aarjaneiro/planetmath huggingface dataset:
height function

height function

Definition 1

Let

A

be an

abelian group

. A

height function

on

A

is a
function

h

:

A

→

R

with the properties:

1.

For all

Q

∈

A

there exists a constant

C

1

, depending on

A

and

Q

, such that for all

P

∈

A

:

h

⁢

(

P

+

Q

)

≤

2

⁢

h

⁢

(

P

)

+

C

1

2.

There exists an

integer

m

≥

2

and a constant

C

2

,
depending on

A

, such that for all

P

∈

A

:

h

⁢

(

m

⁢

P

)

≥

m

2

⁢

h

⁢

(

P

)

-

C

2

3.

For all

C

3

∈

ℝ

, the following set is finite:

{

P

∈

A

:

h

⁢

(

P

)

≤

C

3

}

Examples

:

1.

For

t

=

p

/

q

∈

ℚ

, a

fraction

in lower terms,
define

H

⁢

(

t

)

=

max

⁡

{

∣

p

∣

,

∣

q

∣

}

. Even though this
is not a height function as defined above, this is the prototype
of what a height function should look like.

2.

Let

E

be an

elliptic curve

over

ℚ

. The function on

E

⁢

(

ℚ

)

,
the points in

E

with coordinates in

ℚ

,

h

x

:

E

⁢

(

ℚ

)

→

ℝ

:

h

x

⁢

(

P

)

=

{

log

⁡

H

⁢

(

x

⁢

(

P

)

)

,

i

⁢

f

⁢

P

≠

0

0

,

i

⁢

f

⁢

P

=

0

}

is a height function (

H

is defined as above). Notice that
this depends on the chosen

Weierstrass model

of the curve.

3.

The

canonical height

of

E

/

ℚ

(due to Neron and Tate)
is defined by:

h

C

⁢

(

P

)

=

1

/

2

⁢

lim

N

→

∞

⁡

4

(

-

N

)

⁢

h

x

⁢

(

[

2

N

]

⁢

P

)

where

h

x

is defined as in (2).

Finally we mention the fundamental theorem of “descent”, which
highlights the importance of the height functions:

Theorem 1

(Descent)

Let

A

be an abelian group and let

h

:

A

→

R

be a height function. Suppose that for the integer

m

, as in property (2) of height, the

quotient group

A

/

m

⁢

A

is
finite. Then

A

is

finitely generated

.

References

1

Joseph H. Silverman,

The

Arithmetic of Elliptic Curves

. Springer-Verlag, New York, 1986.

Title

height function

Canonical name

HeightFunction

Date of creation

2013-03-22 13:49:09

Last modified on

2013-03-22 13:49:09

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Definition

Classification

msc 14H52

Related topic

EllipticCurve

Related topic

RankOfAnEllipticCurve

Related topic

ArithmeticOfEllipticCurves

Related topic

CanonicalHeightOnAnEllipticCurve

Defines

height function

Defines

canonical height

Defines

descent theorem

Generated on Fri Feb 9 17:58:47 2018 by

LaTeXML
isogeny

isogeny

Let

E

and

E

′

be

elliptic curves

over a field

k

. An

isogeny

between

E

and

E

′

is a

finite morphism

f

:

E

→

E

′

of

varieties

that preserves

basepoints

.

The two curves are called

isogenous

if there is an isogeny between them. This is an

equivalence relation

,

symmetry

being due to the

existence

of the

dual isogeny

. Every isogeny is an

algebraic

homomorphism

and thus induces homomorphisms of the groups of the elliptic curves for

k

-valued points.

Title

isogeny

Canonical name

Isogeny

Date of creation

2013-03-22 12:52:07

Last modified on

2013-03-22 12:52:07

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

8

Author

mathcam (2727)

