The characteristic-p tag has no wiki summary.

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### Mumford-Ramanujam examples in characteristic p [and in Arakelov geometry]

For a compact Riemann surface $B$ of genus $\geq 2$, it is a consequence of the Narasimhan-Seshadri theorem that there exist rank-$2$ vector bundles $E \to B$ of degree zero, all of whose symmetric ...

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### Applications of alterations

In 1995, de Jong proved the existence of regular alterations in arbitrary characteristic. I would like to have a little survey of important applications of this theorem, e.g. things you could do if ...

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### Extended Deformation Theory (dg-Lie algebra principle in positive characteristic?)

Recently, I looked at articles that make use of Deligne's idea that "in characteristic 0 every deformation problem is governed by a differential graded
Lie algebra" as explained first in ...

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### Invariant theory of $SL_2$ over a field of positive characteristic

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $W$ be a finite dimensional $SL_2$-module over $k$. Let $V$ be the natural representation of $SL_2$.
What can be said - in ...

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### Representations of $SL(2)$ in characteristic 2

In characteristic zero one can use the Clebsch-Gordan rule to decompose tensor products of SL(2)-modules. In characteristic $p$ things are more complicated.
I am interested in the special case ...

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### Nori fundamental group and etale fundamental group in positive characteristic

Let $X$ be a smooth projective surface over an algebraically closed field of char $p > 0$. Suppose that $\pi_{1}^{et}(X) = \{1\}$. Can Nori fundamental group scheme of $X$ be non-trivial?

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### The “Level N modular equation for delta” in characteristics 3, 5, 7 and 13

When $N > 1$, the modular forms $\Delta(z)$ and $\Delta(Nz)$ are algebraically independent over the complexes, and the same then is true of their expansions at infinity. But using the fact that
the ...

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### Katz--Mazur for abelian varieties

Over $\mathbb Z$, there is a smooth DM stack $A_g$ classifying abelian varieties.
Over $\mathbb Z[\frac 1N]$, there is finite etale cover $A_g(N)_{\mathbb Z[\frac 1N]}\to A_g\otimes\mathbb Z[\frac ...

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### Do varieties with ample canonical bundle have finite automorphism group in small characteristic?

Suppose $X$ is a smooth projective variety over a field $k$, with ample canonical bundle. If $\operatorname{char}(k)=0$ or $\operatorname{char}(k)>\dim(X)$ and $X$ lifts to $W_2(k)$ (thanks ...

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### Why was it so difficult to define the relative de Rham-Witt complex?

In Illusie's original article, the de Rham-Witt complex is defined for a smooth scheme over a perfect characteristic $p$ base $S$, without reference to $S$. Some 25 years later, Langer and Zink ...

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### Vanishing theorems that work in positive characteristic

Let $X$ be a smooth projective variety over a field of characteristic $p>0$ of dimension at least $2$. I am looking for some examples when $H^2(\mathcal{O}_X)$ vanishes. Is there any standard way ...

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### Branching rule for classical Lie algebras in positive characteristic

The restriction of an irreducible $\mathfrak{sl}_n(\mathbb{C})$-module to $\mathfrak{sl}_{n-1}(\mathbb{C})$ is described by a branching rule which says that if $L(\lambda)$ is the simple ...

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### Colon property of Gorenstein rings

Let $(R, \mathfrak{m})$ be a Gorenstein local ring of characteistic $p>0$. Let $x_1,...,x_d$ be a system of parameters of $R$. Let $I$ be an $\mathfrak{m}$-primary ideal containg $(x_1,...,x_d)$. ...

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### Descent theory of line bundles on abelian varieties under isogenies (in char p>0)

I have a couple of questions regarding the descent theory of line bundles on abelian varieties under isogenies in positive characteristic.
Let $X$ be an abelian variety and $L\in Pic(X)$ a line ...

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### Are these subspaces of $\mathbb{Z}/3[[x]]$ stable under the shallow Hecke algebra?

This is a characteristic $3$ analog of part of my earlier question, "Are these two subspaces of $\mathbb{Z}/2[[x]]$ the same?"
Notation
Fix a prime $N$ other than $3$. Let $F,G \in ...

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### Is there an analogue of distributions in characteristic p?

Some motivation: When working over $\mathbb{C}$, distributions (in the sense of generalized functions) act as natural generators for $D$-modules (in the sense that any regular holonomic $D$-module is ...

