenstein series

AbstractWe prove that products of at most two vector valued Eisenstein series that originate in level 1 span all spaces of cusp forms for congruence subgroups. This can be viewed as an analogue in the level aspect to a result that goes back to Rankin, and Kohnen and Zagier, which focuses on the weight aspect. The main feature of the proof are vector valued Hecke operators. We recover several classical constructions from them, including classical Hecke operators, Atkin–Lehner involutions, and oldforms. As a corollary to our main theorem, we obtain a vanishing condition for modular forms reminiscent of period relations deduced by Kohnen and Zagier in the context their previously mentioned result.

The moduli spaces of Jacobians isomorphic to a product of two elliptic curves

The purpose of this paper is to study the moduli spaces of curves C of genus 2 with the property that their Jacobians \(J_C\) are isomorphic to a product surface \(E_1\times E_2\). Theorem 1 shows that the set of such curves is the union of infinitely many closed subvarieties T(d), \(d\ge 3\), of the moduli space \(M_2\). Each T(d) is a curve except for finitely many d’s for which T(d) is empty. The precise list of the exceptional d’s is given in Theorem 5 and depends on the validity of a conjecture due to Euler and Gauss. Each T(d) is the union of finitely many irreducible components \(H'(q)\), where q runs over the equivalence classs of certain binary quadratic forms of discriminant \(-16d\); cf. Theorems 2 and 3. The birational structure of the curve \(H'(q)\) (which can be viewed a “generalized Humbert variety”) is determined in Theorem 4. It turns out that \(H'(q)\) is a quotient of the modular curve \(X_0(d)\) modulo certain Atkin–Lehner involutions.

THE SHIMURA CURVE OF DISCRIMINANT 15 AND TOPOLOGICAL AUTOMORPHIC FORMS

We find defining equations for the Shimura curve of discriminant 15 over $\mathbb{Z}[1/15]$. We then determine the graded ring of automorphic forms over the 2-adic integers, as well as the higher cohomology. We apply this to calculate the homotopy groups of a spectrum of ‘topological automorphic forms’ associated to this curve, as well as one associated to a quotient by an Atkin–Lehner involution.

Eisenstein Ideals and the Rational Torsion Subgroups of Modular Jacobian Varieties II

We study the rational torsion subgroup of the modular Jacobian variety $J_0(N)$ when $N$ is square-free. We prove that the $p$-primary part of this group coincides with that of the cuspidal divisor class group for $p\geq 3$ when $3 \nmid N$, and for $p\geq 5$ when $3 \mid N$. We further determine the structure of each eigenspace of such $p$-primary part with respect to the Atkin-Lehner involutions. This is based on our study of the Eisenstein ideals in the Hecke algebras.

Modular forms of half-integral weights on SL(2, ℤ)

AbstractIn this paper, we prove that, for an integer r with (r, 6) = 1 and 0 < r < 24 and a nonnegative even integer s, the setis isomorphic toas Hecke modules under the Shimura correspondence. Here Ms(1) denotes the space of modular forms of weight is the space of newforms of weight 2k on Γ0 (6) that are eigenfunctions with eigenvalues €2 and €3 for Atkin-Lehner involutions W2 and W3, respectively, and the notation ⊕(12/.) means the twist by the quadratic character (12/-). There is also an analogous result for the cases (r, 6) = 3.

Fourier–Mukai transformations on K3 surfaces with [formula omitted] and Atkin–Lehner involutions

We show that there is a surjection from the Fourier–Mukai transformations on projective K3 surfaces with the Picard number ρ(X)=1 to so-called the group of Atkin–Lehner involutions. This was expected in Hosono, Lian, Oguiso and Yau's paper.

Automorphisms and reduction of Heegner points on Shimura curves at Cerednik-Drinfeld primes

The aim of this short note is to show how the interplay of the action of the automorphism group of a Shimura curve on the special fiber of its Cerednik-Drinfeld’s integral model at a prime of bad reduction and its sets of Heegner points, can be exploited to prove some instances of a conjecture that predicts that any automorphism must be an Atkin-Lehner involution.

On polyquadratic twists ofX0(N)

Let K = Q ( d 1 , … , d k ) be a polyquadratic number field and N be a squarefree positive integer with at least k distinct factors. The Galois group, Gal ( K / Q ) is an elementary abelian two-group generated by σ i such that σ i ( d i ) = − d i . Let ζ : Gal ( K / Q ) → Aut ( X 0 ( N ) ) be the cocycle that sends σ i to w m i where w m i are the Atkin–Lehner involutions of X 0 ( N ) . In this paper, we study the Q p -rational points of the twisted modular curve X 0 ζ ( N ) and give an algorithm to produce such curves which has Q p -rational points for all primes p . Then we investigate violations of the Hasse principle for these curves and give an asymptotic for the number of such violations. Finally, we study reasons of such violations.

Points rationnels sur les quotients d’Atkin-Lehner de courbes de Shimura de discriminant pq

Let $p$ and $q$ be two distinct prime numbers, and $X^{pq}/w_q$ be the quotient of the Shimura curve of discriminant $pq$ by the Atkin-Lehner involution $w_q$. We describe a way to verify in wide generality a criterion of Parent and Yafaev to prove that if $p$ and $q$ satisfy some explicite congruence conditions, known as the conditions of the non ramified case of Ogg, and if $p$ is large enough compared to $q$, then the quotient $X^{pq}/w_q$ has no rational point, except possibly special points.

