Some remarks on superspecial and ordinary curves of low genus

2012 ◽  
Vol 286 (1) ◽  
pp. 17-33 ◽  
Author(s):  
Takehiro Hasegawa
Keyword(s):  
Curves Of Low Genus

On the Gaussian map for canonical curves of low genus

1990 ◽  
Vol 61 (2) ◽  
pp. 417-443 ◽  
Author(s):  
Ciro Ciliberto ◽  
Rick Miranda
Keyword(s):  
Canonical Curves ◽  
Gaussian Map ◽  
Curves Of Low Genus

scholarly journals Addition formulae for Abelian functions associated with specialized curves

2011 ◽  
Vol 369 (1939) ◽  
pp. 1245-1263 ◽  
Author(s):  
J. C. Eilbeck ◽  
S. Matsutani ◽  
Y. Ônishi
Keyword(s):  
Genus 2 ◽  
Abelian Functions ◽  
Weierstrass Functions ◽  
Curves Of Low Genus

We discuss a family of multi-term addition formulae for Weierstrass functions on specialized curves of low genus with many automorphisms, concentrating mostly on the case of genus 1 and 2. In the genus 1 case, we give addition formulae for the equianharmonic and lemniscate cases, and in genus 2 we find some new addition formulae for a number of curves.

scholarly journals Quadratic sequences of powers and Mohanty’s conjecture

2018 ◽  
Vol 14 (02) ◽  
pp. 479-507 ◽  
Author(s):  
Natalia Garcia-Fritz
Keyword(s):  
Elliptic Curves ◽  
Arithmetic Progression ◽  
General Type ◽  
Function Fields ◽  
Hyperelliptic Curves ◽  
Arithmetic Progressions ◽  
Surfaces Of General Type ◽  
Genus Zero ◽  
Curves Of Low Genus

We prove under the Bombieri–Lang conjecture for surfaces that there is an absolute bound on the length of sequences of integer squares with constant second differences, for sequences which are not formed by the squares of integers in arithmetic progression. This answers a question proposed in 2010 by Browkin and Brzezinski, and independently by Gonzalez-Jimenez and Xarles. We also show that under the Bombieri–Lang conjecture for surfaces, for every [Formula: see text] there is an absolute bound on the length of sequences formed by [Formula: see text]th powers with constant second differences. This gives a conditional result on one of Mohanty’s conjectures on arithmetic progressions in Mordell’s elliptic curves [Formula: see text]. Moreover, we obtain an unconditional result regarding infinite families of such arithmetic progressions. We also study the case of hyperelliptic curves of the form [Formula: see text]. These results are proved by unconditionally finding all curves of genus zero or one on certain surfaces of general type. Moreover, we prove the unconditional analogues of these arithmetic results for function fields by finding all the curves of low genus on these surfaces.

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