Conductor of an abelian variety

For an abelian variety A defined over a field F as above, with ring of integers R, consider the Néron model of A, which is a 'best possible' model of A defined over R. This model may be represented as a scheme over

:Spec(R)

(cf. spectrum of a ring) for which the generic fibre constructed by means of the morphism

:Spec(F) → Spec(R)

gives back A. Let A⁰ denote the open subgroup scheme of the Néron model whose fibres are the connected components. For a maximal ideal P of R with residue field k, A⁰_k is a group variety over k, hence an extension of an abelian variety by a linear group. This linear group is an extension of a torus by a unipotent group. Let u_P be the dimension of the unipotent group and t_P the dimension of the torus. The order of the conductor at P is

:$f\_P\; =\; 2u\_P\; +\; t\_P\; +\; \backslash delta\_P\; ,\; \backslash ,$

where $\backslash delta\_P\backslash in\backslash mathbb\; N$ is a measure of wild ramification. When F is a number field, the conductor ideal of A is given by

:$f=\; \backslash prod\_P\; P^\{f\_P\}.$

• A has good reduction at P if and only if $u\_P=t\_P=0$ (which implies $f\_P=\backslash delta\_P=\; 0$).
• A has semistable reduction if and only if $u\_P=0$ (then again $\backslash delta\_P=\; 0$).
• If A acquires semistable reduction over a Galois extension of F of degree prime to p, the residue characteristic at P, then δ_P = 0.
• If $p>\; 2d+1$, where d is the dimension of A, then $\backslash delta\_P=0$.
• If $p\backslash le\; 2d+1$ and F is a finite extension of $\backslash mathbb\{Q\}\_p$ of ramification degree $e(F/\backslash mathbb\{Q\}\_p)$, there is an upper bound expressed in terms of the function $L\_p(n)$, which is defined as follows:

: Write $n=\backslash sum\_\{k\backslash ge0\}c\_kp^k$ with

:$(*)\backslash qquad\; f\_P\; \backslash le\; 2d\; +\; e(F/\backslash mathbb\{Q\}\_p)\; \backslash left(\; p\; \backslash left\backslash lfloor\; \backslash frac\{2d\}\{p-1\}\; \backslash right\backslash rfloor\; +\; (p-1)L\_p\backslash left(\; \backslash left\backslash lfloor\; \backslash frac\{2d\}\{p-1\}\; \backslash right\backslash rfloor\; \backslash right)\; \backslash right).$

:Further, for every $d,p,e$ with $p\backslash le\; 2d+1$ there is a field $F/\backslash mathbb\{Q\}\_p$ with $e(F/\backslash mathbb\{Q\}\_p)=e$ and an abelian variety $A/F$ of dimension $d$ so that $(*)$ is an equality.

{{Reflist}}

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• {{cite book | author=S. Lang | authorlink=Serge Lang | title=Survey of Diophantine geometry | url=https://archive.org/details/surveydiophantin00lang_347 | url-access=limited | publisher=Springer-Verlag | year=1997 | isbn=3-540-61223-8 | pages=[https://archive.org/details/surveydiophantin00lang_347/page/n83 70]–71 }}
• {{cite journal | author=J.-P. Serre |author2=J. Tate | title=Good reduction of Abelian varieties | journal=Ann. Math. | volume=88 | year=1968 | pages=492–517 | doi=10.2307/1970722 | issue=3 | publisher=The Annals of Mathematics, Vol. 88, No. 3 | jstor=1970722 }}

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Category:Abelian varieties

Category:Diophantine geometry

Category:Algebraic number theory