Automorphisms of Curves

Integral and modular representation theory

6 July 2016 Karlovassi Samos
Aristides Kontogeorgis
University of Athens Department of Mathematics

Content

Algebraic curves

  • Consider projective algebraic curves $X$ defined over an algebraically closed field $k$ of positive characteristic $p\geq 0$ Instead of the curve one can study the function field $F_X=k(X)$.
  • The Automorphism group $\mathrm{Aut}(X)$ is the group of automorphisms $\sigma:X\rightarrow X$.
  • The genus $g$ of a curve is defined as $\dim H^0(X,\Omega_X)$

Group of Automorphisms

  • $g=0$, $X \cong \mathbb{P}^1(k)$, $\mathrm{Aut}(X)=\mathrm{PGL}(2,k)$

  • $g=1$, $X$ is an elliptic curve and $E \subset \mathrm{Aut}(X)$

  • $g\geq 2$, the automorphism group is finite:

Hurwitz Bound

Characteristic zero

(1)
\[|\mathrm{Aut}(X)| \leq 84 (g-1) \]

Characteristic $p$

(2)
\[|\mathrm{Aut}(X)| \leq 16g^4 \]

Automorphisms groups of Fermat curves

\[X:x^n+y^n+z^n=0 \]

  • $n-1$ is not a power of the characteristic $\mathrm{Aut}(X)=(\mathbb{Z}/n\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z}) \rtimes S_3$

  • $n-1=p^h$,

    \[x^n+y^n+z^n=(x,y,z) \begin{pmatrix} x \\ y \\ z \end{pmatrix}^{p^h} \]

    and $\mathrm{Aut}(X)=\mathrm{PGU}(3,p^{2h})$.

Action of the automorphism group

The automorphism group acts on every object of the curve, in particular on $H^0(X,\Omega_X^{\otimes n})$, inducing a representation

(3)
\[\rho: G=\mathrm{Aut}(X) \longrightarrow H^0(X,\Omega_X^{\otimes n}) \]

Aim: Describe the irreducible - indecomposable summands of the above representation.

Consider the ramified covering $X \rightarrow X/G$, or equivalently the Galois extension $k(X)/k(X)^G$.

Classical problem

Determine the decomposition of $H^0(X,\Omega_X)$ into its indecomposable direct summands.

Problem posed by Hecke and solved in by Chevalley and Weil using character theory.

Definition 1.
For each point $x\in X$ let $m_{X,x}$ be the maximal ideal of the local ring $\mathcal{O}_{X,x}$ and $k(X)$ the residue field of $x$. The fundamental character is the character

$\theta_x:G(x) \rightarrow k(x)^*=\mathrm{Aut}(m_{X,x}/m^2_{X,x}).$

Theorem 1.
If $k$ is algebraically closed and $p \nmid |\mathrm{Aut}(X)|$ then fundamental characters describe the indecomposable summands.

Positive Characteristic

More difficult problem because:

  • Modular representation theory: Irreducible representation is a different than indecomposable. $\mathbb{Z}/p\mathbb{Z}=\langle \sigma \rangle \rightarrow \mathrm{GL}(2,k)\qquad \sigma \mapsto \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$

  • Wild ramification: The decomposition groups $G(x)=\{\sigma \in G: \sigma(P)=P \}$ are not cyclic groups anymore.

  • There is no classification for indecomposable modules unless $G$ is cyclic.

  • If $G=\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p \mathbb{Z}$, $p$ prime $p>2$ then the classification of indecomposable representations is considered to be impossible

Higher ramification groups

Definition 2.
Ramification filtration For $i\geq 0$ the $i$-th lower ramification group $G_{x,i}$ of $G$ at $x$ is the subgroup of elements $\sigma$ which fix $x$ and act trivially on $\mathcal{O}_{X,x}/m_{X,x}^{i+1}$. There is a sequence of groups:

\[G_{x,0} \rhd G_{x,1} \rhd G_{x,2} \rhd \cdots G_{x,n} \rhd \]

such that $G_{x,0}/G_{x,1}$ is a cyclic group of order prime to $p$ and $G_{x,i}/G_{x,i+1}$ are elementary $p$-abelian groups.

