Atkin-Lehner Involutions Research Articles

On average theta functions of certain quadratic forms as sums of Eisenstein series

Abstract Let Q Q be an integral positive definite quadratic form of level N N in 2 k 2k variables. Further, we assume that ( − 1 ) k N {\left(-1)}^{k}N is a fundamental discriminant. We express the average theta function of Q Q as an explicit sum of Eisenstein series, which are Hecke eigenforms by applying Atkin-Lehner involution.

Modular curves with many points over finite fields

We describe an algorithm to compute the number of points over finite fields on a broad class of modular curves: we consider quotients XH/W for H a subgroup of GL2(Z/nZ) such that for each prime p dividing n, the subgroup H at p is either a Borel subgroup, a Cartan subgroup, or the normalizer of a Cartan subgroup of GL2(Z/peZ), and for W any subgroup of the Atkin-Lehner involutions of XH. We applied our algorithm to more than ten thousand curves of genus up to 50, finding more than one hundred record-breaking curves, namely curves X/Fq with genus g that improve the previously known lower bound for the maximum number of points over Fq of a curve with genus g. As a key technical tool for our computations, we prove the generalization of Chen's isogeny to all the Cartan modular curves of composite level.

Bielliptic quotient modular curves of 𝑋₀(𝑁)

Let N ≥ 1 N\geq 1 be a non-square free integer and let W N W_N be a nontrivial subgroup of the group of the Atkin-Lehner involutions of X 0 ( N ) X_0(N) such that the modular curve X 0 ( N ) / W N X_0(N)/W_N has genus at least two. We determine all pairs ( N , W N ) (N,W_N) such that X 0 ( N ) / W N X_0(N)/W_N is a bielliptic curve and the pairs ( N , W N ) (N,W_N) such that X 0 ( N ) / W N X_0(N)/W_N has an infinite number of quadratic points over Q \mathbb {Q} .

Automorphisms of Cartan modular curves of prime and composite level

We study the automorphisms of modular curves associated to Cartan subgroups of $\mathrm{GL}_2(\mathbb Z/n\mathbb Z)$ and certain subgroups of their normalizers. We prove that if $n$ is large enough, all the automorphisms are induced by the ramified covering of the complex upper half-plane. We get new results for non-split curves of prime level $p\ge 13$: the curve $X_{\text{ns}}^+(p)$ has no non-trivial automorphisms, whereas the curve $X_{\text{ns}}(p)$ has exactly one non-trivial automorphism. Moreover, as an immediate consequence of our results we compute the automorphism group of $X_0^*(n):=X_0(n)/W$, where $W$ is the group generated by the Atkin-Lehner involutions of $X_0(n)$ and $n$ is a large enough square.

Dimension formulas for modular form spaces with character for Fricke groups

We first obtain the dimension formulas for the spaces of holomorphic modular forms with character for the Fricke group $\Gamma _0^+(N)$, and then for $\Gamma _0^*(N)$ with all Atkin–Lehner involutions added in a particular case.

Weierstrass points on modular curves X0(N) fixed by the Atkin–Lehner involutions

PurposeThe authors have determined whether the points fixed by all the full and the partial Atkin–Lehner involutions WQ on X0(N) for N ≤ 50 are Weierstrass points or not.Design/methodology/approachThe design is by using Lawittes's and Schoeneberg's theorems.FindingsFinding all Weierstrass points on X0(N) fixed by some Atkin–Lehner involutions. Besides, the authors have listed them in a table.Originality/valueThe Weierstrass points have played an important role in algebra. For example, in algebraic number theory, they have been used by Schwartz and Hurwitz to determine the group structure of the automorphism groups of compact Riemann surfaces of genus g ≥ 2. Whereas in algebraic geometric coding theory, if one knows a Weierstrass nongap sequence of a Weierstrass point, then one is able to estimate parameters of codes in a concrete way. Finally, the set of Weierstrass points is useful in studying arithmetic and geometric properties of X0(N).

Atkin–Lehner theory for Drinfeld modular forms and applications

The present paper deals with Atkin–Lehner theory for Drinfeld modular forms. We provide an equivalent definition of \(\mathfrak {p}\)-newforms (which makes computations easier) and commutativity results between Hecke operators and Atkin–Lehner involutions. As applications, we show a criterion for a direct sum decomposition of cusp forms, we exhibit \(\mathfrak {p}\)-newforms arising from lower levels and we provide \(\mathfrak {p}\)-adic Drinfeld modular forms of level greater than 1.

The automorphism group of the modular curve $X_0^*(N)$ with square-free level

We determine the automorphism group of the modular curve $X_0^*(N)$, obtained as the quotient of the modular curve $X_0(N)$ by the group of its Atkin-Lehner involutions, for all square-free values of $N$.

