Engineering: Reportshttp://hdl.handle.net/10092/6112017-07-09T18:03:58Z2017-07-09T18:03:58ZCurves that change genus can have arbitrarily many rational pointsVoloch JFhttp://hdl.handle.net/10092/134572017-05-12T15:01:30Z1995-01-01T00:00:00ZCurves that change genus can have arbitrarily many rational points
Voloch JF
A singular curve over a non-perfect field K may not have a smooth model over
K. Those are said to "change genus". If K is a global field of positive
characteristic and C/K a curve that change genus, then C(K) is known to be
finite. The purpose of this note is to give examples of curves with fixed
relative genus, defined over K for which #C(K) is arbitrarily large. The
motivation for considering this problem comes from the work of Caporaso et al.
[CHM], where they show that a conjecture of Lang implies that, for a number
field K, #C(K) can be bounded in terms of g and K only for all curves C/K of
genus g > 1.
1995-01-01T00:00:00ZThe Sato-Tate Distribution in Thin Parametric Families of Elliptic CurvesBretèche RDLSha MShparlinski IEVoloch JFhttp://hdl.handle.net/10092/134562017-05-12T15:01:29Z2015-01-01T00:00:00ZThe Sato-Tate Distribution in Thin Parametric Families of Elliptic Curves
Bretèche RDL; Sha M; Shparlinski IE; Voloch JF
We obtain new results concerning the Sato-Tate conjecture on the distribution
of Frobenius traces over single and double parametric families of elliptic
curves. We consider these curves for values of parameters having prescribed
arithmetic structure: product sets, geometric progressions, and most
significantly prime numbers. In particular, some families are much thinner than
the ones previously studied.
2015-01-01T00:00:00ZVisible Points on Curves over Finite FieldsShparlinski IEVoloch JFhttp://hdl.handle.net/10092/134552017-05-12T15:01:26Z2013-01-01T00:00:00ZVisible Points on Curves over Finite Fields
Shparlinski IE; Voloch JF
For a prime 𝑝 and an absolutely irreducible modulo 𝑝 polynomial 𝑓(U,V)
∈ ℤ[U,V] we obtain an asymptotic formulas for the number of solutions to
the congruence 𝑓(𝑥,𝑦) ≡ a (mod 𝑝) in positive integers 𝑥 ⩽ X, 𝑦 ⩽
Y, with the additional condition 𝗀cd(𝑥,𝑦)=1. Such solutions have a natural
interpretation as solutions which are visible from the origin. These formulas
are derived on average over 𝑎 for a fixed prime 𝑝, and also on average over
𝑝 for a fixed integer 𝑎.
2013-01-01T00:00:00ZOn the number of rational points on special families of curves over function fieldsUlmer DVoloch JFhttp://hdl.handle.net/10092/134512017-05-12T15:01:25Z2016-01-01T00:00:00ZOn the number of rational points on special families of curves over function fields
Ulmer D; Voloch JF
We construct families of curves which provide counterexamples for a uniform
boundedness question. These families generalize those studied previously by
several authors. We show, in detail, what fails in the argument of Caporaso,
Harris, Mazur that uniform boundedness follows from the Lang conjecture. We
also give a direct proof that these curves have finitely many rational points
and give explicit bounds for the heights and number of such points.
2016-01-01T00:00:00Z