Below I highlight four of my main research contributions, each illustrating a different aspect of my work in Arakelov geometry, automorphic forms, and related areas. A complete list of my publications can be found here.
Arithmetic Riemann–Roch with singular metrics
Together with G. Freixas i Montplet (CNRS Paris), I extended the arithmetic Riemann–Roch theorem to settings involving singular metrics. As an application, we obtained an explicit formula for the special value at s=1 of Selberg’s zeta function for PSL_2(Z), a value that had not been accessible before. This connects techniques from Arakelov geometry, spectral theory, and number theory.
Asymptotics of Arakelov invariants on modular curves
In joint work with Miguel Grados, I studied the asymptotic behavior of the self-intersection of the dualizing sheaf on modular curves of growing level, complementing earlier results of Abbes–Ullmo and Michel–Ullmo. In further collaboration with Priyanka Majumder, we established bounds for canonical Green’s functions at cusps, a central analytic ingredient in this asymptotic analysis.
Eisenstein series beyond the parabolic case
In collaborations with J. Jorgenson and J. Kramer, I introduced and investigated hyperbolic and elliptic Eisenstein series, generalizations of the classical parabolic Eisenstein series that play a role in spectral theory. This work highlighted analogies with Kronecker’s limit formula and led to further connections with Borcherds lifts and automorphic forms.
Higher Green’s functions on hyperbolic 3-space
Building on work of Gross, Kohnen, and Zagier for modular curves, I studied (with Ö. Imamoğlu, S. Herrero, and M. Schwagenscheidt) special values of higher Green’s functions in hyperbolic 3-space. We showed that their averages can be expressed in terms of logarithms of primes and units in real quadratic fields, and that twisted averages yield algebraicity results. These results contribute to the broader understanding of the arithmetic nature of higher Green’s functions.