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In mathematics, a Jacobi sum is a type of character sum formed with Dirichlet characters. Simple examples would be Jacobi sums J(χ, ψ) for Dirichlet characters χ, ψ modulo a prime number p, defined by
J
(
χ
,
ψ
)
=
∑
χ
(
a
)
ψ
(
1
−
a
)
,
{\displaystyle J(\chi ,\psi )=\sum \chi (a)\psi (1-a)\,,}
where the summation runs over all residues a = 2, 3, ..., p − 1 mod p (for which neither a nor 1 − a is 0). Jacobi...
In algebraic geometry, the Néron model (or Néron minimal model, or minimal model)
for an abelian variety AK defined over the field of fractions K of a Dedekind domain R is the "push-forward" of AK from Spec(K) to Spec(R), in other words the "best possible" group scheme AR defined over R corresponding to AK.
They were introduced by André Néron (1961, 1964) for abelian varieties over the quotient field of a Dedekind domain R with perfect residue fields, and Raynaud (1966) extended this construction to semiabelian varieties over all Dedekind domains.
== Definition ==
Suppose that R is a Dedekind...
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I have a question about proof that Tamagawa number, $[E (K): E_0 (K)]$ is finite.
Could you please tell (correct) me any strange parts about my understanding of the outline of the proof ?
My understanding (outline)
"It is difficult to prove elementaryly that $E (K) / E_0 (K)$ is finite. Therefore...
In mathematics, the Tamagawa number
τ
(
G
)
{\displaystyle \tau (G)}
of a semisimple algebraic group defined over a global field k is the measure of
G
(
A
)
/
G
(
k
)
{\displaystyle G(\mathbb {A} )/G(k)}
, where
A
{\displaystyle \mathbb {A} }
is the adele ring of k. Tamagawa numbers were introduced by Tamagawa (1966), and named...
In mathematics, the Weil conjecture on Tamagawa numbers is the statement that the Tamagawa number
τ
(
G
)
{\displaystyle \tau (G)}
of a simply connected simple algebraic group defined over a number field is 1. In this case, simply connected means "not having a proper algebraic covering" in the algebraic group theory sense, which is not always the topologists' meaning.
== History ==
Weil (1959) calculated the Tamagawa number in many cases of classical groups and observed that it is an integer in all considered cases and that it was...