$k[G]$-module structure

Known results:

  • $X\rightarrow X/G$ ramified, or $(|G|,p)=1$ [Tamagawa, Valentini]

  • Wild Ramification: Weak ramification [B. Koeck]

  • Cyclic Group case: [Valentini-Madan, Karanikolopoulos K]

Finding Bases

  • Consider elements of the form $f dg$, for $f,g\in k(X)$.
  • Find linear independend elements such that $\mathrm{div}(f dg) \geq 0$.
  • Boseck method: Works for cyclic covers of $\mathbb{P}^1$.
  • Consider the action of the automorphism on the constructed basis.

Boseck method

Consider the function field, corresponding to a curve $X$:

\[K(x,y): y^p-y=r(x)=\frac{f(x)}{\prod_{i=1} p_i(x)^{\lambda_i} } \]

Theorem 2.
The following differentials form a basis of $H^0(X,\Omega_X)$

\[x^\nu g_\mu^{-1} y^\mu dx, \nu=0,\ldots,t^{(\mu)}, 0 \leq \mu \leq p-1 \]
\[g_\mu(x)=\prod_{i=1}^r p_i(x)^{\lf \frac{(p-1-\mu)\lambda_i + p-1}{p} \rf} \]
\[t_\mu =\sum_{i=1}^r \mathrm{deg}(p_i) \lf \frac{(p-1-\mu)\lambda_i + p-1}{p} \rf \]

A similar formula holds for coverings $y^n=f(x)$.

Group action on Basis

The action of $\mathrm{Gal}(K(x,y):k(x))$ is given by $x\mapsto x$, $y\mapsto y+1$.

Fix $\nu$.

\[x^\nu g_\mu^{-1} y^\mu dx, \mapsto x^\nu g_\mu^{-1} (y+1)^{\mu} dx. \]

Find Jordan blocks

p-adic uniformization method

Mumford $p$-adic analytic method:

$K$ is a a complete discrete valued field $\mathbb{Q}_p$ or $k[[t]]$. Curves with split analytic reduction are isomorphic to the algebraization of rigid analytic curves of the form $\Gamma\backslash(\mathbb{P}^{1,\mathrm{an}}_K - \mathcal{L}_\Gamma)$

Here, $\Gamma$ is a finitely generated torsion-free discrete subgroup of $\mathrm{PGL}(2,K)$, called a Schottky group, and $\mathcal{L}_\Gamma$ is the set of limit points.

A smooth projective curve obtained in this way, denoted by $X_{\Gamma}$, is called a Mumford curve, and the uniformization just described provides us with a set of tools similar to those coming from the uniformization theory of Riemann surfaces.

Polydifferentials on Mumford curves

Theorem 3.
[P. Schneider, J. Teitelbaum]

\[H^0 (X_\Gamma,\Omega_{X_\Gamma}^{\otimes n}) \cong H^1(\Gamma, P_n), \]

where $P_n$ denotes the space of polynomials of one variable of degree $\leq 2n-2$

In particular

\[H^0(X_\Gamma,\Omega_{X_\Gamma}) \cong \mathrm{Hom} (\Gamma,K). \]

Theorem 4.
We have $\mathrm{Aut}(X_\Gamma)=N/\Gamma$, where $N$ is the normalizer of $\Gamma$ in $\mathrm{PGL}(2,K)$. $N/\Gamma$ defines a well defined action on $H^1(\Gamma,P_n)$ by the action $\delta \mod \Gamma \in N/\Gamma$:

\[(d^{\delta})(\gamma)=[d(\delta\gamma\delta^{-1})]^{\delta} \]

Example: Artin-Schreier Mumford curves

[A.K. - F. Kato]

Let $K$ be a complete non-archimedean valued field of characteristic $p>0$, and $q$ a power of $p$. For $\lambda\in K$ with $0<|\lambda|<1$, the smooth projective model of the affine plane curve defined by the equation

\[(x^q-x)(y^q-y)=\lambda \]

Automorphism Groups ASM-curves

Proposition 1.
The Artin-Schreier-Mumford $X_\Gamma$ where the group $\Gamma$ is, up to conjugacy, given by the commutator group $\Gamma=[A,B]$ of the cyclic subgroups $A,B \subset \mathrm{PGL}(2,K)$ of order $p$ generated by

\[\epsilon_A=\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}\quad\textrm{and}\quad\epsilon_B=\begin{pmatrix} 1 & 0 \\ s & 1 \end{pmatrix}, \]

respectively, where $s\in K^{\times}$ and $|s|>1$.

