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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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### Lifting varieties from char. $p$ to char. 0 after alterations

The question is related to this MO question:
Lifting varieties to characteristic zero.
Let $X$ be a projective smooth variety over $k$ alg. closed field of char. $p.$ Does there always exist an ...

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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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### What's known about the mod 2 reduction of the level l Jacobi modular equation?

Motivation:
Let $\ell$ be an odd prime. Let $A$ in ${\mathbb Z}/2[[x]]$ be $x+x^9+x^{25}+x^{49}+...$, and $B=A(x^\ell)$. One can use the level $\ell$ Jacobi modular equation to get a polynomial ...

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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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### Lifting abelian varieties in (the closed fiber of) a fixed Neron model

Suppose that $R$ is a dvr with field of fractions $K$ and residue field $k$ and that $A_K$ is an abelian variety over $K$ with Neron model $A$ over $R$. Then the closed fiber $A_k$ is a smooth ...

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### Function fields of characteristic p modular curves, and mod p reductions of the classical modular equation

Let l and p be distinct primes, l>2. There are "characteristic p modular curves" X_0(l) and X(l), defined over an algebraic closure, K, of Z/p, solving moduli problems for elliptic curves with some ...

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### Do the Standard Conjectures imply parts of the “Weil II” Riemann Hypothesis?

It is known that Grothendieck's Standard Conjectures on algebraic cycles imply the Riemann Hypothesis of the original Weil Conjectures. However, do they also say something about the version of the ...

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### Degeneration of wildly ramified local monodromy representations - near or far from Deligne?

Suppose we have a surface $S$ mapping to a curve $C$ and a finite cover $Y/S$ that is ramified at a divisor $D$. For each point $x \in C$ we get a ramified cover of the curve $S_x$, and we can study ...

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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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### Are curves over imperfect fields defined over a smaller field?

Let $C$ be regular projective curve defined over a field $K$. Let $K/L$ be a totally inseparable finite extension. Does there exist a regular projective curve $C'$ over $L$ such that that the pullback ...

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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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### 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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### 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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### Resolution of singularities in positive characteristic

I am currently trying to make some small parts of the minimal model program work for some very explicit varities in positive characteristic. I have such a variety $X_1$ and I know that there is a ...

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### Beilinson-Bernstein localization in positive characteristic

This is a follow-up to this question; in particular, I'm wondering if anyone can expand upon the interesting answers given by Kevin McGerty and David Ben-Zvi there. (In particular, in this question ...

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### Ample vector bundles, $H^1=0$ and global generation in characteristic $p$

This is a follow up from this question, in characteristic $p>0$. First, some background. It is known that in characteristic $0$, a vector bundle $E$ on say a projective curve $X$ is ample if and ...

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### Alterations of regular varieties

Let $X$ be a regular quasi-projective variety over a perfect field $k$. The existence of a "good compactification" of $X$, i.e. a regular projective variety $\bar{X}$ with an embedding ...

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### On periods of algebraic integers modulo rational primes

I run, somewhat indirectly, into the following problem and I have no hints where to look in the literature in search for answers or clues.
Let $K$ be a number field, which we may assume Galois if it ...

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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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### 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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### Modular reduction of exceptional complex reflection groups

I am interested in reducing reflection representations of complex reflection groups modulo a prime $p$. For the infinite family $G(m,r,n)$, it is straightforward to get "good reduction" provided that ...

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### Partial differential Equation over characteristic p

I want some references on partial differential equations over characteristic $p$. If we have a first order partial differential equation, how can we check whether there exists polynomials or rational ...

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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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### 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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### 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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### 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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### 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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### Inequalities between numerical invariants of nonsingular projective Varieties in positive Characteristic

It is well-known that Miyaoka and Yau-type inequalities do not hold in positive characteristic. In "a note on Bogomolov-Gieseker’s inequality in positive characteristic", however, we can ...

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### Given two linear operators A and B over a finite field, is there a third operator C whose kernel is the intersection of kernels of A and B?

Let $V$ be a finite dimensional linear space over a finite field $k$. Let $A$ and $B$ be two endomorphisms of $V$.
Question 1. Is there an endomorphism $C$ of $V$, which is expressed in terms of ...

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### Hochschild Kostant Rosenberg theorem for varieties in positive characteristic?

Is there is a known version of the HKR theorem as proved in say Swan's paper "Hochschild Cohomology of Quasiprojective Varieties" in positive characteristic? I assume something is known about this as ...

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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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### 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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### 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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### 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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### What would be a characteristic-$p$ analogue for $C^{\infty}$-fiber bundles?

I'd like to know a notion for a morphism between algebraic varieties in characteristic $p$ that plays the role of a $C^{\infty}$-fiber bundle. It should be, in particular, flat. I'm not assuming the ...

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### Mod 2 modular forms in levels 5 and 25--how to account for this Hecke isomorphism?

The space $P1$ of my earlier question 203755 "Two spaces attached to mod 2 level 9 modular forms...", is essentially the space of mod 2 level 3 modular forms. That such a space should appear inside ...

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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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### 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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### 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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### Dimension of fibres of moment maps in characteristic $p$

Suppose $G$ is a connected semisimple linear algebraic group with Lie algebra $\mathfrak{g}$ and $X$ is a homogeneous $G$-space with isotropy subgroup $H$ (associated Lie algebra $\mathfrak{h}$) that ...

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### Two spaces attached to mod 2 level 9 modular forms--a conjectural Hecke isomorphism

MOTIVATION
Nicolas and Serre have analyzed the structure of the space of mod $2$ modular forms of level $1$, viewed as a "Hecke-module". They show that for each $p>2$, the operator $T_p$ acting on ...

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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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### “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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### 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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### Liftability in positive characteristic

What clsses of algebraic varieties over field of positive characteristic can be lift to $W_2(k)$?

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### DeRham cohomology

The Poincare lemma fails in positive characteristic, since pth powers vanish under differention. My question is : is there still some kind of resolution of the local system k by considering some ...