Modular algorithms for Gross-Stark units and Stark-Heegner points

In recent work, Darmon, Pozzi and Vonk explicitly construct a modular form whose spectral coefficients are $p$-adic logarithms of Gross-Stark units and Stark-Heegner points. Here we describe how this construction gives rise to a practical algorithm for explicitly computing these logarithms to specified precision, and how to recover the exact values of the Gross-Stark units and Stark-Heegner points from them. Key tools are overconvergent modular forms, reduction theory of quadratic forms and Newton polygons. As an application, we tabulate Brumer-Stark units in narrow Hilbert class fields of real quadratic fields with discriminants up to $10000$, for primes less than $20$, as well as Stark-Heegner points on elliptic curves.Comment: 23 pages, 4 tables, 2 figure

Motivic action on coherent cohomology of Hilbert modular varieties

We propose an action of a certain motivic cohomology group on coherent cohomology of Hilbert modular varieties, extending conjectures of Venkatesh, Prasanna, and Harris. The action is described in two ways: on cohomology modulo $p$ and over $\mathbb C$, and we conjecture that they both lift to an action on cohomology with integral coefficients. The latter is supported by theoretical evidence based on Stark's conjecture on special values of Artin $L$-functions and by numerical evidence in base change cases

Derived Hecke Operators on Unitary Shimura Varieties

We propose a coherent analogue of the non-archimedean case of Venkatesh's conjecture on the cohomology of locally symmetric spaces for Shimura varieties coming from unitary similitude groups. Let G be a unitary similitude group with an indefinite signature at at least one archimedean place. Let Π be an automorphic cuspidal representation of G whose archimedean component Π∞ is a non-degenerate limit of discrete series and let be an automorphic vector bundle such that Π contributes to the coherent cohomology of its canonical extension. We produce a natural action of the derived Hecke algebra of Venketesh with torsion coefficients via cup product coming from étale covers and show that under some standard assumptions this action coincides with the conjectured action of a certain motivic cohomology group associated to the adjoint representation Adπ of the Galois representation attached to Π. We also prove that if the rank of G is greater than two, then the classes arising from the \'etale covers do not admit characteristic zero lifts, thereby showing that previous work of Harris-Venkatesh and Darmon-Harris-Rotger-Venkatesh is exceptional

The Harris-Venkatesh conjecture for derived Hecke operators

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 CM dihedral weight-one modular forms. This reproves results of Darmon-Harris-Rotger-Venkatesh, extends their work to the adelic setting, and removes all assumptions on primality and ramification from the imaginary dihedral case of the Harris-Venkatesh conjecture. This is done by introducing the Harris-Venkatesh period on cuspidal one-forms on modular curves, introducing two-variable optimal modular forms, evaluating GL(2) × GL(2) Rankin-Selberg convolutions on optimal forms and newforms, and proving a modulo-ℓᵗ comparison theorem between the Harris-Venkatesh and Rankin-Selberg periods. Furthermore, these methods explicitly describe local factors appearing in the constant of proportionality prescribed by the Harris-Venkatesh conjecture. We also look at the application of our methods to non-dihedral forms