Automorphism Groups ASM-curves

The groups $A$ and $B$ generate a discrete subgroup $N\subseteq\mathrm{PGL}(2,K)$, which is isomorphic to the free product $A\ast B$.
$\Gamma$ is a normal subgroup of $N$ and $N/\Gamma\cong A\times B;$ $\Gamma$ is a free group of rank $(p-1)^2$ with the basis given by the commutators $e_{i,j}=[\epsilon^i_A,\epsilon^j_B]$ $(=\epsilon^i_A\epsilon^j_B\epsilon^{-i}_A\epsilon^{-j}_B)$ for $i,j=1,\ldots,p-1$.

Galois module structure

Theorem 5.

* As an $A$-module:

\[H^0(X_\Gamma,\Omega_X)\cong L^{p-1}\otimes_\mathbb{Z} K, \]

where $L$ is the integral representation corresponding to the matrix

\[M= \begin{pmatrix} -1 & -1 & -1 & \cdots & -1 \\ 1 & 0 & \cdots & \cdots & 0 \\ 0 & 1 & 0 & & \vdots \\ \vdots & \ddots& \ddots & \ddots & \vdots \\ 0 & \cdots & 0 & 1 & 0\\ \end{pmatrix} \in \mathrm{GL}(p-1,\mathbb{Z}) \]
  • As a $K[G]$-module $H^0(X_\Gamma,\Omega_X)$ is an indecomposable module.

Groups with cyclic $p$-Syllow

[Bleher, Chinburg, K.]

Theorem 6.
Suppose that $G$ has non trivial cyclic Syllow $p$-subgroups. Then then $k[G]$-module structure of $H^0(X,\Omega_X)$ is uniquely determined by the lower ramification groups and the fundamental charcters of closed points $x$ which are ramified on the cover $X \rightarrow X/G$.

Green Ring

GreenRing

Let $a(k[G])$ be the Green ring: the ring consisting of $\mathbb{Z}$-linear combinations of symbols $[M]$, one for each isomorphism class of finitely generated $k[G]$-modules, with relations:

\[[M]+[M']=[M \oplus M'] \]
\[[M]\cdot [M']=[M\otimes_k M'] \]

where $G$ acts diagonally on $M\otimes_k M'$.

Define the representation algebra

\[A(k[G])=\mathbb{Q}\otimes_\mathbb{Z} a(k[G]). \]

Green Ring

\[ a(K[G]) \subset A(k[G]) \]

and both rings share the same unit element: $k_G$, the trivial simple $k[G]$-module.

We also have induction maps:

\[a(k[H]) \rightarrow a(k[G]) \qquad A(k[H]) \rightarrow A(k[G]) \]

Also for finitely generated $k[G]$-modules $[N]=[N']$ if and only if $N\cong N'$ as $k[G]$-modules.

Conlon Induction theorem

In $A(k[G])$ we have for certain $\alpha_H\in \mathbb{Q}$:

\[[k_G]=\sum_{H \in \mathcal{H'} } \alpha_H [\mathrm{Ind}_H^G(k_H)] \]

So

\[M \otimes_k \mathrm{Ind_H^G}(k_H) \cong \mathrm{Ind}_H^G(M_H \otimes_k k_H)\cong \mathrm{Ind}_H^G(M_H) \]
\[[M]=\sum_{H \in \mathcal{H}'} \alpha_H [\mathrm{Ind}_H^G M_H] \]

Study of $p$-hypo-elementary groups

$H=P\rtimes_{\chi} C$, $\chi:C \rightarrow \mathbb{F}_p^*$.

  • Description of indecomposable summands: $V(\lambda,k)$ is a $k$-dimensional $H$-module, $1\leq k \leq p$) where $e$ is an eigenvector for $\tau$, $\tau e=\zeta_n^\lambda e$, and a basis is formed by $(\sigma-1)^\nu e, \nu=0,\ldots, k-1$.
\[V=\bigoplus_{\lambda=0}^{n-1} \bigoplus_{k=1}^{p^h} V(\lambda,k)^{d(\lambda,k)}. \]

Aim: Compute the exponents $d(\lambda,k)$.

