Atkin operator
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| Atkin operator | |
|---|---|
| Name | Atkin operator |
| Field | Number theory |
| Introduced | 1970s |
| Inventors | A. O. L. Atkin |
| Related | Hecke operators, Atkin–Lehner involutions, modular forms, newforms |
Atkin operator The Atkin operator is a linear operator arising in the theory of modular forms and arithmetic algebraic geometry, introduced by A. O. L. Atkin in work on congruence subgroups and computational techniques. It refines the action of Hecke algebra elements on spaces associated to congruence subgroups such as Gamma_0(N), interacts with Atkin–Lehner involution theory, and plays a role in the decomposition into oldforms and newforms used in explicit arithmetic and algorithmic studies.
Definition and basic properties
The Atkin operator is defined on spaces of cusp forms for a congruence subgroup like Gamma_0(N), often denoted U_p or V_p depending on normalization; it commutes with some elements of the Hecke algebra while failing to commute with others, so its algebraic properties give a finer module structure over rings connected to Hecke algebra. Its basic properties include linearity, compactness in p-adic completions related to the Coleman operator framework, and compatibility with level-raising and level-lowering maps used in the study of modular curves such as X_0(N). In many settings the Atkin operator intertwines with maps induced by degeneracy morphisms between Jacobians like J_0(N) and with correspondences used in the work of Shimura and Deligne.
Action on modular forms and q-expansions
On a q-expansion of a modular form f(z)=sum a_n q^n, the Atkin U_p operator typically maps coefficients via a_n ↦ a_{pn} or averages them across cosets associated to p-adic decomposition; this action can be compared with the action of classical Hecke operator T_n on Fourier coefficients. The operator preserves spaces such as S_k(Gamma_0(N)) and interacts with operators defined by Nebentypus characters appearing in the theory of Dirichlet characters and L-functions. Computations of q-expansions under U_p are central to algorithms that compute bases of modular forms used in the work of Cremona, Stein, and William Stein's software projects and build on structural results by Iwaniec and Kohnen.
Relationship with Hecke operators and Atkin–Lehner theory
The Atkin operator sits in the web of relations among Hecke operators and Atkin–Lehner involutions, providing criteria for distinguishing oldforms from newforms in the decomposition established by Atkin and Lehner. It interacts with the Eichler–Shimura relation and the multiplicity one theorem in contexts involving automorphic representations studied by Jacquet and Langlands. The compatibility and commutation relations with Hecke operators at primes not dividing the level are integral to proofs of level-structure results by Mazur and to modularity lifting theorems used by Wiles and Taylor.
Spectral theory and eigenforms
Spectral properties of the Atkin operator are studied in the context of eigenvalues and eigenspaces corresponding to p-adic and complex spectra; eigenforms for U_p correspond to p-stabilized newforms and to points on eigencurves constructed by Coleman and Buzzard. The operator is compact in many p-adic Banach space formulations used by Kisin and Emerton, leading to discrete spectra that underpin deformation theoretic approaches in the work of Mazur and to connections with Galois representations as explored by Serre and Fontaine.
Applications in computational number theory
Atkin operators are implemented in computational packages developed by projects like SageMath (originally by William Stein), PARI/GP (by Henri Cohen and collaborators), and software used by John Cremona for elliptic curve tables. They are used in calculating Fourier coefficients, computing spaces of modular forms, and verifying congruences appearing in databases such as the L-functions and Modular Forms Database (LMFDB). Algorithms leveraging U_p accelerate computations of modular symbols by methods influenced by Manin and optimizations introduced by Atkin and Morain in primality testing and point-counting for elliptic curves used in cryptography.
Generalizations and variants
Variants include V_p operators, Fricke involutions associated to primes dividing the level, and p-adic generalizations appearing in the theory of overconvergent modular forms by Buzzard and Coleman. Higher-rank analogues appear in the study of automorphic forms on GL_n and unitary groups investigated by Arthur and Harris, and in the work on eigenvarieties by Chenevier and Bellaïche. The operator also generalizes to correspondences on Shimura varieties as in research by Kottwitz and Harris–Taylor, and to p-adic families relevant to the Iwasawa theory studied by Greenberg and Kato.
Historical development and key contributors
Key contributors include A. O. L. Atkin and J. Lehner who formalized the decomposition of modular form spaces; later advances were made by John Tate in p-adic methods, Robert Coleman in overconvergent theory, and Kevin Buzzard on eigencurves. Foundational contexts were provided by Goro Shimura, Pierre Deligne, and Jean-Pierre Serre, while computational implementations and algorithmic refinements came from John Cremona, William Stein, Henri Cohen, François Morain, and others. Deep connections to modularity and arithmetic geometry were developed through work by Andrew Wiles, Richard Taylor, Barry Mazur, Kazuya Kato, and Mark Kisin.