Leiden University Student Repository

Lawson, Dominique (2025)

Derivatives of Rankin–Selberg L-functions via the Goldfeld–Zhang method

Master thesis | Mathematics (MSc)

We use a method by Goldfeld and Zhang to compute values and derivatives of the Rankin–Selberg L-function associated to a pair of modular forms. Central to this approach is the holomorphic kernel, which relates the L-function to an inner product of modular forms. Using the Petersson formula, the Fourier coefficients of the holomorphic kernel can be understood. We solidify the Goldfeld–Zhang method by providing extra details to proofs and making justified adjustments to their formulas. In particular, we believe that an additional term has been overlooked in their main theorem. For weight k > 2, we verify their final claim that it is possible to recover the analytic part of the Gross–Zagier formulas for heights of Heegner cycles. More generally, we recover the analytic results in recent work by Lilienfeldt and Shnidman on heights of generalized Heegner cycles.

Modular forms

Rankin-Selberg L-functions

Theta series

Poincaré series

Author

Lawson, Dominique

Faculty

Faculty of Science

Specialisation

Algebra, Geometry and Number Theory

Supervisors

Lilienfeldt, D.T.-B.G.

ECTS Credits

40

Language

en