Idea of the proof

Let $I=\langle \tau \rangle$ be the cyclic subgroup of $P$ generated by the Sylow $p$-subgroups of the inertia groups of all closed points of $X$.

For all integers $j\leq 0$ let $\Omega_X^{(j)}$ be the kernel of the action of $\mathcal{J}^j=k[I] (\tau-1)^j$ on $\Omega_X$.

Proposition 2.
For $0\leq j \leq \#I-1$ the action of $\mathcal{O}_Y$ and of $H$ on $\Omega_X$ makes the quotient sheaf $\mathcal{L}_j=\Omega_X^{(j+1)}/\Omega_X^{(j})$ into a sheaf of $\mathcal{O}_Y[H]$-modules.

\[\mathcal{L}_j \cong S_{\chi^{-j}} \otimes_k \Omega_Y(D_j), \]

for an effective divisor $D_j$ on $Y$.

Idea Use Riemann-Roch theorem on the quotient curve $Y$, to compute $\mathrm{dim}(\Omega_Y(D_j))$

An Example - Reduction of modular curves in char 3.

  • $X_\ell(p)$ modulo $\ell\neq p$ has an automorphism group which contains $\mathrm{PSL}(2,p)$, which has order $(p^2-1)p/2$.

  • The cover $X_3(p)\rightarrow X_3(p)/\mathrm{PSL}(2,p)$ is branched at $2$ points with inertia groups $S_3$ and $\mathbb{Z}/p$ - moreover, in the first case the second ramification group is trivial.

Some decompositions

\[\begin{array}{ccc} p= 7 & \oplus V(1,3) & g= 3 \\ p= 11 & V(0,3)^{ 4 } \oplus V(1,3)^{ 4 } \oplus V(0,2) & g= 26 \\ p= 13 & V(0,3)^{ 8 } \oplus V(1,3)^{ 8 } \oplus V(0,2) & g= 50 \\ p= 17 & V(0,3^ 2 )^{ 5 }\oplus V(1,3^ 2 )^{ 9 } \oplus V(0, 7 ) & g= 133 \\ p= 19 & V(0,3^ 2 )^{ 8 }\oplus V(1,3^ 2 )^{ 13 } \oplus V(0, 7 ) & g= 196 \\ p= 23 & V(0,3)^{ 61 } \oplus V(1,3)^{ 62 } \oplus V(0,2)^{ 2 } \oplus V(1,2) & g= 375 \\ p= 29 & V(0,3)^{ 130 } \oplus V(1,3)^{ 136 } \oplus V(0,2)^{ 2 } \oplus V(1,2)^{ 2 } & g= 806 \\ p= 31 & V(0,3)^{ 162 } \oplus V(1,3)^{ 169 } \oplus V(0,2)^{ 2 } \oplus V(1,2)^{ 2 } & g= 1001 \\ \end{array} \]

Lift to Characteristic zero

  • Modular representation theory: Brauer Characters, lift the representation to characteristic zero.
  • Lift the whole curve together with the automorphism group, not always possible!
  • Oort conjecture, cyclic group actions always lift.

Question: How the representation theory could help us understand deformations and liftings?

Deformation Theory

A deformation of the curve $X$ is a relative curve (proper, smooth) over a local ring $R$ with maximal ideal $m$

$\mathcal{X} \rightarrow \mathrm{Spec}(R)$ such that $X \cong \mathcal{X} \times_{\mathrm{Spec} R} \mathrm{Spec} R/m$

\[\xymatrix{ X\cong \mathcal{X} \times_{\mathrm{Spec} R} \mathrm{Spec} R/m \ar[r] \ar[d] & \mathcal{X} \ar[d] \\ \mathrm{Spec}(k)\cong R/m \ar[r] & \mathrm{Spec}(R) } \]

such that $\psi$ gives the identity on the special fibres.

