# Galois representations with quaternion multiplication associated to noncongruence modular forms

@article{Atkin2010GaloisRW,
title={Galois representations with quaternion multiplication associated to noncongruence modular forms},
author={A. O. L. Atkin and Wen-Ching Winnie Li and Tongyin Liu and Ling Long},
journal={Transactions of the American Mathematical Society},
year={2010},
volume={365},
pages={6217-6242}
}
• A. Atkin
• Published 22 May 2010
• Mathematics
• Transactions of the American Mathematical Society
In this paper we study the compatible family of degree-4 Scholl representations � associated with a space S of weight � > 2 noncongruence cusp forms satisfying Quaternion Mul- tiplications over a biquadratic extension of Q. It is shown that � is automorphic, that is, its associated L-function has the same Euler factors as the L-function of an automorphic form for GL4 over Q. Further, it yields a relation between the Fourier coefficients of noncongruence cusp forms in S and those of certain…
10 Citations
On ℓ-adic representations for a space of noncongruence cuspforms
• Mathematics
• 2010
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).
Fourier coefficients of noncongruence cuspforms
• Mathematics
• 2012
Given a finite index subgroup of SL2(ℤ) with modular curve defined over ℚ, under the assumption that the space of weight k (⩾2) cuspforms is one‐dimensional, we show that a form in this space with
Atkin and Swinnerton-Dyer congruences and noncongruence modular forms (Algebraic Number Theory and Related Topics 2012)
• Mathematics
• 2013
Atkin and Swinnerton-Dyer congruences are special congruence recursions satisfied by coefficients of noncongruence modular forms. These are in some sense $p$-adic analogues of Hecke recursion
Potentially $GL_2$-type Galois representations associated to noncongruence modular forms
• Mathematics
Transactions of the American Mathematical Society
• 2018
In this paper, we consider Galois representations of the absolute Galois group $\text{Gal}(\overline {\mathbb Q}/\mathbb Q)$ attached to modular forms for noncongruence subgroups of
Moduli interpretations for noncongruence modular curves
We consider the moduli of elliptic curves with G-structures, where G is a finite 2-generated group. When G is abelian, a G-structure is the same as a classical congruence level structure. There is a
A Whipple $$_7F_6$$ Formula Revisited
• Mathematics
La Matematica
• 2022
A well-known formula of Whipple relates certain hypergeometric values $_7F_6(1)$ and $_4F_3(1)$. In this paper we revisit this relation from the viewpoint of the underlying hypergeometric data $HD$,

## References

SHOWING 1-10 OF 63 REFERENCES
On Atkin and Swinnerton-Dyer congruence relations (2)
• Mathematics
• 2005
In this paper we give an example of a noncongruence subgroup whose three-dimensional space of cusp forms of weight 3 has the following properties. For each of the four residue classes of odd primes
On ℓ-adic representations for a space of noncongruence cuspforms
• Mathematics
• 2010
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).
On Some l‐Adic Representations of Gal( (Q¯/Q) Attached to Noncongruence Subgroups
The l‐adic parabolic cohomology groups attached to noncongruence subgroups of SL2(Z) are finite‐dimensional l‐adic representations of Gal(Q¯/K) for some number field K. We exhibit examples (with K =
Modularity of abelian surfaces with quaternionic multiplication
We prove that any abelian surface defined over Q of GL2-type having quaternionic multiplication and good reduction at 3 is modular. We generalize the result to higher dimensional abelian varieties
On the Brauer Class of Modular Endomorphism Algebras
• Mathematics
• 2005
In this paper we study the Brauer class of the endomorphism algebra Xf of the motive attached to a primitive elliptic modular cusp form f without complex multiplication (CM). Our study includes the
Potential automorphy and change of weight
• Mathematics
• 2010
We prove an automorphy lifting theorem for l-adic representations where we impose a new condition at l, which we call \potentential diagonalizability." This result allows for \change of weight" and
Modularity of 2-dimensional Galois representations
Our aim is to explain some recent results on modularity of 2-dimensional potentially Barsotti-Tate Galois representations. That such representations should arise from modular forms is a special case
Fourier coefficients of noncongruence cuspforms
• Mathematics
• 2012
Given a finite index subgroup of SL2(ℤ) with modular curve defined over ℚ, under the assumption that the space of weight k (⩾2) cuspforms is one‐dimensional, we show that a form in this space with
Vanishing cycles and non-classical parabolic cohomology
The work described here began as an attempt to understand the structure of the cohomology groups associated to a subgroup ? of nite index of SL 2 (Z) which are not congruence subgroups. One knows