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, W. Li, L. Long
  • 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… 
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).
Fourier coefficients of noncongruence cuspforms
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)
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
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
The unbounded denominator conjecture for the noncongruence subgroups of index 7
A Whipple $$_7F_6$$ Formula Revisited
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)
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
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
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
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
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
...
...