Entry type

Definition

Classification

msc 14H52

Classification

msc 14A15

Classification

msc 14A10

Classification

msc 14-00

Synonym

isogenous

Related topic

EllipticCurve

Related topic

ArithmeticOfEllipticCurves

Generated on Fri Feb 9 17:58:48 2018 by

LaTeXML
j-invariant

j-invariant

Let

E

be an

elliptic curve

over

ℚ

with

Weierstrass
equation

:

y

2

+

a

1

⁢

x

⁢

y

+

a

3

⁢

y

=

x

3

+

a

2

⁢

x

2

+

a

4

⁢

x

+

a

6

with

coefficients

a

i

∈

ℚ

. Let:

b

2

=

a

1

2

+

4

⁢

a

2

,

b

4

=

2

⁢

a

4

+

a

1

⁢

a

3

,

b

6

=

a

3

2

+

4

⁢

a

6

,

b

8

=

a

1

2

⁢

a

6

+

4

⁢

a

2

⁢

a

6

-

a

1

⁢

a

3

⁢

a

4

+

a

3

2

⁢

a

2

-

a

4

2

,

c

4

=

b

2

2

-

24

⁢

b

4

,

c

6

=

-

b

2

3

+

36

⁢

b

2

⁢

b

4

-

216

⁢

b

6

Definition 1

.

1.

The

discriminant

of

E

is defined to be

Δ

=

-

b

2

2

⁢

b

8

-

8

⁢

b

4

3

-

27

⁢

b

6

2

+

9

⁢

b

2

⁢

b

4

⁢

b

6

2.

The

j-invariant

of

E

is

j

=

c

4

3

Δ

3.

The

invariant differential

is

ω

=

d

⁢

x

2

⁢

y

+

a

1

⁢

x

+

a

3

=

d

⁢

y

3

⁢

x

2

+

2

⁢

a

2

⁢

x

+

a

4

-

a

1

⁢

y

Example

:

If

E

has a Weierstrass equation in the simplified form

y

2

=

x

3

+

A

⁢

x

+

B

then

Δ

=

-

16

⁢

(

4

⁢

A

3

+

27

⁢

B

2

)

,

j

=

-

1728

⁢

(

4

⁢

A

)

3

Δ

Note

: The discriminant

Δ

coincides in this case with the usual notion of discriminant of the

polynomial

(

http://planetmath.org/Discriminant

)

x

3

+

A

⁢

x

+

B

.

Title

j-invariant

Canonical

name

Jinvariant

Date of creation

2013-03-22 13:49:54

Last modified on

2013-03-22 13:49:54

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

9

Author

alozano (2414)

Entry type

Definition

Classification

msc 14H52

Synonym

discriminant

Synonym

j

-invariant

Synonym

j invariant

Related topic

EllipticCurve

Related topic

BadReduction

Related topic

ModularDiscriminant

Related topic

Discriminant

Related topic

ArithmeticOfEllipticCurves

Defines

j-invariant

Defines

discriminant of an elliptic curve

Defines

invariant differential

Generated on Fri Feb 9 17:58:51 2018 by

LaTeXML
Mazur’s theorem on torsion of elliptic curves

Mazur’s theorem on torsion of elliptic curves

Theorem 1

(Mazur).

Let

E

/

Q

be an

elliptic curve

. Then the

torsion subgroup

E

torsion

⁢

(

Q

)

is exactly one of the
following groups:

ℤ

/

N

⁢

ℤ

1

≤

N

≤

10

o

⁢

r

N

=

12

ℤ

/

2

⁢

ℤ

⊕

ℤ

/

2

⁢

N

⁢

ℤ

1

≤

N

≤

4

Note: see

Nagell-Lutz theorem

for an efficient algorithm to compute the torsion subgroup of an elliptic curve defined over

ℚ

.

References

1

Joseph H. Silverman,

The

Arithmetic of Elliptic Curves

. Springer-Verlag, New York, 1986.

2

Barry Mazur,

Modular curves and the
Eisenstein ideal

, IHES Publ. Math. 47 (1977), 33-186.

3

Barry Mazur,

Rational

isogenies

of prime
degree

, Invent. Math. 44 (1978), 129-162.

Title

Mazur’s theorem on

torsion

of elliptic curves

Canonical name

MazursTheoremOnTorsionOfEllipticCurves

Date of creation

2013-03-22 13:51:59

Last modified on

2013-03-22 13:51:59

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Theorem

Classification

msc 14H52

Related topic

EllipticCurve

Related topic

MordellWeilTheorem

Related topic

RankOfAnEllipticCurve

Related topic

TorsionSubgroupOfAnEllipticCurveInjectsInTheReductionOfTheCurve

Related topic

ArithmeticOfEllipticCurves

Defines

Mazur’s theorem

Generated on Fri Feb 9 17:58:55 2018 by

LaTeXML
Mordell curve

Mordell curve

A

Mordell curve

is an

elliptic curve

E

/

K

, for some field

K

, which admits a model by a

Weierstrass equation

of the form:

y

2

=

x

3

+

k

,

k

∈

K

Examples

:

1.

