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Nonvanishing of Central Derivatives of Modular -series in Level
Brian Lawrence
A quadratic twist of the L-function associated with a modular form is known to satisfy a functional equation, which may be even or odd. A result due to Gross and Zagier explicitly…
A counterexample to an optimistic guess about étale local systems
Brian Lawrence, Shizhang Li
In p-adic Hodge theory, it is known that if a Galois representation is de Rham, then it becomes semistable after extension of the base field. Liu and Zhu asked whether a correspond…
Two recent p-adic approaches towards the (effective) Mordell conjecture
Jennifer S. Balakrishnan, Alex J. Best, Francesca Bianchi +4
We give an introductory account of two recent approaches towards an effective proof of the Mordell conjecture, due to Lawrence--Venkatesh and Kim. The latter method, which is usual…
On the -adic distribution of torsion values for a section of an abelian scheme
Brian Lawrence, Umberto Zannier
Let be an abelian scheme over a -adic field, and let be a section. We study the torsion locus $\bigcup \limits_{n \geq 1} s^{-1}(A[n…
Diophantine problems and -adic period mappings
Brian Lawrence, Akshay Venkatesh
We give an alternative proof of Faltings's theorem (Mordell's conjecture): a curve of genus at least two over a number field has finitely many rational points. Our argument utilize…