# On a BSD-type formula for L-values of Artin twists of elliptic curves

## TL;DR

This paper explores a BSD-type formula for L-functions of elliptic curves twisted by Artin representations, revealing new arithmetic properties and peculiar behaviors of the Tate-Shafarevich group that challenge traditional methods.

## Contribution

It proposes a conjectural BSD-type formula for Artin-twisted elliptic curves and uncovers novel properties of the Tate-Shafarevich group related to these twists.

## Key findings

- Tate-Shafarevich group components can behave dependently.
- L-values can differ in arithmetically identical settings.
- New conjectural relationships between L-functions and arithmetic invariants.

## Abstract

This is an investigation into the possible existence and consequences of a Birch-Swinnerton-Dyer-type formula for L-functions of elliptic curves twisted by Artin representations. We translate expected properties of L-functions into purely arithmetic predictions for elliptic curves, and show that these force some peculiar properties of the Tate-Shafarevich group, which do not appear to be tractable by traditional Selmer group techniques. In particular we exhibit settings where the different p-primary components of the Tate-Shafarevich group do not behave independently of one another. We also give examples of "arithmetically identical" settings for elliptic curves twisted by Artin representations, where the associated L-values can nonetheless differ, in contrast to the classical Birch-Swinnerton-Dyer conjecture.