Let

E

1

/

ℚ

:

y

2

=

x

3

+

2

, this is a Mordell
curve with Mordell-Weil group

E

1

⁢

(

ℚ

)

≃

ℤ

and generated by

(

-

1

,

1

)

.

2.

Let

E

2

/

ℚ

:

y

2

=

x

3

+

109858299531561

, then

E

2

⁢

(

ℚ

)

≃

ℤ

/

3

⁢

ℤ

⁢

⊕

ℤ

5

. See

http://math.bu.edu/people/alozano/

Torsion

.html

generators

here.

3.

In general, a Mordell curve of the form

y

2

=

x

3

+

n

2

has
torsion group

isomorphic

to

ℤ

/

3

⁢

ℤ

generated by

(

0

,

n

)

.

4.

Let

E

3

/

ℚ

:

y

2

=

x

3

+

496837487681

then this is
a Mordell curve with

E

3

⁢

(

ℚ

)

≃

ℤ

8

. See

http://math.bu.edu/people/alozano/Mordell.html

generators
here.

5.

http://www.maths.nott.ac.uk/personal/pmxtow/mordellc.htm

Here
you can find a list of the minimal-known

positive

and negative k
for Mordell curves of given rank, and the Mordell curves with
maximum rank known (see

BS-D conjecture

).

Title

Mordell curve

Canonical name

MordellCurve

Date of creation

2013-03-22 13:49:57

Last modified on

2013-03-22 13:49:57

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Definition

Classification

msc 14H52

Related topic

EllipticCurve

Related topic

BirchAndSwinnertonDyerConjecture

Related topic

ArithmeticOfEllipticCurves

Defines

Mordell curve

Generated on Fri Feb 9 17:58:59 2018 by

LaTeXML
Mordell-Weil theorem

Mordell-Weil theorem

Let

K

be a

number field

and let

E

be an

elliptic curve

over

K

. By

E

⁢

(

K

)

we denote the set of points in

E

with coordinates
in

K

.

Theorem 1

(Mordell-Weil).

E

⁢

(

K

)

is a

finitely generated

abelian
group

.

Proof.

The proof of this theorem is fairly involved. The
main two ingredients are the so called

weak Mordell-Weil theorem

(

http://planetmath.org/WeakMordellWeilTheorem

), the concept of

height function

for abelian groups and
the “descent (

http://planetmath.org/HeightFunction

)” theorem.

See

[

2

]

, Chapter VIII, page
189.
∎

References

1

James Milne,

Elliptic Curves

, online course notes.

http://www.jmilne.org/math/CourseNotes/math679.html

http://www.jmilne.org/math/CourseNotes/math679.html

2

Joseph H. Silverman,

The

Arithmetic of Elliptic Curves

. Springer-Verlag, New York, 1986.

3

Joseph H. Silverman,

Advanced Topics in
the Arithmetic of Elliptic Curves

. Springer-Verlag, New York,
1994.

4

Goro Shimura,

Introduction to the
Arithmetic Theory of Automorphic Functions

. Princeton University
Press, Princeton, New Jersey, 1971.

Title

Mordell-Weil theorem

Canonical name

MordellWeilTheorem

Date of creation

2013-03-22 12:16:35

Last modified on

2013-03-22 12:16:35

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

10

Author

alozano (2414)

Entry type

Theorem

Classification

msc 14H52

Related topic

WeakMordellWeilTheorem

Related topic

MazursTheoremOnTorsionOfEllipticCurves

Related topic

EllipticCurve

Related topic

RankOfAnEllipticCurve

Related topic

ArithmeticOfEllipticCurves

Generated on Fri Feb 9 17:59:00 2018 by

LaTeXML
Nagell-Lutz theorem

Nagell-Lutz theorem

The following theorem, proved independently by E. Lutz and T.
Nagell, gives a very efficient method to compute the

torsion
subgroup

of an

elliptic curve

defined over

ℚ

.