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### $(\varphi, \Gamma)$-module of dimension 2 modulo $p$

Let $p$ be a prime number $\geq 3$. Let $V$ be a representation of $Gal(\bar{\mathbb{Q}}_p/ \mathbb{Q}_p)$ with coefficients in $\mathbb{F}_p$. Assume $V$ is a non-split extension of two characters ...

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### Status of the Euler characteristic in characteristic p

In the introduction to the Asterisque 82-83 volume on `Caractérisque d'Euler-Poincaré, Verdier writes:
Enfin signalons que la situation en caractéristique positive est loin
d'être aussi ...

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### Higher level analogs of Nicolas-Serre theory

NICOLAS-SERRE THEORY
Let $F \in Z/2[[x]]$ be $x+x^9+x^{25}+...$, the exponents being the odd squares, and $V$ be the space spanned by the $F^k$ with $k$ odd. Nicolas and Serre define formal Hecke ...

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### Does this space of mod 2 modular forms admit a (Z/8)* degree decomposition?

Fix an odd N>0. Let M consist of all odd elements of Z/2[[x]] that are the mod 2 reductions of elements of Z[[x]] arising as the Fourier expansions of modular forms for (Gamma_0)(N); it's easy to see ...

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### Gauss mapping in finite characteristic

Suppose that $X\subset\mathbb P^n$ is a $d$-dimentional smooth projective variety (not a linear subspace) over an algebraically closed field. If $\gamma\colon X\to\mathrm{Gr}(d,\mathbb P^n)$ is Gauss ...

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### A subring of the Serre Swinnerton -Dyer ring of level N modular power series

Suppose ell is prime and (N,ell)=1. Consider those power series over Z that are expansions at infinity of modular forms for gamma_0 (N) of weight a multiple of ell-1. I'll say that an element of ...

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### Quotient of a reductive group by a non-smooth subgroup

This is a continuation of my question Quotient of a reductive group by a non-smooth central finite subgroup.
Let $G$ be a smooth, connected, reductive $k$-group over a field $k$ of characteristic ...

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### Have we ever proved any non-solvable case of reciprocity without the Langlands program ?

The reciprocity of the title is the following not completely well-posed problem:
Fix $P(X)$ a monic irreducible polynomial of degree $n$, with coefficients in $\mathbb Z$. "Describe"
(in some sense) ...

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### Decomposition theorem for polarized abelian varieties in positive characteristic

In characteristic zero we have the following decomposition theorem for polarized abelian varieties: it gives an isomorphism between a PPAV and a product of PPAV's of lower dimension and is valid (as ...

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### Are these empirical discoveries about the Serre Swinnerton-Dyer ring of prime level modular power series actual theorems?

In this question Joel Bellaiche constructed an algebra, M, of modular forms for
gamma_0 (N) in finite characteristic (which he called p, but I'll call ell) and asked to know its structure. Matt ...

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### Lifting vector fields to its resolution in char $p$

In this paper 4.7, the authors showed that on a normal variety $X$, if there is a tangent vector field on its smooth locus, then it can be lifted as a logarithmic tangent vector field on a log ...

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### Level p characteristic 2 modular forms and thetas

BACKGROUND
Let p be an odd prime. An element of Z/2[[x]] is "modular of level p" if it is the mod 2 reduction of a g in Z[[x]] with g the Fourier expansion of a modular form for gamma_0(p). In ...

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### Dimension of faithful irreducible representations of $\mathbb{Z}_q\rtimes \mathbb{Z}_{p^2}$ in characteristic p,q

Let $p,q$ be primes s.t. $q=np+1$. Denote $m=p^2$. Then $\mathbb{Z}_p$ acts non-trivially on $\mathbb{Z}_q$, so we have a non-abelian semi-direct product $\mathbb{Z}_q\rtimes \mathbb{Z}_m$, with the ...

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### Dimension of irreducible representations in characteristic p

Hi, I know that over $\mathbb{C}$ the dimension of an irreducible representation of a finite group $G$ must divide the order of the group. I've read somewhere that if $p$ does not divide the order of ...

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### Intuitive pictures in characteristic p

This is a tough one, but does anyone know of any images that recall characteristic p geometry (over algebraically closed fields) in some sense? It is not enough if it is some picture that can be also ...

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### Foliation over characteristic positive

Ekedahl wrote about foliation in characteristic positive, over the field $/frac{Z}{pZ}$ as a subsheaft of the tangent sheaft, that is closed over involution and $p$-power, My question is if there ...

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### “Higher” Tangent spaces in char-p geometry - definition?

Hi, everyone!
I have some construction that requires exact definition.
I considered polynomial homomorphisms from $\Bbbk$ to $U_n(\Bbbk)$ (unitriangular matrices), where $char \Bbbk = p>0$ (more ...

