October 3, 2025Open Access

Analytic ranks of twists of Carlitz modules -- a survey of results

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The analytic ranks of twists of Carlitz modules can vary, but their bounds remain uncertain, indicating areas for further exploration.
Sets of polynomials with specific analytic ranks are represented by affine varieties defined over finite fields, yet their dimensions remain unknown.
Research reveals that for certain values of q, irreducible components of varieties can be linked to weighted binary trees.
Detailed properties of the L-function's behavior at infinity also lead to significant insights on twists in relation to Carlitz modules.

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This survey compiles previous findings on analytic ranks of twists, highlighting challenges with l-functions and finite fields.

Grishkov et al. (2025) studied this question.

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