Atkin-Lehner Involutions Research Articles

Algebraic hypergeometric transformations of modular origin

It is shown that Ramanujan's cubic transformation of the Gauss hypergeometric function 2 F 1 arises from a relation between modular curves, namely the covering of X 0 (3) by X 0 (9). In general, when 2≤ N≤ 7, the N-fold cover of X 0 (N) by X 0 (N 2 ) gives rise to an algebraic hypergeometric transformation. The N = 2,3,4 transformations are arithmetic-geometric mean iterations, but the N = 5,6,7 transformations are new. In the final two cases the change of variables is not parametrized by rational functions, since X 0 (6),X 0 (7) are of genus 1. Since their quotients X 0 + (6),X 0 + (7) under the Fricke involution (an Atkin-Lehner involution) are of genus 0, the parametrization is by two-valued algebraic functions. The resulting hypergeometric transformations are closely related to the two-valued modular equations of Fricke and H. Cohn.

Broué–Enguehard maps and Atkin–Lehner involutions

Let ℓ be one of the ten integers such that the sum of their divisors divide 24. For each such ℓ , (except 15) we give a map from an algebra of polynomial invariants of some finite group to the algebra of modular forms invariant under the Atkin–Lehner group of level ℓ . These maps are motivated and inspired by constructions of modular lattices from self-dual codes over rings. This work generalizes Broué–Enguehard work in level one and three obtained from binary and ternary codes.

Trigonal quotients of modular curves $X_{0}(N)$

Let $W(N)$ be the group of Atkin-Lehner involutions on the modular curve $X_0(N)$. The purpose of this article is to give complementary result to [7, 8, 9]; namely, we determine trigonal curves of the form $X_0(N)/W'$, where $W'$ is a subgroup of $W(N)$ such that $2< |W'| < |W(N)|$.

Failure of the Hasse Principle for Atkin–Lehner Quotients of Shimura Curves over Q

We show how to construct counter-examples to the Hasse principle over the field of rational numbers on Atkin–Lehner quotients of Shimura curves and on twisted forms of Shimura curves by Atkin–Lehner involutions. A particular example is the quotient of the Shimura curve X23∙107 attached to the indefinite rational quaternion algebra of discriminant 23∙107 by the Atkin–Lehner involution ω107. The quadratic twist of X23∙107 by Q( √ −23) with respect to this involution is also a counter-example to the Hasse principle over Q.

A CLASSICAL CHARACTERIZATION OF NEWFORMS WITH EQUIVALENT EIGENFORMS IN ${S_{k + 1/2}}(4N, \chi)$

The paper continues the investigation of the Hecke structure of spaces of half-integral weight cusp forms ${S_{k + 1/2}}(4N, \chi)$ , where $k$ and $N$ are positive integers with $N$ odd, and $\chi$ is an even quadratic Dirichlet character modulo $4N$ . In the Hecke decomposition of these spaces, contributions are determined arising from newforms which are quadratic twists of newforms at lower levels. Combining this result with its counterpart in a previous paper regarding non-twists gives this paper's main result: necessary and sufficient conditions under which a given newform has equivalent cusp forms in ${S_{k + 1/2}}(4N, \chi)$ . The result reformulates, in classical terms, the representation-theoretic conditions given by Flicker and Waldspurger. The conditions involve easily-verified information about the primes dividing the level of the newform, and about the behavior of the newform under certain quadratic twists and Atkin–Lehner involutions. The theorem is applied to give explicit examples of twisted newforms having no equivalent half-integral weight cusp forms in any space ${S_{k + 1/2}}(4N, \chi)$ as above.

Modular Shimura varieties and forgetful maps

In this note we consider several maps that occur naturally between modular Shimura varieties, Hilbert-Blumenthal varieties and the moduli spaces of polarized abelian varieties when forgetting certain endomorphism structures. We prove that, up to birational equivalences, these forgetful maps coincide with the natural projection by suitable abelian groups of Atkin-Lehner involutions.

On the automorphism groups of some modular curves

Let $N$ be a prime number, and $X$ a curve that is intermediate to the cover $X_1(N)\rightarrow X_0(N)$. We study the automorphism group of $X$, and prove that in most cases it is generated by the Galois group of the cover $X\rightarrow X_0(N)$, and any lift of the Atkin–Lehner involution to $X$.

On Drinfeld Modular Curves with Many Rational Points over Finite Fields

It is known that Drinfeld modular curves can be used to construct asymptotically optimal towers of curves over finite fields. Using reductions of the Drinfeld modular curves X 0( n ), we try to find individual curves over finite fields with many rational points. The main idea is to divide by an Atkin–Lehner involution which has many fixed points in order to obtain a quotient with a better ratio #{rational points}/genus. In a few cases we can improve the known records of rational points.

On Drinfeld Modular Curves with Many Rational Points over Finite Fields

It is known that Drinfeld modular curves can be used to construct asymptotically optimal towers of curves over finite fields. Using reductions of the Drinfeld modular curves X 0 ( n ), we try to find individual curves over finite fields with many rational points. The main idea is to divide by an Atkin–Lehner involution which has many fixed points in order to obtain a quotient with a better ratio #{rational points}/genus. In a few cases we can improve the known records of rational points.

Rational and Elliptic Parametrizations of Q-Curves

We describe explicit parametrizations of the rational points of X*( N), the algebraic curve obtained as quotient of the modular curve X 0( N) by the group B( N) generated by the Atkin–Lehner involutions, whenever Nis square-free and the curve is rational or elliptic. By taking into account the moduli interpretation of X*( N), along with a standard “boundedness” conjecture, we obtain all the Q-isogeny classes of Q-curves except for a finite set.

On the Q-rational cuspidal subgroup and the component group ofJ 0(p r )

Forp≥3 a prime, we compute theQ-rational cuspidal subgroupC(p r ) of the JacobianJ 0(p r ) of the modular curveX 0(p r ). This result is then applied to determine the component group Φ p r of the Neron model ofJ 0(p r ) overZ p . This extends results of Lorenzini [7]. We also study the action of the Atkin-Lehner involution on thep-primary part ofC(p r ), as well as the effect of degeneracy maps on the component groups.

On type numbers of split orders of definite quaternion algebras

In this paper, we shall give a new relation between the arithmetic of quaternion algebras and modular forms; we shall express the type numberT q, N of a split order of type (q, N) as the sums of dimensions of some subspaces of the space of cusp forms of weight 2 with respect to Γ0(qN) which are common eigenspaces of Atkin-Lehner's involutions.

Atkin-Lehner involutions and class number residuality

Atkin-Lehner involutions and class number residuality