The maximal discrete extension of the Hermitian modular group

Let \Gamma_n(\mathcal{O}_{\mathbb{K}}) denote the Hermitian modular group of degree n over an imaginary-quadratic number field \mathbb{K} . In this paper we determine its maximal discrete extension in \operatorname{SU}(n, n; \mathbb{C}) , which coincides with the normalizer of \Gamma_n(\mathcal{O}_{\mathbb{K}}) . The description involves the n -torsion subgroup of the ideal class group of \mathbb{K} . This group is defined over a particular number field \widehat{\mathbb{K}}_n and we can describe the ramified primes in it. In the case n=2 we give an explicit description, which involves generalized Atkin-Lehner involutions. Moreover we find a natural characterization of this group in \operatorname{SO}(2,4) .

An intriguing hyperelliptic Shimura curve quotient of genus 16

Let $F$ be the maximal totally real subfield of $\mathbf{Q}(\zeta_{32})$, the cyclotomic field of $32$nd roots of unity. Let $D$ be the quaternion algebra over $F$ ramified exactly at the unique prime above $2$ and 7 of the real places of $F$. Let $\mathcal{O}$ be a maximal order in $D$, and $X_0^D(1)$ the Shimura curve attached to $\mathcal{O}$. Let $C = X_0^D(1)/\langle w_D \rangle$, where $w_D$ is the unique Atkin-Lehner involution on $X_0^D(1)$. We show that the curve $C$ has several striking features. First, it is a hyperelliptic curve of genus $16$, whose hyperelliptic involution is exceptional. Second, there are $34$ Weierstrass points on $C$, and exactly half of these points are CM points; they are defined over the Hilbert class field of the unique CM extension $E/F$ of class number $17$ contained in $\mathbf{Q}(\zeta_{64})$, the cyclotomic field of $64$th roots of unity. Third, the normal closure of the field of $2$-torsion of the Jacobian of $C$ is the Harbater field $N$, the unique Galois number field $N/\mathbf{Q}$ unramified outside $2$ and $\infty$, with Galois group $\mathrm{Gal}(N/\mathbf{Q})\simeq F_{17} = \mathbf{Z}/17\mathbf{Z} \rtimes (\mathbf{Z}/17\mathbf{Z})^\times$. In fact, the Jacobian $\mathrm{Jac}(X_0^D(1))$ has the remarkable property that each of its simple factors has a $2$-torsion field whose normal closure is the field $N$. Finally, and perhaps the most striking fact about $C$, is that it is also hyperelliptic over $\mathbf{Q}$.

Bielliptic quotient modular curves with N square-free

Let N≥1 be an square free integer and let WN be a non-trivial subgroup of the group of the Atkin-Lehner involutions of X0(N) such that the modular curve X0(N)/WN has genus at least two. We determine all pairs (N,WN) such that X0(N)/WN is a bielliptic curve and the pairs (N,WN) such that X0(N)/WN has an infinite number of quadratic points over Q.

Shimura and Shintani liftings of certain cusp forms of half-integral and integral weights

In [4], W. Kohnen constructed explicit Shintani lifts from the space of cusp forms of weight $2k$ on $Γ_0(N)$, which are mapped into the plus space $S^+_{k+1/2}(Γ_0(4N))$, where $N$ is an odd integer. This lifting is adjoint to the (modified) Shimura map defined by him with respect to the Petersson scalar product. Using this construction along with the theory of newforms on $S^+_{k+1/2}(Γ_0(4N))$ (where $N$ is an odd square-free natural number) developed in [3], he derived explicit Waldspurger theorem for the newforms belonging to the space $S^+_{k+1/2}(Γ_0(4N))$. Further in this work, Kohnen considered the $\pm 1$ eigen subspaces corresponding to the Atkin-Lehner involution and showed that the intersection of this $\pm$ spaces with the corresponding newform spaces are isomorphic under the Shimura correspondence. A natural question is whether a similar result can be obtained for the intersection of this $(\pm1)$ subspaces with the space of oldforms. In this direction, S. Choi and C. H. Kim [2] considered the case where $N$ is an odd prime $p$ and constructed similar Shimura and Shintani maps between subspaces of forms of half-integral and integral weights. The subspaces considered by Choi and Kim are nothing but $+1$ eigenspace under the Fricke involution. In this paper, we generalise the work of Choi and Kim to the case where $N$ is an odd square-free natural number.

Newforms mod 𝑝 in squarefree level with applications to Monsky’s Hecke-stable filtration

We propose an algebraic definition of the space of ℓ \ell -new mod- p p modular forms for Γ 0 ( N ℓ ) \Gamma _0(N\ell ) in the case that ℓ \ell is prime to N N , which naturally generalizes to a notion of newforms modulo p p in squarefree level. We use this notion of newforms to interpret the Hecke algebras on the graded pieces of the space of mod- 2 2 level- 3 3 modular forms described by Paul Monsky. Along the way, we describe a renormalized version of the Atkin-Lehner involution: no longer an involution, it is an automorphism of the algebra of modular forms, even in characteristic p p .