Theorem 1

(Nagell-Lutz Theorem).

Let

E

/

Q

be an elliptic curve with

Weierstrass equation

:

y

2

=

x

3

+

A

⁢

x

+

B

,

A

,

B

∈

ℤ

Then for all non-zero

torsion

points

P

we have:

1.

The coordinates of

P

are in

ℤ

, i.e.

x

⁢

(

P

)

,

y

⁢

(

P

)

∈

ℤ

2.

If

P

is of order greater than

2

, then

y

⁢

(

P

)

2

d

⁢

i

⁢

v

⁢

i

⁢

d

⁢

e

⁢

s

4

⁢

A

3

+

27

⁢

B

2

3.

If

P

is of order

2

then

y

⁢

(

P

)

=

0

a

⁢

n

⁢

d

x

⁢

(

P

)

3

+

A

⁢

x

⁢

(

P

)

+

B

=

0

References

1

E. Lutz,

Sur l’

equation

y

2

=

x

3

-

A

⁢

x

-

B

dans
les corps p-adic

, J. Reine Angew. Math. 177 (1937), 431-466.

2

T. Nagell,

Solution de quelque problemes
dans la theorie arithmetique des cubiques planes du premier
genre

, Wid. Akad. Skrifter Oslo I, 1935, Nr. 1.

3

James Milne,

Elliptic Curves

,

http://www.jmilne.org/math/CourseNotes/math679.html

online course
notes.

4

Joseph H. Silverman,

The

Arithmetic of Elliptic Curves

. Springer-Verlag, New York, 1986.

Title

Nagell-Lutz theorem

Canonical name

NagellLutzTheorem

Date of creation

2013-03-22 13:52:02

Last modified on

2013-03-22 13:52:02

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

4

Author

alozano (2414)

Entry type

Theorem

Classification

msc 14H52

Related topic

EllipticCurve

Related topic

MordellWeilTheorem

Related topic

RankOfAnEllipticCurve

Related topic

TorsionSubgroupOfAnEllipticCurveInjectsInTheReductionOfTheCurve

Related topic

ArithmeticOfEllipticCurves

Defines

Nagell-Lutz theorem

Generated on Fri Feb 9 17:59:02 2018 by

LaTeXML
supersingular

supersingular

An

elliptic curve

E

over a field of

characteristic

p

defined by the

cubic equation

f

⁢

(

w

,

x

,

y

)

=

0

is called

supersingular

if the

coefficient

of

(

w

⁢

x

⁢

y

)

p

-

1

in

f

⁢

(

w

,

x

,

y

)

p

-

1

is zero.

A

supersingular elliptic curve

is said to have Hasse invariant

0

; an ordinary (i.e. non-supersingular) elliptic curve is said to have Hasse invariant

1

.

This is

equivalent

to many other conditions.

E

is supersingular iff the

invariant differential

is exact.
Also,

E

is supersingular iff

F

*

:

H

1

⁢

(

E

,

𝒪

E

)

→

H

1

⁢

(

E

,

𝒪

E

)

is nonzero where

F

*

is

induced

from the

Frobenius morphism

F

:

E

→

E

.

Title

supersingular

Canonical name

Supersingular

Date of creation

2013-03-22 12:18:30

Last modified on

2013-03-22 12:18:30

Owner

nerdy2 (62)

Last modified by

nerdy2 (62)

Numerical id

5

Author

nerdy2 (62)

Entry type

Definition

Classification

msc 14H52

Generated on Fri Feb 9 17:59:09 2018 by

LaTeXML
genus

genus

“Genus” has number of distinct but compatible

definitions

.

In

topology

, if

S

is an

orientable surface

, its genus

g

⁢

(

S

)

is the number of “handles” it has.
More precisely, from the classification of

surfaces

, we know that any orientable
surface is a sphere, or the

connected sum

of

n

tori. We say the sphere
has genus 0, and that the connected sum of

n

tori has genus

n

(alternatively, genus is additive with respect to connected sum, and the genus of a torus is 1).
Also,

g

⁢

(

S

)

=

1

-

χ

⁢

(

S

)

/

2

where

χ

⁢

(

S

)

is the

Euler characteristic

of

S

.

