190,129 research outputs found
On ℓ-adic representations for a space of noncongruence cuspforms
No full textThis 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
Atkin, P, VX6402
No full textThis record was harvested from a previous catalogue system and will be withdrawn in 2025. Information in this record may be superseded or incomplete. Visit this record in UMA's new catalogue at: https://archives.library.unimelb.edu.au/nodes/view/369281Surname: ATKIN
Given Name(s) or Initials: P
Military Service Number or Last Known Location: VX6402
Missing, Wounded and Prisoner of War Enquiry Card Index Number: 6370179323
Item: [2016.0049.01608] "Atkin, P, VX6402
Atkin, M N P (Malcolm Northcde Palaster), NX55753
No full textThis record was harvested from a previous catalogue system and will be withdrawn in 2025. Information in this record may be superseded or incomplete. Visit this record in UMA's new catalogue at: https://archives.library.unimelb.edu.au/nodes/view/369277Surname: ATKIN
Given Name(s) or Initials: M N P (MALCOLM NORTHCDE PALASTER)
Military Service Number or Last Known Location: NX55753
Missing, Wounded and Prisoner of War Enquiry Card Index Number: 16580179319
Item: [2016.0049.01604] "Atkin, M N P (Malcolm Northcde Palaster), NX55753
Atkin Nicholas,Pétain
No full textKitson Simon. Atkin Nicholas,Pétain. In: Vingtième Siècle, revue d'histoire, n°58, avril-juin 1998. p. 186
On Atkin and Swinnerton-Dyer congruence relations (2)
No full textIn 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 modulo 8 there is a basis whose Fourier coefficients at infinity satisfy a three-term Atkin and Swinnerton-Dyer congruence relation, which is the p-adic analogue of the three-term recursion satisfied by the coefficients of classical Hecke eigenforms. We also show that there is an automorphic L-function over mathbb{Q} whose local factors agree with those of the l-adic Scholl representations attached to the space of noncongruence cusp forms. © 2007 Springer-Verlag
On the distribution of Atkin and Elkies primes for reductions of elliptic curves on average
No full textFor an elliptic curve E/Q without complex multiplication we study the distribution of Atkin and Elkies primes ℓ, on average, over all good reductions of E modulo primes p. We show that, under the generalized Riemann hypothesis, for almost all primes p there are enough small Elkies primes ℓ to ensure that the Schoof-Elkies-Atkin point-counting algorithm runs in (log p)⁴⁺⁰⁽¹⁾ expected time.15 page(s
On Atkin and Swinnerton-Dyer congruence relations (3)
No full textAbstractIt is now well known that Hecke operators defined classically act trivially on genuine cuspforms for noncongruence subgroups of SL2(Z). Atkin and Swinnerton-Dyer speculated the existence of p-adic Hecke operators so that the Fourier coefficients of their eigenfunctions satisfy three-term congruence recursions. In the previous two papers with the same title ([W.C. Li, L. Long, Z. Yang, On Atkin and Swinnerton-Dyer congruence relations, J. Number Theory 113 (1) (2005) 117–148] by W.C. Li, L. Long, Z. Yang and [A.O.L. Atkin, W.C. Li, L. Long, On Atkin and Swinnerton-Dyer congruence relations (2), Math. Ann. 340 (2) (2008) 335–358] by A.O.L. Atkin, W.C. Li, L. Long), the authors have studied two exceptional spaces of noncongruence cuspforms where almost all p-adic Hecke operators can be diagonalized simultaneously or semi-simultaneously. Moreover, it is shown that the l-adic Scholl representations attached to these spaces are modular in the sense that they are isomorphic, up to semisimplification, to the l-adic representations arising from classical automorphic forms.In this paper, we study an infinite family of spaces of noncongruence cuspforms (which includes the cases in [W.C. Li, L. Long, Z. Yang, On Atkin and Swinnerton-Dyer congruence relations, J. Number Theory 113 (1) (2005) 117–148; A.O.L. Atkin, W.C. Li, L. Long, On Atkin and Swinnerton-Dyer congruence relations (2), Math. Ann. 340 (2) (2008) 335–358]) under a general setting. It is shown that for each space in this family there exists a fixed basis so that the Fourier coefficients of each basis element satisfy certain weaker three-term congruence recursions. For a new case in this family, we will exhibit that the attached l-adic Scholl representations are modular and the p-adic Hecke operators can be diagonalized semi-simultaneously
The group structure of the normalizer of Γ0(N) after Atkin-Lehner
No full textWe determine the group structure of the normalizer of Γ0 (N) in SL2(R) modulo Γ0 (N). These results correct the Atkin-Lehner statement (Atkin and Lehner, 1970, Theorem 8)
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