On ℓ-adic representations for a space of noncongruence cuspforms

Article English OPEN
Hoffman, Jerome William ; Long, Ling ; Verrill, Helena (2012)

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:\ud \ud 1.\ud 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.\ud 2.\ud 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.\ud The key point is that the representation ρℓ admits a quaternion multiplication structure in the sense of Atkin, Li, Liu, and Long.
  • References (21)
    21 references, page 1 of 3

    [ALLL10] A. O. L. Atkin, W. C. Li, T. Liu, and L. Long, Galois representations with quaternion multiplications associated to noncongruence modular forms, arXiv:1005.4105 (2010).

    [ALL08] A. O. L. Atkin, W. C. Li, and L. Long, On Atkin and Swinnerton-Dyer congruence relations. II, Math. Ann. 340 (2008), no. 2, 335-358. MR2368983 (2009a:11102)

    [Cli37] A. H. Clifford, Representations induced in an invariant subgroup, Ann. of Math. (2) 38 (1937), no. 3, 533-550. MR1503352

    [DeRa] P. Deligne and M. Rapoport, Les sch´emas de modules de courbes elliptiques. Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), pp. 143-316. Lecture Notes in Math., Vol. 349, Springer, Berlin, 1973. MR0337993 (49:2762)

    [Del68] P. Deligne, Formes modulaires et repr´esentations l-adiques, S´em. Bourbaki, 355, 139- 172.

    [DS75] P. Deligne and J.-P. Serre, Formes modulaires de poids 1. Ann. Sci. E´cole Norm. Sup. (4) 7 (1974), 507-530 (1975). MR0379379 (52:284)

    [DS05] F. Diamond and J. Shurman, A first course in modular forms, Graduate Texts in Mathematics, vol. 228, Springer-Verlag, New York, 2005. MR2112196 (2006f:11045)

    [FHL08] L. Fang, J. W. Hoffman, B. Linowitz, A. Rupinski, and H. Verrill, Modular forms on noncongruence subgroups and Atkin-Swinnerton-Dyer relations, Experimental Mathematics 19, no. 1 (2010), 1-27. MR2649983

    [KM85] N. M. Katz and B. Mazur, Arithmetic moduli of elliptic curves. Annals of Mathematics Studies, 108. Princeton University Press, Princeton, NJ, 1985. MR772569 (86i:11024)

    [Lan72] R. P. Langlands, Modular forms and -adic representations. Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture Notes in Math., Vol. 349, Springer, Berlin, 1973, pp. 361-500. MR0354617 (50:7095)

  • Similar Research Results (1)
  • Metrics
    0
    views in OpenAIRE
    0
    views in local repository
    54
    downloads in local repository

    The information is available from the following content providers:

    From Number Of Views Number Of Downloads
    Warwick Research Archives Portal Repository - IRUS-UK 0 54
Share - Bookmark