In

algebraic geometry

, the genus of a smooth

projective curve

X

over a field

k

is the

dimension

over

k

of the vector space

Ω

1

⁢

(

X

)

of global

regular

differentials on

X

. Recall that a smooth complex curve is also a

Riemann surface

,
and hence topologically a surface. In this case, the two definitions of genus coincide.

Title

genus

Canonical

name

Genus

Date of creation

2013-03-22 12:03:45

Last modified on

2013-03-22 12:03:45

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

10

Author

mathcam (2727)

Entry type

Definition

Classification

msc 14H99

Generated on Fri Feb 9 17:59:17 2018 by

LaTeXML
Hurwitz genus formula

Hurwitz genus formula

The following formula, due to Hurwitz, is extremely useful when trying to compute the genus of an

algebraic curve

. In this entry

K

is a

perfect field

(i.e. every

algebraic extension

of

K

is

separable

). Recall that a non-constant map of curves

ψ

:

C

1

→

C

2

over

K

is separable if the extension of function fields

K

⁢

(

C

1

)

/

ψ

∗

⁢

K

⁢

(

C

2

)

is a

separable extension

of fields.

Theorem

(Hurwitz Genus Formula).

Let

C

1

and

C

2

be two

smooth curves

defined over

K

of genus

g

1

and

g

2

, respectively. Let

ψ

:

C

1

→

C

2

be a non-constant and separable map. Then

2

⁢

g

1

-

2

≥

(

deg

⁡

ψ

)

⁢

(

2

⁢

g

2

-

2

)

+

∑

P

∈

C

1

(

e

ψ

⁢

(

P

)

-

1

)

where

e

ψ

⁢

(

P

)

is the

ramification index

of

ψ

at

P

. Moreover, there is equality if and only if either

char

⁡

(

K

)

=

0

or

char

⁡

(

K

)

=

p

>

0

and

p

does not divide

e

ψ

⁢

(

P

)

for all

P

∈

C

1

.

Example

.

As an application of the

Hurwitz genus formula

, we show that an

elliptic curve

E

:

y

2

=

x

⁢

(

x

-

α

)

⁢

(

x

-

β

)

defined over a field

K

of

characteristic

0

has genus

1

. Notice that the fact that

E

is an elliptic curve over

K

implies that

0

,

α

and

β

are distinct elements of

K

, otherwise

E

would be a

singular

curve. We define a map:

ψ

:

E

→

ℙ

1

,

[

x

,

y

,

z

]

↦

[

x

,

z

]

and notice that

[

0

,

1

,

0

]

, the “point at infinity” of

E

, maps to

[

1

,

0

]

, the point at infinity of

ℙ

1

. The degree of this map is

2

: generically every point in

ℙ

1

has two preimages, namely

[

x

,

y

,

z

]

and

[

x

,

-

y

,

z

]

. Moreover, the genus of

ℙ

1

is

0

and the map

ψ

is ramified exactly at

4

points, namely

P

1

=

[

0

,

0

,

1

]

,

P

2

=

[

α

,

0

,

1

]

,

P

3

=

[

β

,

0

,

1

]

and the point at infinity. It is easily checked that the ramification index at each point is

e

ψ

⁢

(

P

i

)

=

2

. Hence, the

Hurwitz formula

reads:

2

⁢

g

1

-

2

=

2

⁢

(

2

⋅

0

-

2

)

+

∑

i

=

1

4

(

e

ψ

⁢

(

P

i

)

-

1

)

=

-

4

+

4

=

0

.

We conclude that

g

1

=

1

, as claimed.

Title

Hurwitz genus formula

Canonical name

HurwitzGenusFormula

Date of creation

2013-03-22 15:57:15

Last modified on

2013-03-22 15:57:15

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

6

Author

alozano (2414)

Entry type

Theorem

Classification

msc 14H99

Related topic

RiemannRochTheorem

Related topic

EllipticCurve

Generated on Fri Feb 9 17:59:19 2018 by

LaTeXML
projective curve

projective curve

A

projective curve

over a field

k

is an equidimensional

projective
variety

over

k

of

dimension

1. In other , each of the

irreducible components

of this

variety

must have dimension 1.

