Every BT_1 group scheme appears in a Jacobian Pries, Rachel; Ulmer, Douglas
Proceedings of the American Mathematical Society,
February 1, 2022, Volume:
150, Issue:
2
Journal Article
Peer reviewed
Open access
Let p be a prime number and let k be an algebraically closed field of characteristic p. A BT_1 group scheme over k is a finite commutative group scheme which arises as the kernel of p on a ...p-divisible (Barsotti–Tate) group. Our main result is that every BT_1 group scheme over k occurs as a direct factor of the p-torsion group scheme of the Jacobian of an explicit curve defined over {\mathbb F}_p. We also treat a variant with polarizations. Our main tools are the Kraft classification of BT_1 group schemes, a theorem of Oda, and a combinatorial description of the de Rham cohomology of Fermat curves.
We study abelian varieties defined over function fields of curves in positive characteristic p, focusing on their arithmetic in the system of Artin-Schreier extensions. First, we prove that the ...L-function of such an abelian variety vanishes to high order at the center point of its functional equation under a parity condition on the conductor. Second, we develop an Artin-Schreier variant of a construction of Berger. This yields a new class of Jacobians over function fields for which the Birch and Swinnerton-Dyer conjecture holds. Third, we give a formula for the rank of the Mordell-Weil groups of these Jacobians in terms of the geometry of their fibers of bad reduction and homomorphisms between Jacobians of auxiliary Artin-Schreier curves. We illustrate these theorems by computing the rank for explicit examples of Jacobians of arbitrary dimension g, exhibiting Jacobians with bounded rank and others with unbounded rank in the tower of Artin-Schreier extensions. Finally, we compute the Mordell-Weil lattices of an isotrivial elliptic curve and a family of non-isotrivial elliptic curves. The latter exhibits an exotic phenomenon whereby the angles between lattice vectors are related to point counts on elliptic curves over finite fields. Our methods also yield new results about supersingular factors of Jacobians of Artin-Schreier curves.
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By a result of Moonen, there are exactly 20 positive-dimensional families of cyclic covers of the projective line for which the Torelli image is open and dense in the associated Shimura variety. For ...each of these, we compute the Newton polygons, and the
μ
-ordinary Ekedahl–Oort type, occurring in the characteristic
p
reduction of the Shimura variety. We prove that all but a few of the Newton polygons appear on the open Torelli locus. As an application, we produce multiple new examples of Newton polygons and Ekedahl–Oort types of Jacobians of smooth curves in characteristic
p
. Under certain congruence conditions on
p
, these include: the supersingular Newton polygon for genus 5, 6, 7; fourteen new non-supersingular Newton polygons for genus 5–7; eleven new Ekedahl–Oort types for genus 4–7 and, for all
g
≥
6
, the Newton polygon with
p
-rank
g
-
6
with slopes 1 / 6 and 5 / 6.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OBVAL, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
The authors study the Jacobian $J$ of the smooth projective curve $C$ of genus $r-1$ with affine model $y^r = x^r-1(x + 1)(x + t)$ over the function field $\mathbb F_p(t)$, when $p$ is prime and ...$r\ge 2$ is an integer prime to $p$. When $q$ is a power of $p$ and $d$ is a positive integer, the authors compute the $L$-function of $J$ over $\mathbb F_q(t^1/d)$ and show that the Birch and Swinnerton-Dyer conjecture holds for $J$ over $\mathbb F_q(t^1/d)$.
We prove results about the intersection of the
p-rank strata and the boundary of the moduli space of hyperelliptic curves in characteristic
p
⩾
3
. This yields a strong technique that allows us to ...analyze the stratum
H
g
f
of hyperelliptic curves of genus
g and
p-rank
f. Using this, we prove that the endomorphism ring of the Jacobian of a generic hyperelliptic curve of genus
g and
p-rank
f is isomorphic to
Z
if
g
⩾
4
. Furthermore, we prove that the
Z
/
ℓ
-monodromy of every irreducible component of
H
g
f
is the symplectic group
Sp
2
g
(
Z
/
ℓ
)
if
g
⩾
3
, and
ℓ
≠
p
is an odd prime (with mild hypotheses on
ℓ when
f
=
0
). These results yield numerous applications about the generic behavior of hyperelliptic curves of given genus and
p-rank over finite fields, including applications about Newton polygons, absolutely simple Jacobians, class groups and zeta functions.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
The a-numbers of Jacobians of Suzuki curves FRIEDLANDER, HOLLEY; GARTON, DEREK; MALMSKOG, BETH ...
Proceedings of the American Mathematical Society,
09/2013, Volume:
141, Issue:
9
Journal Article
Peer reviewed
Open access
For m ∈ ℕ, let S
m
be the Suzuki curve defined over
${\mathrm{\mathbb{F}}}_{{2}^{2\mathrm{m}+1}}$
. It is well-known that S
m
is supersingular, but the p-torsion group scheme of its Jacobian is not ...known. The a-number is an invariant of the isomorphism class of the p-torsion group scheme. In this paper, we compute a closed formula for the a-number of S
m
using the action of the Cartier operator on H
0
(S
m
, Ω
1
).
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Suppose X is a smooth projective connected curve defined over an algebraically closed field k of characteristic p>0 and B⊂X(k) is a finite, possibly empty, set of points. The Newton polygon of a ...degree p Galois cover of X with branch locus B depends on the ramification invariants of the cover. When X is ordinary, for every possible set of branch points and ramification invariants, we prove that there exists such a cover whose Newton polygon is minimal or close to minimal.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
The absolute Galois group of the cyclotomic field K=Q(ζp) acts on the étale homology of the Fermat curve X of exponent p. We study a Galois cohomology group which is valuable for measuring an ...obstruction for K-rational points on X. We analyze a 2-nilpotent extension of K which contains the information needed for measuring this obstruction. We determine a large subquotient of this Galois cohomology group which arises from Heisenberg extensions of K. For p=3, we perform a Magma computation with ray class fields, group cohomology, and Galois cohomology which determines it completely.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
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