Geometric Aspects of the Painleve Equations PIII(D-6) and PIII(D-7)

The Riemann-Hilbert approach for the equations PIII(D-6) and PIII(D-7) is studied in detail, involving moduli spaces for connections and monodromy data, Okamoto-Painleve varieties, the Painleve property, special solutions and explicit Backlund transformations.</p

Geometric Aspects of the Painlev\'e Equations PIII(D6){\rm PIII(D_6)} and PIII(D7){\rm PIII(D_7)}

The Riemann-Hilbert approach for the equations ${\rm PIII(D_6)}$ and ${\rm PIII(D_7)}$ is studied in detail, involving moduli spaces for connections and monodromy data, Okamoto-Painlev\'e varieties, the Painlev\'e property, special solutions and explicit B\"acklund transformations

On certain maximal hyperelliptic curves related to Chebyshev polynomials

We study hyperelliptic curves arising from Chebyshev polynomials. The aim of this paper is to characterize the pairs such that the hyperelliptic curve over a finite field given by is maximal over the finite field of cardinality . Here denotes the Chebyshev polynomial of degree d. The same question is studied for the curves given by , and also for . Our results generalize some of the statements in [12]203276293FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULO - FAPESP2017/19190-5The first author was supported by FAPESP/SP-Brazil grant 2017/19190-5. The second author thanks Marco Streng and Nurdagül Anbar Meidl for helpful suggestions. We also thank the referee for various comments which helped us improve the expositio