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### The mod 3 reduction of some powers of delta

Let f in Z/3[[x]] be the mod 3 reduction of the Fourier expansion of the normalized weight 12 cusp form delta for the full modular group. The exponents appearing in f are all 1 mod 3. Fix
k>0 and ...

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### Frobenius splitting of Fano varieties

Dear MO,
Question 1. Do you know of an example of a Fano variety which is not Frobenius split?
Background
(1) A variety $X$ in characteristic $p$ is called Frobenius split if there is a "$p$-th ...

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### Characteristic zero and characteristic $p$ in algebraic geometry

Are there non-trivial (i.e. excluding concepts that can be defined only for $p>0$) statements in algebraic geometry that hold for all fields of characteristic $p$ for all prime $p$ but are known ...

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### Unicritical rational functions on curves in characteristic $p$

Let $k$ be an algebraically closed field of positive characteristic $p$, and let $X_{/k}$ be a smooth projective connected curve. Let $x_0$ be a point of $X(k)$.
How precisely can one describe ...

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### deRham cohomoloy of CM liftings of Jacobians

Let $k$ be field of characteristic $p>0$ and $W=W(k)$ the ring of Witt vectors of $k$. We call a smooth curve over $k$, ordinary, when the Jacobian of $J(X)$ of $X$ is an ordinary abelian variety. ...

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### What are supersingular varieties?

For varieties over a field of characteristic $p$, I saw people talking about supersingular varieties.
I wanted to ask "why are supersingular varieties interesting". However, as I don't want to ask an ...

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### Explicit description of O^{cris}_n in Fontaine/Messing

Let $k$ be a perfect field of characteristic $p$, $W(k)$ the Witt ring and $K$ its quotient field. In their article "$p$-adic periods and $p$-adic etale cohomology" Fontaine and Messing give in II.1.4 ...

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### Is there a really big ring of differential operators in characteristic p?

$k$ is a field of characteristic $p$.
$k[t]$ has canonical first-order differential operator $\partial$
As an endomorphism of $k[t]$, $\partial^p=0$.
First way to fix it:
Use the divided power ...

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### Deformation space of non-ordinary abelian varieties

It is a well known result of Serre and Tate that if $A$ is an ordinary abelian variety over a field $k$ of characteristic $p>0$, then the deformation space $\mathcal{M}$ of $A$ to an abelian ...

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### Proofs in the same vein as Ax-Grothendieck

I would like to see other examples of (ideas of) proofs and results in the same vein as the proof of the Ax-Grothendieck theorem. To explain what I mean by "in the same vein", I will quote from the ...

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### Do there exist weak Lefschetz-type statements that were proved for varieties over complex numbers, but were not proved in finite characteristic?

As far as I understood the situation (reading section 3.5B of Lazarsfeld's "Positivity in algebraic geometry" and also some of the references), some of weak Lefschetz-type statements known rely on ...

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### Minimal Model Program for surfaces over algebraically closed fields of characteristic p

Let $k$ be an algebraically closed field of characteristic $p>0$.
I have been trying to find out unsuccessfully if there is a mmp for algebraic surfaces over $k$. I know minimal surfaces are ...

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### Invertible Hasse-Witt for non-ordinary curves

Assume that $S$ is a smooth curve of over a field $k$ of characteristic $p>0$ and $f\colon X\to S$ is a relative curve over $S$ (i.e., the fibers are curves). It is well-known that when all the ...

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### Finite connected groups over a perfect field of characteristic p

In 14.4 of "Introduction to Affine Group Schemes" it is proved (!) that if $A$ represents a finite connected group scheme over a perfect field $k$ of characteristic $p$ then $A$ has the form $k[X_{1}, ...

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### Frobenius functor and length of local cohomology

Let $(R,\mathfrak{m})$ be a Noetherian local ring of positive prime characteristic $p$ and let $F$ be the Frobenius functor. Write $d$ for dimension of $R$. Assume that for some $0\leq i< d $ the ...

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### If $p=0$ and $df=0$, is $f$ a $p$th power?

This question is a follow-up to When does the relative differential $df=0$ imply that $f$ comes from the base?. There it was asked, for an $A$-algebra $B$, under what conditions does $df=0$ (in the ...

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### Chevalley's Theorem on Constructible Sets

I'm having a hard time understanding the theorem in the title, more specifically the proof of the related fact that the image of a dominant morphism contains a dense open set of it's closure. (My ...