Title

projective curve

Canonical

name

ProjectiveCurve

Date of creation

2013-03-22 12:42:09

Last modified on

2013-03-22 12:42:09

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

7

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 14H99

Related topic

AlgebraicGeometry

Generated on Fri Feb 9 17:59:21 2018 by

LaTeXML
proof of Riemann-Roch theorem

proof of Riemann-Roch theorem

For a

divisor

D

, let

𝔏

⁢

(

D

)

be the associated

line bundle

. By

Serre duality

,

H

0

⁢

(

𝔏

⁢

(

K

-

D

)

)

≅

H

1

⁢

(

𝔏

⁢

(

D

)

)

, so

ℓ

⁢

(

D

)

-

ℓ

⁢

(

K

-

D

)

=

χ

⁢

(

D

)

, the

Euler characteristic

of

𝔏

⁢

(

D

)

. Now, let

p

be a point of

C

, and consider the divisors

D

and

D

+

p

. There is a natural injection

𝔏

⁢

(

D

)

→

𝔏

⁢

(

D

+

p

)

. This is an

isomorphism

anywhere away from

p

, so the quotient

ℰ

is a skyscraper sheaf supported at

p

. Since skyscraper sheaves are flasque, they have trivial higher

cohomology

, and so

χ

⁢

(

ℰ

)

=

1

. Since Euler characteristics add along

exact sequences

(because of the

long exact sequence in cohomology

)

χ

⁢

(

D

+

p

)

=

χ

⁢

(

D

)

+

1

. Since

deg

⁢

(

D

+

p

)

=

deg

⁢

(

D

)

+

1

, we see that if Riemann-Roch holds for

D

, it holds for

D

+

p

, and vice-versa. Now, we need only confirm that the theorem holds for a single line bundle.

𝒪

X

is a line bundle of degree 0.

ℓ

⁢

(

0

)

=

1

and

ℓ

⁢

(

K

)

=

g

. Thus, Riemann-Roch holds here, and thus for all line bundles.

Title

proof of Riemann-Roch theorem

Canonical name

ProofOfRiemannRochTheorem

Date of creation

2013-03-22 13:51:36

Last modified on

2013-03-22 13:51:36

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

6

Author

bwebste (988)

Entry type

Proof

Classification

msc 14H99

Generated on Fri Feb 9 17:59:23 2018 by

LaTeXML
space of functions associated to a divisor

space of functions associated to a divisor

Let

C

/

K

be a curve defined over the field

K

, and

D

a

divisor

for that curve. We define the

space of functions associated to a divisor

by

ℒ

⁢

(

D

)

=

{

f

∈

K

¯

⁢

(

C

)

*

:

div

⁢

(

f

)

≥

-

D

}

∪

{

0

}

,

where

K

¯

⁢

(

C

)

*

denotes the dual to the

function field

of

C

.

For any

D

,

ℒ

⁢

(

D

)

is a finite-dimensional

vector space

over

K

¯

, the

algebraic closure

of

K

, and we denote its dimension by

ℓ

⁢

(

D

)

, a somewhat ubiquitous number that, for example, appears in the

Riemann-Roch theorem for curves

.

Title

space of functions associated to a divisor

Canonical name

SpaceOfFunctionsAssociatedToADivisor

Date of creation

2013-03-22 14:12:25

Last modified on

2013-03-22 14:12:25

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

4

Author

mathcam (2727)

Entry type

Definition

Classification

msc 14H99

Generated on Fri Feb 9 17:59:25 2018 by

LaTeXML
elliptic surface

elliptic surface

Definition 1

.

Let

k

be a
field and let

C

/

k

be a smooth

projective curve

defined over the
field

k

and has genus

g

. The

function field

of

C

/

k

will be
denoted by

K

=

k

⁢

(

C

)

. An

elliptic surface

E

over the curve

C

is, by definition, a two-dimensional

projective variety

together
with:

1.

A morphism

π

:

ℰ

→

C

such that for all but
finitely many points

t

∈

C

⁢

(

k

¯

)

, the fiber

ℰ

t

=

π

-

1

⁢

(

t

)

is a

non-singular

curve of genus

1

,

2.

A

section

to

π

(the zero section)

σ

0

:

C

→

ℰ

.

With this definition,

E

/

K

may be regarded as an

elliptic curve

over the
field

K

.