Equivalence of Deformations

Two deformations $\mathcal{X}_1$ $\mathcal{X}_2$ are considered to be equivalent if there is an isomorphism $\psi$ making the diagram commutative:

\[\xymatrix{ \mathcal{X}_1 \ar[rr]^{\psi} \ar[dr] & & \mathcal{X}_2 \ar[dl] \\ & \mathrm{Spec} R & \\ X \ar[uu] \ar[dr] \ar[rr]^{\qquad\mathrm{Id}} & & X\ar[uu] \ar[dl] \\ & \mathrm{Spec} k \ar[uu]& } \]

The Deformation functor

${D_{\rm gl}}: \mathcal{C} \rightarrow {\rm Sets},$ where $\mathcal{C}$ is the category of local Artin algebras.

(4)
\[A \mapsto\left\{\mbox{\begin{tabular}{l}Equivalence classes \\of deformations of \\couples (X,G) over A\end{tabular}}\right\} \]

Problems:

  • Compute the dimension of the tangent space $T:=D(k[\epsilon]/\langle \epsilon^2\rangle)$, in terms of the ramification data of the cover $X\rightarrow X/G$.

  • For a vector $v\in T$, integrate to a deformation $\mathcal{X} \rightarrow \mathrm{Spec}(A)$ over a complete local ring.

Moduli Space and Tangent deformation functor

ModuliSpace2

Local Deformation functor

There is a representation: \[\rho: G(P) \rightarrow \mathrm{Aut} (k[[t]]) \] expressing the action of the decomposition group to the completed local ring at a point.

The local deformation functor is defined:

\[D_P:\mathcal{C} \rightarrow {\rm Sets}, A \mapsto \left\{ \text{ \begin{tabular}{l} lifts $G(P)\rightarrow \mathrm{Aut} (A[[t]])$ of $\rho$ mod- \\ulo conjugation with an element \\ of $\mathrm{ker}(\mathrm{Aut} A[[t]]\rightarrow k[[t]] )$ \end{tabular} } \right\} \]

Local-Global principle

  • Cohomology theories appear as derived functor of appropriate left exact functors. For example for group cohomology we apply the functor of invariant element of $G$-modules, and for Zarisky cohomology the functor of global sections.

  • Grothendieck's Tohoku paper: Cohomology of composition of two left exact functors. In particular global sections and group invariants. Machinery for computations: 5-term exact sequence.

  • The study of the functor $D_{\mathrm{gl}}$ can be reduced to the study of the following deformation functors attached to each wild ramification point $P$ of the cover $X \rightarrow X/G$:
\[{\tiny 0 \rightarrow H^1\left(\frac{X}{G},\pi_*^G(T_X) \right) \rightarrow H^1(G,X,T_X) \rightarrow H^0\left( \frac{X}{G}, R^1 \pi_*^G (T_X) \right) \rightarrow 0.} \]

Representation theory again

Idea: Serre duality:

\[H^1(X,T_X)\cong H^0(X,\Omega^{\otimes 2})^* \]

Equivariant version of it: [A.K.]

\[{D_{\rm gl}}(k[\epsilon]/\langle \epsilon^2 \rangle)=H^1(X,T_X)^G \cong H^0(X,\Omega_X^{\otimes 2})_G. \]

For a $G$-module $A$,

$A^G:=\{a \in A: a^g=a \}$

$A_G:=A/\langle ga-a: a\in A, g\in G \rangle.$

Computations

  • If $G=\mathbb{Z}/p$, then $A^G=A_G$.

  • If $G=\mathbb{Z}/p \times \cdots \times \mathbb{Z}/p$, then we can have $A^G \neq A_G$.

Idea: The knowledge of $k[G]$-module structure can lead to a computation of $\mathrm{dim} D_{\mathrm{gl}}(k[\epsilon])$.

