Weil Reciprocity in Analytic GeometryTynan Ochse
3. Rigid difficulties
4. Current Work + Hopes for the Future
Theory Algebraic Complex analytic Rigid analytic Basic object Spec(A) Dn Spa(A)
Functions polynomial holomorphic “holomorphic”
Cohomology etale Deligne-Beilinson TBD
I K (C ) the function field of C
I D is a closed disk
I D is an open disk
(Algebraic) Weil Reciprocity
Version 1: Let C be a curve over an algebraically closed field K . For a divisor D and function f ∈ K (C ), define
f (D) = ∏ P
f (P)vP(D)
Then for any f , g ∈ K (C ) (with disjoint support), one has
f ((g)) = g((f ))
Version 2: Let C be a curve over an algebraically closed field K . Given invertible functions f and g in K (C ), we define a local symbol (f , g)P at each point P on C as follows:
(f , g)P = (−1)vP(f )·vP(g) f vP(g)
g vP(f ) (P)
where in this context vP is the order of a function at a point P. One then has that ∏
P∈C (f , g)P = 1
Reciprocity Laws Lightning Round
(−1) p−1
2 q−1
−1 else
Theorem (Gauss)
a, b ∈ Z[i ] distinct primes with a ≡ b ≡ 1 (mod 2 + 2i). Then
(−1) N(a)−1
4 N(b)−1
4
[ a
b
][ b
a
] = 1
For all a, b ∈ Z[ √ −1] not dividing 2 there is a unique[
a b
Theorem (Eisenstein)
l odd prime, α ∈ Z[ζl ] congruent some rational integer mod (1− ζl)2. Then (α
a
) l
for all a ∈ Z prime to l .
For α - p in Z[ζl ], there is a unique l th root of unity such that( α
p
Theorem (Hilbert)
k an algebraic number field. Then ∀a, b ∈ k×, one has∏ p
( a, b
Weil Reciprocity is King
The result is still true over non-algebraically closed fields: Let C = P1 and f , g polynomials in K (C ). Write f =
∏n i=1 fi and
g = ∏m
i=1 gi . Set Fi = k[x ]/fi and Gi = k[x ]/gi . Then
m∏ i=1
i=1
pn−1 p−1
(f mod g) pd−1
2 = (−1) ed(p−1)
2
2 = ( α p
Gm ×Gm → Gm
en : (Pic0C )[n]× (Pic0C )[n]→ µn ⊂ k×
Pic0C is the degree 0 divisors on C (isomorphism classes of line bundles with vanishing first Chern class).
en(D1,D2) = f1(D2) f2(D1) is not a-priori well-defined... but Weil
reciprocity assures it is.
If C = E is an elliptic curve, then E ∼= Pic0(E ).
en : E [n]× E [n]→ µn
Poincare duality in etale cohomology is a pairing
H i et(XK ,Z/nZ)× H2n−i
et (XK ,Z/nZ)→ µn
H1 et(E ,Z/lZ)× H1
Weil reciprocity = tame symbol = Poincare duality
.
Gm(K )×Gm(K )→ Gm(k)
Hence an isomorphism
We note Gm(K ) ∼= K0, and then dualize:
1 DGm(K )dR DGm(K ) ∼= BGm(K ) B1 K 0
d log
Qcoh(LocSysGm(D∗)) Qcoh(Gm(K )dR) Dmod(Gm(K ))∼ ∼
Variant in Characteristic p
Suppose k is finite and Γ a finite abelian group. The Tame symbol gives an isomorphism
Homgp(K×,BΓ) ∼= Homsch(D∗,BΓ)
The RHS is identified with Γ-torsors on D, i.e. Hom(πet 1 D∗, Γ).
On the other hand, the Lang isogeny gives an isomorphism
Hom(K×,BΓ) ∼= Hom(K×, Γ)
Take aways...
1. implies interesting modularity relations,
2. produces a well-defined pairing on algebraic group schemes, and, up to *insert theory here,*
3. is a crucial piece of geometric local class field theory
Now let X be an open Riemann surface.
(f , g)x = (−1)vx (f )vx (g) g vx (f )
f vx (g) (x)
( −1
2πi
∫ γ
log(f )d log(g)
) · g vx (f )
General idea is to compute the residue of some line bundle on X (monodromy). Monodromy away from ∂X is automatically trivial, so suffices to consider holomorphic functions defined near ∂X .
Robba Rings
O the structure sheaf on X , i : X → X .
Define RX as the subsheaf of i∗O consisting of germs of holomorphic functions that have a value on ∂X .
RX = ∏ π0(∂X )RX ,γ
For each γ ∈ π0(∂X ), have a local symbol (f , g)γ given by the monodromy formula.
Theorem (Deligne)
For all f ∈ H0(∂X ,R∗), g ∈ H0(X ,R∗) one has
1 = (f , g)∂X = ∏
} ∼= { π1(X )→ Gm}
Given functions f and g , and s ∈ π1(X ), we get a number
χf ,g (s) = (f , g)s ∈ Gm(C)
π1(X ) has a single relation ∏
i [ai , bi ] ∏
j γj = 1.