Yoshida lifts and Selmer groups

Let $f$ and $g$, of weights $k' > k \geq 2$, be normalised newforms for $\Gamma_0(N)$, for square-free $N > 1$, such that, for each Atkin-Lehner involution, the eigenvalues of $f$ and $g$ are equal. Let $\lambda\mid\ell$ be a large prime divisor of the algebraic part of the near-central critical value $L(f\otimes g,(k+k'-2)/2)$. Under certain hypotheses, we prove that $\lambda$ is the modulus of a congruence between the Hecke eigenvalues of a genus-two Yoshida lift of (Jacquet-Langlands correspondents of) $f$ and $g$ (vector-valued in general), and a non-endoscopic genus-two cusp form. In pursuit of this we also give a precise pullback formula for a genus-four Eisenstein series, and a general formula for the Petersson norm of a Yoshida lift. Given such a congruence, using the 4-dimensional $\lambda$-adic Galois representation attached to a genus-two cusp form, we produce, in an appropriate Selmer group, an element of order $\lambda$, as required by the Bloch-Kato conjecture on values of $L$-functions.

COMPUTATIONS OF BASES FOR THE SPACES OF CUSPFORMS OF WEIGHT 2

In this paper, we present a explicit procedure to compute a basis for the spaces of cuspforms of weight 2 on which consists of eigenforms for the Atkin-Lehner involutions.

The cuspidal class number formula for certain quotient curves of the modular curve X 0 ( M ) by Atkin-Lehner involutions

We calculate the cuspidal class number of a certain quotient curve of the modular curve X 0 ( M ) with M square-free. For each factor r of M , let w r denote the Atkin-Lehner type involution of X 0 ( M ) . Let M 0 be a divisor of M , and W 0 the subgroup of the automorphism group of X 0 ( M ) consisting of all w r with r dividing M 0 . Our object is the quotient of X 0 ( M ) by W 0 . In this paper, we consider the case where M is odd.

Parametrization of Abelian K-surfaces with quaternionic multiplication

We prove that the Abelian K-surfaces whose endomorphism algebra is a rational quaternion algebra are parametrized, up to isogeny, by the K-rational points of the quotient of certain Shimura curves by the group of their Atkin–Lehner involutions. To cite this article: X. Guitart, S. Molina, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Minimal resolution of Atkin–Lehner quotients of [formula omitted

Let X 0 ( N ) be the classic modular curve of level N over Z . Let W M be the Atkin–Lehner involution of X 0 ( N ) associated to a divisor M with ( M , N / M ) = 1 . In this paper an explicit description is given for the minimal resolution over Z [ 1 / 6 ] of the Atkin–Lehner quotient X 0 ( N ) / W M . As an application a new proof of Deuring's formula on the number of supersingular j-invariants in F p is given. In certain cases it is also shown that the action of Hecke operators on the component group of the Jacobian of the Atkin–Lehner quotient is Eisenstein.

On the non-existence of exceptional automorphisms on Shimura curves

We study the group of automorphisms of Shimura curves X0(D, N) attached to an Eichler order of square-free level N in an indefinite rational quaternion algebra of discriminant D>1. We prove that, when the genus g of the curve is greater than or equal to 2, Aut (X0(D, N)) is a 2-elementary abelian group which contains the group of Atkin–Lehner involutions W0(D, N) as a subgroup of index 1 or 2. It is conjectured that Aut (X0(D, N))=W0(D, N) except for finitely many values of (D, N) and we provide criteria that allow us to show that this is indeed often the case. Our methods are based on the theory of complex multiplication of Shimura curves and the Cerednik–Drinfeld theory on their rigid analytic uniformization at primes p| D.

Weierstrass points on X 0(p ℓ) and arithmetic properties of Fourier coefficients of cusp forms

Generally it is unknown, whether or not ∞ is a Weierstrass point on the modular curve X0(N) if N is squarefree. A classical result of Atkin and Ogg states that ∞ is not a Weierstrass point on X0(N), if N=pM with p prime, p∤M and the genus of X0(M) zero. We use results of Kohnen and Weissauer to show that there is a connection between this question and the p-adic valuation of cusp forms under the Atkin–Lehner involution. This gives, in a sense, a generalization of Ogg’s Theorem in some cases.

Determination of Hauptmoduls and Construction of Abelian Extensions of Quadratic Number Fields

AbstractWe obtain Hauptmoduls of genus zero congruence subgroups of the type (p) := Γ0(p) + wp, where p is a prime and wp is the Atkin–Lehner involution. We then use the Hauptmoduls, along with modular functions on Γ1(p) to construct families of cyclic extensions of quadratic number fields. Further examples of cyclic extension of bi-quadratic and tri-quadratic number fields are also given.

Proving the triviality of rational points on Atkin–Lehner quotients of Shimura curves

In this paper we give a method for studying global rational points on certain quotients of Shimura curves by Atkin–Lehner involutions. We obtain explicit conditions on such quotients for rational points to be “trivial” (coming from CM points only) and exhibit an explicit infinite family of such quotients satisfying these conditions.