Global long root Apackets for G_{2}: the dihedral case [PDF] [arXiv]
with Siyan Daniel LiHuerta, Petar Bakić, and Naomi Sweeting submitted (2024) Cuspidal automorphic representations τ of PGL_{2} correspond to global long root Aparameters for G_{2}. Using an exceptional theta lift between PU_{3} and G_{2}, we construct the associated global Apacket and prove the Arthur multiplicity formula for these representations when τ is dihedral and satisfies some technical hypotheses. We also prove that this subspace of the discrete automorphic spectrum forms a full near equivalence class. Our construction yields new examples of quaternionic modular forms on G_{2}. 

Balanced triple product padic Lfunctions and Stark points [PDF] [arXiv]
with Luca Dall'Ava submitted (2024) We use padic methods to investigate the equivariant BSD conjecture for an elliptic curve and two odd 2dimensional Artin representations. When the rank of the relevant MordellWeil group is two, DarmonLauderRotger used a dominant triple product padic Lfunction to study it, and gave an Elliptic Stark Conjecture which relates its value outside of the interpolation range to two Stark points and one Stark unit. Our paper achieves a similar goal in the rank one setting. We first generalize Hsieh's construction of a 3variable balanced triple product padic Lfunction in order to allow Hida families with classical weight one specializations. We then give an Elliptic Stark Conjecture relating its value outside of the interpolation range to a Stark point and two Stark units. We prove our conjecture for dihedral representations associated with the same imaginary quadratic field. This requires a generalization of the results of BertoliniDarmonPrasanna which we prove in the appendix. 

Motivic action for Siegel modular forms [PDF] [arXiv]
with Kartik Prasanna submitted (2023)
Inspired by the work of PrasannaVenkatesh on singular cohomology of locally symmetric spaces, we propose a conjecture that explains all the contributions of Siegel modular forms of low weight to coherent cohomology in terms of the action of a motivic cohomology group. Under some technical conditions, we prove that our conjecture is equivalent to Beilinson's conjecture for the adjoint Lfunction of f. We also consider two special cases: for lifts of Hilbert modular forms, we prove an unconditional result towards the conjecture, and for lifts of Bianchi modular forms, we show that our conjecture is compatible with the conjecture of PrasannaVenkatesh. The latter establishes a connection between the motivic action conjectures for locally symmetric spaces of nonhermitian type and those for coherent cohomology of Shimura varieties.


A database of basic numerical invariants of Hilbert modular surfaces [PDF] [Journal] [arXiv]
with Eran Assaf, Angelica Babei, Ben Breen, Edgar Costa, Avinash Kulkarni, Grant Molnar, Sam Schiavone, and John Voight AMS Contemporary Mathematics 796 (LMFDB, Computation, and Number Theory) (2023) We describe algorithms for computing geometric invariants for Hilbert modular surfaces, and we report on their implementation. 

Motivic action on coherent cohomology of Hilbert modular varieties [PDF] [Journal] [arXiv]
IMRN, Volume 2023, Issue 12, June 2023, Pages 1043910531 (2023)
We propose an action of a certain motivic cohomology group on the 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 C, and we conjecture that they both lift to an action on cohomology with integral coefficients. The conjecture is supported by theoretical evidence based on Stark's conjecture on special values of Artin Lfunctions, and by numerical evidence in base change cases.