Example:Certain Elementary Abelian extensions

$F/K(x)$ with $y^{p^n}-y =\frac{g(x)}{(x-a_1)^{\Phi(1)} \cdots (x-a_s)^{\Phi(s)}}$

\[\Omega(m) \cong \bigoplus_{j=1}^{p^n} W_j^{d_j}. \]
\[\Gamma_k=\sum_{i=1}^s \lf \frac{m(p^n-1)(\Phi(i)+1)-k\Phi(i)}{p^n}\rf \]
\[d_{p^n}=\Gamma_{p^n-1}(m)-2m+1,$$ $$d_j=\Gamma_{j-1}(m)-\Gamma_j(m), \qquad j=1,\ldots,p^n-1, \]
\[W_j=\langle \theta_0,\ldots, \theta_{j-1} \rangle_K, \qquad \sigma_\alpha (\theta_i)=\sum_{\ell=0}^i \binom{i}{\ell} \alpha^{i-\ell} \theta_\ell. \]

Example: Certain Elementary Abelian extensions

Theorem 7.
Assume that $j$ has the $p$-adic expansion $j=\sum_{i=1}^n a_i p^i$. Let $\chi$ be the map

$\chi: \{0,\ldots,p-1\} \rightarrow \{0,1\}$

defined by:

$\chi(a):=\begin{cases}1 & \mbox{ if } a \neq 0,\\0 & \mbox{ if } a =0\end{cases}$

then $\mathrm{dim}( (W_j)_G)=\sum_{i=1}^n \chi(a_i).$

Example: Certain Elementary Abelian extensions

Theorem 8.
$\mathrm{dim}(H^1(X,G,T_X))=\begin{cases}s(n+2)-3 & \mbox{ if } p>3 \\s(n+1)-3 & \mbox{ if } p=3 \\sn-3 & \mbox{ if } p=2\end{cases}$

Action on L(D) for G-invariant D.

Let $P$ be a fully ramified point of $X\rightarrow X/G_1(P)$.

Theorem 9.
Assume that $g_X \geq 2$, $p\geq 2,3$. Consider the Weierstrass semigroup at $P$ up to the first pole number $m_r$ not divisible by $p$:

$0=m_0< \cdots < m_{r-1}< m_r,$ and select functions in $k(X)$ $f_0,\ldots,f_r$ with $(f_i)_\infty=m_i P$. Then the natural representation

\[\rho: G_1(P) \rightarrow \mathrm{GL}(L(m_r P)) \]

is faithful.

Explicit description of actions on $Aut(k[[t]])$.

  • We can select the uniformizer $t$ such that $f_r=t^{-m}$, $m=m_r$
  • The action is now given in closed form:
\[\sigma(t) =t \left(1 +t^m \sum_{\nu=1}^r a_{\nu,r} f_\nu\right)^{-1/m}. \]
  • Idea: Work with General Linear group instead of $\mathrm{Aut}(k[[t]])$

Representation Filtration

For each $0 \leq i \leq r$, consider the representations:

\[\rho_i: G_1(P) \rightarrow \mathrm{GL}(L(m_iP)), \]

which give rise to the decreasing sequence of groups:

\[G_1(P)=\ker \rho_0 \supseteq \ker \rho_1 \supseteq \ker \rho_2 \supseteq \cdots \supseteq \ker \rho_r =\{1\}. \]

We also obtain a tower of function fields:

\[F^{G_1(P)}=F^{\ker \rho_0} \subseteq F^{\ker \rho_1} \subseteq \cdots \subseteq F^{\ker \rho_r}=F. \]

Theorem 10.
[A.K. - S. Karanikolopoulos]

If $X \rightarrow X/G$ is a HKG-cover, then the representation and the ramification filtrations coincide.

Harbater-Katz-Gabber Covers

Definition 3.
A HKG-cover is a cover $X \rightarrow X/G=\mathbb{P}^1$, where $G$ is a $p$-group and there is only one branch point point $P\in X/G$.

The Harbater–Katz–Gabber compactification theorem asserts that given an action of a $p$-group $G$ on $\mathcal{O}=k[[t]]$, there is a HKG-cover $X_{\mathrm{HKG}}\rightarrow \mathbb{P}^1$ ramified only at one point $P$ of $X$ with Galois group $G=\mathrm{Gal}(X_{\mathrm{HKG}}/\mathbb{P}^1)=G_0$ such that $G_0(P)=G_0$ and the action of $G_0$ on the completed local ring $\hat{\mathcal{O}}_{X_{\mathrm{HKG}},P}$ coincides with the original action of $G_0$ on $\mathcal{O}$.

