Mathematicshttp://hdl.handle.net/10211.3/1288322022-05-18T00:43:32Z2022-05-18T00:43:32ZPeriod, index, and potential ShaClark, PeteSharif, Shahedhttp://hdl.handle.net/10211.3/1597362019-10-12T00:12:39Z2010-01-01T00:00:00ZPeriod, index, and potential Sha
Clark, Pete; Sharif, Shahed
We present three results on the period-index problem for genus-one curves over
global fields. Our first result implies that for every pair of positive integers (P, I)
such that I is divisible by P and divides P
2
, there exists a number field K and a
genus-one curve C/K with period P and index I. Second, let E/K be any elliptic
curve over a global field K, and let P > 1 be any integer indivisible by the characteristic
of K. We construct infinitely many genus-one curves C/K with period
P, index P
2
, and Jacobian E. Our third result, on the structure of Shafarevich–
Tate groups under field extension, follows as a corollary. Our main tools are
Lichtenbaum–Tate duality and the functorial properties of O’Neil’s period-index
obstruction map under change of period.
2010-01-01T00:00:00ZPeriod and index of genus one curves over number fieldsSharif, Shahedhttp://hdl.handle.net/10211.3/1597352019-10-12T00:12:39Z2012-11-01T00:00:00ZPeriod and index of genus one curves over number fields
Sharif, Shahed
The period of a curve is the smallest positive degree of Galoisinvariant
divisor classes. The index is the smallest positive degree of rational
divisors. We construct examples of genus one curves with prescribed period
and index over certain number fields. The final publication is available at Springer via http://dx.doi.org/10.1007/s00208-011-0745-1
2012-11-01T00:00:00ZA descent map for curves with totally degenerate semi-stable reductionSharif, Shahedhttp://hdl.handle.net/10211.3/1597342019-10-12T00:12:39Z2013-01-01T00:00:00ZA descent map for curves with totally degenerate semi-stable reduction
Sharif, Shahed
Let $K$ be a local field of residue characteristic $p$. Let $C$ be a curve over $K$ whose minimal proper regular model has totally degenerate semi-stable reduction. Under certain hypotheses, we compute the prime-to-$p$ rational torsion subgroup on the Jacobian of $C$. We also determine divisibility of line bundles on $C$, including rationality of theta characteristics and higher spin structures. These computations utilize arithmetic on the special fiber of $C$.The final publication is available through the Journal de Theorie des Nombres de Bordeaux http://jtnb.cedram.org/jtnb-bin/item?id=JTNB_2013__25_1_211_0
2013-01-01T00:00:00Z