Conductors and minimal discriminants of hyperelliptic curves Padmavathi Srinivasan

advertisement
Conductors and minimal discriminants of
hyperelliptic curves
Padmavathi Srinivasan
MIT
AMS Summer Institute in Algebraic Geometry, Salt Lake City
July 31, 2015
Outline
1
Introduction
2
Conductors and minimal discriminants
3
Comparing conductors and discriminants
What are conductors and minimal discriminants?
Degenerating family
of hyperelliptic curves
Measures of
degeneracy
Artin conductor
Minimal discriminant
How are these related?
y 2 = (x − 1)(x 2 − t)
y2 = x3 − t
Outline
1
Introduction
2
Conductors and minimal discriminants
3
Comparing conductors and discriminants
Notation
R : complete discrete valuation ring
K : fraction field of R
k : residue field of R, algebraically closed, char 6= 2
ν : discrete valuation K → Z ∪ {∞}
t : a uniformizer of R, i.e., ν(t) = 1.
Examples: C[[t]], Zunr
p
Notation
f (x) : monic, squarefree, even degree ≥ 4 polynomial in R[x]
X : smooth projective model of the plane curve y 2 = f (x) over K
g:
genus of X
If f (x) = x d + ad−1 x d−1 + . . . + a0 factors as (x − α1 ) . . . (x − αd )
in K [x], then
Y
disc(f ) :=
(αi − αj )2
i<j
∈ K [a0 , . . . , ad−1 ]
Minimal discriminant of X
f (x) = x d + ad−1 x d−1 + . . . + a0 = (x − α1 ) . . . (x − αd )
The naive disciminant of y 2 = f (x) is defined to be
Y
∆f := ν(disc(f )) = ν
(αi − αj )2 .
i<j
The minimal discriminant ∆min of X is given by
!
f (x) ∈ R[x] such that y 2 = f (x)
∆min = ∆min (X ) := min ∆f is birational to X over K
∆min = 0 ⇐⇒ X has good reduction.
Artin conductor
X : a hyperelliptic curve over K
X : a proper, flat, regular R-scheme with XK ' X
XK : geometric generic fiber of X
Xk : special fiber of X
Fix a prime ` 6= char k. For any curve C over an algebraically
closed field of char 6= `, let
P
i (C , Q )
χ(C ) := 2i=0 (−1)i dim Hét
`
δ : Swan conductor for the representation H 1 (XK , Q` )
(integer, ≥ 0, measure of wild ramification).
Artin Conductor
X : a hyperelliptic curve over K
X : a proper, flat, regular R-scheme with XK ' X
XK : geometric generic fiber of X
Xk : special fiber of X
χ : `-adic Euler Poincaré characteristic
δ : Swan conductor for the representation H 1 (XK , Q` )
− Art(X /R) = χ(Xk ) − χ(XK ) + δ
=
(Saito)
Deligne discriminant.
Properties of the Artin Conductor
Art(X /R) is independent of `.
− Art(X /R) ≥ 0.
− Art(X /R) = 0 ⇐⇒ X is smooth or g = 1 and (Xk )red
is smooth.
Let n be the number of components of Xk and let be the
tame conductor. Then,
− Art(X /R) = (n − 1) + + δ
≥ n − 1.
When X is regular and semi-stable,
− Art(X /R) = # singular points of Xk .
Outline
1
Introduction
2
Conductors and minimal discriminants
3
Comparing conductors and discriminants
Earlier results (Ogg, Saito, Liu)
Let X be the minimal proper regular model of X .
If g = 1, then − Art(X /R) = ∆min [Ogg-Saito formula].
This also holds when char k = 2.
If g = 2, Liu showed that − Art(X /R) ≤ ∆min . He showed
that equality does not always hold.
Question: Does − Art(X /R) ≤ ∆min hold for hyperelliptic curves
of arbitrary genus?
Main result
Theorem ( )
Let X be a hyperelliptic curve over K and let X be the minimal
proper regular model of X . Assume that
(a) the Weierstrass points of X are K -rational, and,
(b) that the residue characteristic of K is not 2.
Then,
− Art(X /R) ≤ ∆min .
Explicit construction of a regular model X 0
The first step in the proof is the explicit construction of a regular
model X 0 for X (not necessarily minimal).
Lemma
Let Bl P1R be an arithmetic surface birational to P1R .
Let f be an element of the function field of P1R .
Assume that the odd multiplicity components of the divisor of f on
Bl P1R are disjoint.
p
Then, the normalization of Bl P1R in K (x, f (x)) is a proper
regular model for the hyperelliptic curve given by y 2 = f (x).
Proof strategy
1. Explicitly construct a regular model X 0 (not necessarily
minimal).
P
2. X 0 is strict simple normal crossings, Xk0 = mi Γi and
mi ∈ {1, 2}.


 X
X
X
Γi .Γj .
(1 − mi )χ(Γi ) +
(mj − 1)Γi .Γj +
− Art(X 0 /R) =


i
j6=i
3. Similarly decompose ∆min into local terms.
4. Prove a local comparison inequality.
5. Add local inequalities.
− Art(X /R) ≤ − Art(X 0 /R) ≤ ∆min .
i<j
Finally . . .
Thank you!
Download