$G$-module structure for HKG-covers

  • The HKG-compactification allows us to attach global invariants to the local case, like genus, Jacobian, $p$-rank, differentials etc.
  • Study of $H^0(X,\Omega_X^{\otimes m})$.

Select a function $f_{i_0} \in k(X)$ such that $k(X)^{G}=k(f_{i_0})$.

\[\mathrm{div}(df_{i_0}^{\otimes m})=\left(-2m p^{h_0} + m \sum_{i=1}^n (b_i-b_{i-1})(p^{h-1}-1)\right) P, \]

where $b_0=-1,p^{h_0}=|G_1(P)|,p^{h_i}=|\ker \rho_{c_{i+1}}|=|G_{b_{i+1}}|, \mbox{ for } i \geq 1.$

$G$-module structure for HKG-covers

Theorem 11.
For every pole number $\mu$ select a function $f_\mu$ such that $(f_\mu)_\infty=\mu P$. The set

$\{f_\mu df_{i_0}^{\otimes m}: \deg(f_i) \leq m (2g_X-2)\}$

forms a basis for the space of $m$-holomorphic (poly)differentials of $X$.

Theorem 12.
The module $H^0(X,\Omega_X^{\otimes m})$ is a direct sum of $N=\lf \frac{m(2g-2)}{p^{h_0}}\rf$ direct indecomposable summands.

Corollary: If $|G_1(P)| \geq m(2g-2)$, then $N=1$. In particular curves with big-action (in the sense of M.Matignon-M.Rocher) have one indecomposable summand.

Integral Representation Theory

The problem

Lemma 1.
Let $R$ be a local integral domain with residue field $k$ and quotient field $L$. Every finitely generated $R$ module $M$ so that $\dim_k M \otimes_R k=\dim_L M\otimes_R L=r$ is free $R$-module of rank $r$.

Let $\mathcal{X} \rightarrow \mathrm{Spec} R$ be a deformation of $(X,G)$. The spaces $M_n=H^0(\mathcal{X},\Omega_\mathcal{X}^{\otimes n})$ are free $R$-modules.

Problem: Describe the module structure of $M_n$ within the theory of integral representations

Remark Usually the term integral representation is reserved for $\mathbb{Z}[G]$-modules. Our situation is a little bit easier since we work over complete local rings, and we can add the eigenvalues $R=W(k)(\zeta_n)$.

Example: Cyclic groups

Description of the modules.

We extend the binomial coefficient $\binom{i}{j}$ by zero for $i < j$. For $a\in \mathbb{Z}$ we consider the matrix $A_a=(a_{ij})$, defined by $a_{ij}=\binom{j-1}{i-1}.$ This is a lower triangular matrix defined over $\mathbb{Z}$, and in positive characteristic has order $p$.

The representation $\rho:\sigma^i \rightarrow A_a^i$ is indecomposable.

Idea: Over $k$ we consider the vector space with basis $\{1,x,x^2,\ldots,x^{a-1}\}$, with the natural action of the generator $x\mapsto x+1$. (so $x^i \mapsto (x+1)^i$).

Description of the modules

  • $S$ is an integral domain which is a $W(k)[\zeta_p]$-algebra

  • $\lambda=\zeta_p-1$ ($\lambda \equiv 0 \mod m$)

  • For $a_0,a_1 \in \mathbb{Z}$ we define the $S$-module

$\tilde{V}_{a_0,a_1}:=_S \langle (\lambda X+1)^i: a_0 \leq i \leq a_0+a_1 \rangle \subset S(\lambda X+1).$

The modules $\tilde{V}_{a_0,a_1},\tilde{V}_{a_0+p,a_1}$ are isomorphic.

Remark: The reduction of this basis is trivial.

Consider instead the following module: $V_{a_0,a_1}:=_S \langle (\lambda X+1)^{a_0} X^i : 0 \leq i < a_1 \rangle.$

The modules $\tilde{V}{a_0,a_1}$ and $V_{a_0,a_1}$ are $\mathrm{GL}_a(\mathrm{Quot(S)})$-isomorphic but not $\mathrm{GL}_a(S)$-isomorphic.

Description of the modules

  • The modules $V_{a_0,a_1}$ are indecomposable. Indeed, their reduction is the indecomposable module ${1,x,\ldots,x^{a-1}}$.