The maximal discrete extension of $$SL_2(\mathcal {\scriptstyle {O}}_K)$$ S L 2 ( O K ) for an imaginary quadratic number field K

Let $$\mathcal {\scriptstyle {O}}_K$$ be the ring of integers of an imaginary quadratic number field K. In this paper we give a new description of the maximal discrete extension of the group $$SL_2(\mathcal {\scriptstyle {O}}_K)$$ inside $$SL_2(\mathbb {C})$$ , which uses generalized Atkin–Lehner involutions. Moreover we find a natural characterization of this group in SO(1, 3).

Ramification of the Eigencurve at Classical RM Points

AbstractJ. Bellaïche and M. Dimitrov showed that the$p$-adic eigencurve is smooth but not étale over the weight space at$p$-regular theta series attached to a character of a real quadratic field$F$in which$p$splits. In this paper we prove the existence of an isomorphism between the subring fixed by the Atkin–Lehner involution of the completed local ring of the eigencurve at these points and a universal ring representing a pseudo-deformation problem. Additionally, we give a precise criterion for which the ramification index is exactly 2. We finish this paper by proving the smoothness of the nearly ordinary and ordinary Hecke algebras for Hilbert modular forms over$F$at the overconvergent cuspidal Eisenstein points, being the base change lift for$\text{GL}(2)_{/F}$of these theta series. Our approach uses deformations and pseudo-deformations of reducible Galois representations.

On Bhargava’s Heuristics for GL2(𝔽 p )-Number Fields and the Number of Elliptic Curves of Bounded Conductor

We propose a modular forms-based model for counting -number fields having the same local properties as the splitting field of the mod p-Galois representation associated with an elliptic curve over the rational numbers. We explain how this modular forms-based model and Bhargava’s local-to-global heuristics for counting -number fields both shed light on the problem of estimating the number of elliptic curves over the rational numbers of squarefree conductor The modular forms-based model predicts the existence of significantly more -number fields with the desired local properties than does the local-to-global model when N has a large number of distinct prime factors, owing to degeneracies caused by Atkin–Lehner involutions. We describe computational evidence supporting the modular forms-based model.

Computing $\protect \mathcal{L}$-invariants via the Greenberg–Stevens formula

In this article, we describe how to compute slopes of p-adic ℒ-invariants of Hecke eigenforms of arbitrary weight and level by means of the Greenberg–Stevens formula. Our method is based on the work of Lauder and Vonk on computing the reverse characteristic series of the U p -operator on overconvergent modular forms. Using higher derivatives of this series, we construct a polynomial whose roots are precisely the ℒ-invariants appearing in the corresponding space of modular forms with fixed sign of the Atkin–Lehner involution at p. In addition, we describe how to compute this polynomial efficiently. In the final section, we give computational evidence for relations between slopes of ℒ-invariants of different levels and weights for small primes p.

Rational torsion in elliptic curves and the cuspidal subgroup

Let A be an elliptic curve over ℚ of square free conductor N that has a rational torsion point of prime order r such that r does not divide 6N. We show that then r divides the order of the cuspidal subgroup C of J 0 (N). If A is optimal, then viewing A as an abelian subvariety of J 0 (N), our proof shows more precisely that r divides the order of A∩C. Also, under the hypotheses above minus the hypothesis that r does not divide N, we show that for some prime p that divides N, the eigenvalue of the Atkin–Lehner involution W p acting on the newform associated to A is -1.

Nonvanishing of Central Values of L-functions of Newforms in S2(Γ0(dp2)) Twisted by Quadratic Characters

AbstractWe prove that for d ∊ {2, 3, 5, 7, 13} and K a quadratic (or rational) field of discriminant D and Dirichlet character 𝜒, if a prime p is large enough compared to D, there is a new form f ∊ S2Γ0(dp2)) with sign (+1) with respect to the Atkin–Lehner involution wp2 such that L( f ⊗ 𝜒, •)≠ 0. This result is obtained through an estimate of a weighted sum of twists of L-functions that generalises a result of Ellenberg. It relies on the approximate functional equation for the L-functions L( f ⊗ 𝜒, · ) and a Petersson trace formula restricted to Atkin–Lehner eigenspaces. An application of this nonvanishing theorem will be given in terms of existence of rank zero quotients of some twisted jacobians, which generalises a result of Darmon and Merel.

Twisting of Siegel paramodular forms

Let Sk(Γpara(N)) be the space of Siegel paramodular forms of level N and weight k. Fix an odd prime p ∤ N and let χ be a nontrivial quadratic Dirichlet character mod p. Based on [Twisting of paramodular vectors, Int. J. Number Theory 10 (2014) 1043–1065], we define a linear twisting map 𝒯χ : Sk(Γpara(N)) → Sk(Γpara(Np4)). We calculate an explicit expression for this twist, give the commutation relations of this map with the Hecke operators and Atkin–Lehner involution for primes ℓ ≠p, and prove that the L-function of the twist has the expected form.