Overconvergent generalised eigenforms of weight one and class fields of real quadratic fields
This article examines the Fourier expansions of certain non-classical <em>p</em>-adic modular forms of weight one: the eponymous <em>generalised eigenforms</em> of the title, so called because they lie in a generalised eigenspace for the Hecke operators. When this generalised...
| Main Authors: | , , |
|---|---|
| Format: | Journal article |
| Published: |
Elsevier
2015
|
| _version_ | 1797062501527977984 |
|---|---|
| author | Lauder, A Darmon, H Rotger, V |
| author_facet | Lauder, A Darmon, H Rotger, V |
| author_sort | Lauder, A |
| collection | OXFORD |
| description | This article examines the Fourier expansions of certain non-classical <em>p</em>-adic modular forms of weight one: the eponymous <em>generalised eigenforms</em> of the title, so called because they lie in a generalised eigenspace for the Hecke operators. When this generalised eigenspace contains the theta series attached to a character of a real quadratic field <em>K</em> in which the prime <em>p</em> splits, the coe!cients of the attendant generalised eigenform are expressed as <em>p</em>-adic logarithms of algebraic numbers belonging to an idoneous ring class field of <em>K</em>. This suggests an approach to “explicit class field theory” for real quadratic fields which is simpler than the one based on Stark’s conjecture or its <em>p</em>-adic variants, and is perhaps closer in spirit to the classical theory of singular moduli. |
| first_indexed | 2024-03-06T20:46:25Z |
| format | Journal article |
| id | oxford-uuid:3610fdac-ae77-40eb-a6e2-c17081575c9f |
| institution | University of Oxford |
| last_indexed | 2024-03-06T20:46:25Z |
| publishDate | 2015 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | oxford-uuid:3610fdac-ae77-40eb-a6e2-c17081575c9f2022-03-26T13:35:31ZOverconvergent generalised eigenforms of weight one and class fields of real quadratic fieldsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:3610fdac-ae77-40eb-a6e2-c17081575c9fSymplectic Elements at OxfordElsevier2015Lauder, ADarmon, HRotger, VThis article examines the Fourier expansions of certain non-classical <em>p</em>-adic modular forms of weight one: the eponymous <em>generalised eigenforms</em> of the title, so called because they lie in a generalised eigenspace for the Hecke operators. When this generalised eigenspace contains the theta series attached to a character of a real quadratic field <em>K</em> in which the prime <em>p</em> splits, the coe!cients of the attendant generalised eigenform are expressed as <em>p</em>-adic logarithms of algebraic numbers belonging to an idoneous ring class field of <em>K</em>. This suggests an approach to “explicit class field theory” for real quadratic fields which is simpler than the one based on Stark’s conjecture or its <em>p</em>-adic variants, and is perhaps closer in spirit to the classical theory of singular moduli. |
| spellingShingle | Lauder, A Darmon, H Rotger, V Overconvergent generalised eigenforms of weight one and class fields of real quadratic fields |
| title | Overconvergent generalised eigenforms of weight one and class fields of real quadratic fields |
| title_full | Overconvergent generalised eigenforms of weight one and class fields of real quadratic fields |
| title_fullStr | Overconvergent generalised eigenforms of weight one and class fields of real quadratic fields |
| title_full_unstemmed | Overconvergent generalised eigenforms of weight one and class fields of real quadratic fields |
| title_short | Overconvergent generalised eigenforms of weight one and class fields of real quadratic fields |
| title_sort | overconvergent generalised eigenforms of weight one and class fields of real quadratic fields |
| work_keys_str_mv | AT laudera overconvergentgeneralisedeigenformsofweightoneandclassfieldsofrealquadraticfields AT darmonh overconvergentgeneralisedeigenformsofweightoneandclassfieldsofrealquadraticfields AT rotgerv overconvergentgeneralisedeigenformsofweightoneandclassfieldsofrealquadraticfields |