8,501 research outputs found
Hypergeometric evaluation identities and supercongruences
In this article, we provide an application of hypergeometric evaluation
identities, including a strange valuation of Gosper, to prove several
supercongruences related to special valuations of truncated hypergeometric
series. In particular, we prove a conjecture of van Hamme
On Atkin and Swinnerton-Dyer congruence relations (3)
In the previous two papers with the same title ([LLY05] by W.C. Li, L. Long,
Z. Yang and [ALL05] by A.O.L. Atkin, W.C. Li, L. Long), the authors have
studied special families of cuspforms for noncongruence arithmetic subgroups.
It was found that the Fourier coefficients of these modular forms at infinity
satisfy three-term Atkin and Swinnerton-Dyer congruence relations which are the
-adic analogue of the three-term recursions satisfied by the coefficients of
classical Hecke eigenforms.
In this paper, we first consider Atkin and Swinnerton-Dyer type congruences
which generalize the three-term congruences above. These weaker congruences are
satisfied by cuspforms for special noncongruence arithmetic subgroups. Then we
will exhibit an infinite family of noncongruence cuspforms, each of which
satisfies three-term Atkin and Swinnerton-Dyer type congruences for almost
every prime . Finally, we will study a particular space of noncongruence
cuspforms. We will show that the attached -adic Scholl representation is
isomorphic to the -adic representation attached to a classical automorphic
form. Moreover, for each of the four residue classes of odd primes modulo 12
there is a basis so that the Fourier coefficients of each basis element satisfy
three-term Atkin and Swinnerton-Dyer congruences in the stronger original
sense
On modular forms for some noncongruence subgroups of SL_2(Z) II
In this paper we show two classes of noncongruence subgroups satisfy the
so-called unbounded denominator property. In particular, we establish our
conjecture in [KL08] which says that every type II noncongruence character
group of Gamma^0(11) satisfies the unbounded denominator property
On ℓ-adic representations for a space of noncongruence cuspforms
This paper is concerned with a compatible family of 4-dimensional ℓ-adic representations ρℓ of GQ := Gal(Q/Q) attached to the space of weight-3 cuspforms S3(Γ) on a noncongruence subgroup Γ ⊂ SL2(Z). For this representation we prove that:
1.
It is automorphic: the L-function L(s,ρℓ∨) agrees with the L-function for an automorphic form for GL4(AQ), where ρℓ∨ is the dual of ρℓ.
2.
For each prime p≥5 there is a basis hp = {hp+, hp-} of S3(Γ) whose expansion coefficients satisfy 3-term Atkin and Swinnerton-Dyer (ASD) relations, relative to the q-expansion coefficients of a newform f of level 432. The structure of this basis depends on the class of p modulo 12.
The key point is that the representation ρℓ admits a quaternion multiplication structure in the sense of Atkin, Li, Liu, and Long
On l-adic representations for a space of noncongruence cuspforms
This paper is concerned with a compatible family of 4-dimensional \ell-adic
representations \rho_{\ell} of G_\Q:=\Gal(\bar \Q/\Q) attached to the space of
weight 3 cuspforms S_3 (\Gamma) on a noncongruence subgroup \Gamma \subset \SL.
For this representation we prove that: 1.)It is automorphic: the L-function
L(s, \rho_{\ell}^{\vee}) agrees with the L-function for an automorphic form for
\text{GL}_4(\mathbb A_{\Q}), where \rho_{\ell}^{\vee} is the dual of
\rho_{\ell}. 2.) For each prime p \ge 5 there is a basis h_p = \{h_p ^+, h_p ^-
\} of S_3 (\Gamma) whose expansion coefficients satisfy 3-term Atkin and
Swinnerton-Dyer (ASD) relations, relative to the q-expansion coefficients of a
newform f of level 432. The structure of this basis depends on the class of p
modulo 12. The key point is that the representation admits a
quaternion multiplication structure in the sense of a recent work of Atkin, Li,
Liu and Long.Comment: Second revised version. To appear: Proceedings of the American
Mathematical Societ
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