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The Harris-Venkatesh conjecture for derived Hecke operators | | Robin Zhang
; | | Date: |
2 Jan 2023 | | Abstract: | The Harris-Venkatesh conjecture posits a relationship between the action of
derived Hecke operators on weight-one modular forms and Stark units. We prove
the full Harris-Venkatesh conjecture for all dihedral weight-one modular forms
arising as definite theta series. This reproves results of
Darmon-Harris-Rotger-Venkatesh, extends their work to the adelic setting, and
removes all primality and ramification assumptions from the imaginary case of
the Harris--Venkatesh conjecture. This is accomplished by introducing the
Harris-Venkatesh pairing on cuspidal one-forms on modular curves, introducing
optimal modular forms, evaluating $mathrm{GL}(2) imes mathrm{GL}(2)$
Rankin-Selberg convolutions on optimal forms and newforms, and proving a
modulo-$ell^t$ comparison theorem between the Harris-Venkatesh and
Rankin-Selberg pairings. We also look at the application of our methods to
non-dihedral forms. | | Source: | arXiv, 2301.00570 | | Services: | Forum | Review | PDF | Favorites | |
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