χf ,g ( ∏ i
[ai , bi ] ∏ j
C∗ is commutative, so χf ,g ( ∏
j γj) = ∏
0 Z(1) O O∗ 0 exp
0 G (C) G (O) 1 ⊗ Lie(G ) 0 dlog(g)
Hence a morphism
] (1)
G (C)[1] ∼−→ [ G (O)→ Lie(G )⊗ 1
] Yields
Passing to cohomology gives
H i (X ,O∗)⊗ H j(X ,G (O)) Ui,j−−→ H i+j+1(X ,G (C)(1))
For i = j = 0, this map attaches a G (C)(1)-torsor (f , g) to functions f and g .
We also have a residue map
Res : H1(X\{x},G (C)(1))→ A
If f is invertible on X\{x}, and g is a function on X\{x} with values in G , then set
Res U0,0 = (f , g)x ∈ G (C)
Let γ be a closed curve about x . Then (f , g)x is the monodromy of (f , g) about γ.
We can also specialize to various G : when G = Ga,
(f , g)x = Res(g · d log(f ))
And when G = Gm,
f vx (g) (x)
In general, provided g admits a logarithm in Lie(G ), one has
(f , g)x = exp(Res(g · d log(f )))
A choice of log(f ) defines a trivialization of (f , g), denoted (log(f ), g). Let γ be an oriented curve about x and fix y ∈ γ. Then a local horizontal section of (f , g) can be written as
(log(f ), g) · exp( −1
p−1 X → 0
And define Hq D(X ,Z(p)) = Hq(X ,Z(p)).
Just like in the algebraic case, the cup product on Deligne-Beilinson cohomology
H1 D(X ,Z(1))× H1
Rigid Geometry
Roughly, a rigid analytic space X is like a complex analytic space, but defined over a non-archimedian field k.
Non-archimedean means
2. |xy | = |x | · |y | 3. |x + y | ≤ max{|x |, |y |}
Definition The Tate algebra of dimension n is the subring kt1, · · · , tn of k[[t1, · · · , tn]] defined by
Tn(k) = { ∑ i
i→∞ |ai | = 0}
Definition An affinoid algebra A is an algebra isomorphic to Tn/I for some n and some closed ideal I . An affinoid is Spa(A) for some affinoid algebra A.
Definition A rigid analytic space over k is a locally ringed space (X ,OX ) with an admissible cover by affinoids.
Examples
2. Gm,k =
t , t qn )
3. Gm,k/qZ, q ∈ k∗, |q| < 1 (Tate elliptic curve).
4. XFF = ⊕
Rigid Difficulties p1
2. Covers by admissible opens (G-topology)
3. Want theory to be familiar (coherent sheaves, sheaf cohomology, GAGA) and robust (base changes, stalks, quasicoherent sheaves).
Rigid Difficulties p2
Now let X be a rigid analytic curve over a complete nonarchimedean field k . If one wants to produce an analogue of Deligne’s symbol, immediate questions are
1. What space of functions should we consider?
2. Do log, exp, ∫
, etc. even make sense in this setup?
3. Can the theory be set up to work uniformly for characteristic (0, 0), (0, p), and (p, p) situations?
I Characteritic (0, 0): R((t)) (|x | = q−ordt(x))
I Characteristic (0, p): Qp
I Characteristic (p, p): Fp((t)).
Robba Ring
Let Γ(r , 1) be the ring of convergent power series on the annulus {t | r < t < 1}.
Definition The Robba ring on D is
R = r<1
i→−∞ |ai |δi = 0
Replace D by the disk of radius p
−1 p−1 in characteristic p.
Rigid Analytic Tame Symbol
The goal is to produce a nondegenerate, perfect local symbols
R∗ ×R∗ (·,·)x−−−→ Gm
Unfortunately, log, exp and ∫
The upshot is one can extract monodromy-type information from 2-determinants [Garcia-Lopez, Clausen].
Linear algebra of Frechet/Tate-analytic spaces
Given f , g ∈ R∗X ,x , we get lines 2det(f ) and 2det(g) and two isomorphisms
2det(f )⊗ 2det(g) 2det(g)⊗ 2det(f )
RX = ∏ RX ,x and (f , g) =
∏ x(fx , gx)x
1 = (f , g) = ∏ x
Moreover, there exists a subquotient H of RX such that
H × H R/k ×R/k
Gm
log
exp(Res(fdg))
Local Global
Multiplicative R∗X ×R∗X → k∗ BunGm(X )× BunGm,c(X )→ · · · Additive RX ×RXdt → k BunGa(X )× BunGa,c,(X )→ · · ·
R∗X = Gm × Z× H+ × H−
Hopes for the Future
2. Rigid Class Field Theory
3. Matching the Rigid Tame Symbol with the cup product of some cohomology theory
Thank you!