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Abstract
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Let
be a finite field
of characteristic
.
For any power
of
, consider the
elliptic curve
defined by
over
. We describe several
arithmetic invariants of
such as the rank of its Mordell–Weil group
, the size of its Néron–Tate
regulator Reg,
and the order of its Tate–Shafarevich group
Ш (which we prove
is finite). These invariants have radically different behaviors depending on the congruence class
of
modulo 6. For
instance Ш either
has trivial
-part
or is a
-group.
On the other hand, we show that the product
Ш Reg
has size comparable to
as
, regardless
of
.
Our approach relies on the BSD conjecture, an explicit expression for the
-function of
, and a geometric analysis
of the Néron model of
.
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Keywords
elliptic curves over function fields, Mordell–Weil rank,
Néron–Tate regulator, Tate–Shafarevich group, L-function
and BSD conjecture
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Mathematical Subject Classification 2010
Primary: 11G05, 14J27
Secondary: 11G40, 11G99, 14G10, 14G99
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Milestones
Received: 29 May 2019
Revised: 11 November 2019
Accepted: 13 November 2019
Published: 29 April 2020
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