Example 1

.

The

surface

y

2

=

x

3

+

t

is an elliptic surface over the curve

ℙ

1

⁢

(

ℚ

)

. It may be regarded as an elliptic curve over the function field

ℚ

⁢

(

t

)

.

References

1

R. Miranda,

The basic theory of elliptic surfaces

,
Dottorato di Ricerca in Matematica, Dipartimento di Mathematica dell’ UniversitÃÂ di Pisa, ETS Editrice Pisa, 1989.

2

J. Silverman,

Advanced Topics in the

Arithmetic of Elliptic Curves

, Graduate Texts in Mathematics 151,
Springer-Verlag, New York.

Title

elliptic surface

Canonical name

EllipticSurface

Date of creation

2013-03-22 15:34:16

Last modified on

2013-03-22 15:34:16

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Definition

Classification

msc 14J27

Related topic

EllipticCurve

Generated on Fri Feb 9 17:59:27 2018 by

LaTeXML
Shioda-Tate formula

Shioda-Tate formula

The main references for this part are the works of Shioda and Tate

[

2

]

,

[

4

]

,

[

5

]

.

Let

k

be a field and let

k

¯

be
a fixed

algebraic closure

of

k

. Let

ℰ

be an

elliptic surface

over a curve

C

/

k

and let

K

=

k

⁢

(

C

)

be the

function field

of

C

. Let

ℰ

¯

=

ℰ

⁢

(

k

¯

)

(or more precisely

ℰ

¯

=

ℰ

×

Spec

⁡

k

Spec

⁡

k

¯

). The Néron-Severi group of

ℰ

¯

, denoted by

NS

⁡

(

ℰ

¯

)

, is by

definition

the group of

divisors

on

ℰ

¯

modulo

algebraic

equivalence. Under
the previous assumptions,

NS

⁡

(

ℰ

¯

)

is a

finitely generated

abelian
group

(this is a consequence of the so-called ‘

theorem

of the
base’ which can be found in

[

1

]

). The Néron-Severi group
of

ℰ

, denoted by

NS

⁡

(

ℰ

)

, is simply the image of the group of
divisors on

ℰ

in

NS

⁡

(

ℰ

¯

)

. Let

T

⊂

NS

⁡

(

ℰ

)

be the

subgroup

generated by the image of the zero-section

σ

0

and all the

irreducible components

of the fibers of

π

.

T

is sometimes
called the “trivial part” of

NS

⁡

(

ℰ

)

.

Theorem

(Shioda-Tate formula).

For each

t

∈

C

let

n

t

be the number of
irreducible components on the fiber at

t

, i.e.

π

-

1

⁢

(

t

)

.
Then:

rank

ℤ

⁡

(

ℰ

/

K

)

=

rank

ℤ

⁡

(

NS

⁡

(

ℰ

)

)

-

rank

ℤ

⁡

(

T

)

=

rank

ℤ

⁡

(

NS

⁡

(

ℰ

)

)

-

2

-

∑

t

∈

C

(

n

t

-

1

)

.

References

1

S. Lang,

Fundamentals of Diophantine
Geometry

, Springer-Verlag (1983).

2

T. Shioda,

On elliptic modular

surfaces

, J. Math. Soc. Japan 24 (1972), 20-59.

3

T. Shioda,

An Explicit Algorithm for Computing the Picard Number of Certain Algebraic
Surfaces

, Amer. J. Math. 108 (1986), 415-432.

4

T. Shioda,

On the Mordell-Weil Lattices

, Commentarii Mathematici Universitatis Sancti Pauli,
Vol 39, No. 2, 1990, pp. 211-239.

5

J. Tate,

On the

conjectures

of Birch and Swinnerton-Dyer and a geometric analog

,
Séminaire Bourbaki, 9, Soc. Math. France, Paris, 1966, Exp. No.
306, 415-440, 1995.

Title

Shioda-Tate formula

Canonical

name

ShiodaTateFormula

Date of creation

2013-03-22 15:34:22

Last modified on

2013-03-22 15:34:22

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

4

Author

alozano (2414)

Entry type

Theorem

Classification

msc 14J27

Generated on Fri Feb 9 17:59:29 2018 by

LaTeXML