  • Set $V_a:=V_{1-p,a}$

  • Set $R=\begin{cases}W(k)[\zeta_p][x_1,\ldots,x_q] & \mbox{ if } l=1 \\W(k)[\zeta_p][x_1,\ldots,x_{q-1}] & \mbox{ if } l>1\end{cases}$

Remark: The ring $R$ is called the Oort-Sekiguchi-Suwa factor of the versal deformation ring $R_\sigma$.

The decomposition

Theorem 13.
Let $\sigma$ be an automorphism of $\mathcal{X}$ of order $p\neq 2$ and conductor $m$ with $m=pq-l$, $1 \leq q$, $1\leq l \leq p-1$. The free $R$-module $H^0(\mathcal{X},\Omega_{\mathcal{X}})$ has the following $R[G]$ structure:

\[H^0(\mathcal{X},\Omega_{\mathcal{X}})=\bigoplus_{\nu=0}^{p-2} V_\nu^{\delta_\nu}, \]

where

\[\delta_\nu= \begin{cases} q+ \lc \frac{(\nu+1)l}{p} \rc - \lc \frac{(2+\nu)l}{p}\rc & \mbox{ if } \nu \leq p-3,\\ q-1 & \mbox{ if } \nu=p-2. \end{cases} \]

The Bertin-Mezard model

  • Kummer extension $(X+\lambda^{-1})^p=x^{-m}+ \lambda^{-p}$.
  • $m=pq-l$, $0 < l \leq p-1$ and set $\lambda X+1=y/x^q$. The model becomes $y^p=(\lambda^p + x^m)x^l =\lambda^p x^l +x^{qp}$.
  • More generally we can replace $x^q$ by $a(x)=x^q+x_1x^{q-1}+\cdots+x_q$, where $x_q=0$ if $l\neq 1$, and consider the Kummer extension

$(\lambda \xi +a(x))^p=\lambda^p x^l+a(x)^p,$ where $\xi=Xa(x)$, $y=\lambda \xi+a(x)=a(x) (\lambda X +1)$ we have
$y^p=\lambda^p x^l+a(x)^p=x^l(\lambda^p +a(x)^p x^{-l}).$

Remark: The assumption made for $x_q \Rightarrow a(x)^p x^{-l}$ is a polynomial.

Kummer extensions

SpecialGeneric

Description of a basis

Proposition 3.
The set of differentials of the form

\[ x^{N} a(x)^a \frac{(\lambda X+1)^{a}}{ a(x)^{p-1} (\lambda X+ 1)^{p-1}} dx, \]

where

$0 \leq a < p-1 \mbox{ and } l-\lc\frac{(1+a)l}{p} \rc \leq N \leq (p-1-a) q-2,$ forms a basis of holomorphic differentials.

This base is not suitable for taking the reduction modulo the maximal ideal of the ring $S=W(k)[\zeta]$.

Description of the basis

Define

\[c_a= \begin{cases} q+ \lc \frac{(a+1)l}{p}\rc - \lc \frac{(2+a)l}{p} \rc & \mbox{ if } a\leq p-3 \\ q-1 & \mbox{ if } a=p-2. \end{cases} \]

\[\Omega_X=\bigoplus_{a=0}^{p-2} V_\nu^{c_\nu} \]

There are some polynomials $f_{\kappa}^{(\nu)}\in R[x]$ such that

\[\left\{ {f}_\kappa^{(\nu)} a(x)^{a} \frac{ X^a}{a(x)^{p-1}(\lambda X+1)^{p-1}}dx: 1\leq \kappa \leq c_\nu, 0\leq a \leq \nu \right\} \]

is a basis for $V_\nu^{c_\nu}$.

Use the special fibre Boseck Basis to show that the reductions are holomorphic.

Ingredients of the proof

  • Knowledge of the Galois module structure and ramification in both the special and the generic fibre.

  • Knowledge of the relative curve.

Conclusions - Further study

  • Representation theoretic obstructions.

  • Compute integral representations of all the groups that are known to lift.

  • Applications to relative Weierstrass points and their deformations.

Thank you

kerkis Kerkis Mountain 31 January 2005