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Given an irreducible homogeneous polynomial $F \in \mathbb{Z}[x_1,\cdots,x_n]$ (which can be assumed to be irreducible over $\bar{\mathbb{Q}}$ if necessary),

  1. Is it true that there exists arbitrarily large prime such that $F$ modulo $p$ is irreducible? What about geometrically irreducible?

  2. Is there anything we can say about the smallest prime $p$ such that $F$ modulo $p$ is irreducible? Same question for geometric irreducibililty.

Motivation: I am trying to understand the Lang-Weil bound for varieties over finite field. It seems like one needs some kind of geometric irreducibility to apply the bound, so I would like to know the answer to especially question 2, as it is related to how I want to use the bound. Thanks!

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1 Answer 1

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First Lang-Weil estimate is for any algebraic variety $X$ over $\mathbb F_q$. The number of points of $X$ in $\mathbb F_{q^n}$ is equivalent to $c(X)(q^n)^{\dim X}$ if $c(X)$ is the number of geometric irreducible components of $X$ of dimension $\dim X$.

  1. Yes. This is quite general if the generic fiber is geometrically integral. Indeed, let $X\to Y$ be a morphism of finite type of irreducible noetherian schemes with geometrically integral generic fiber. Then there exists a dense open subset of $Y$ over which all fibers are geometrically integral, see EGA, IV, 9.7.7(iii) and (iv). In particular, if your $F$ is irreducible over $\bar{\mathbb Q}$, then for all $p$ big enough, $F$ mod $p$ is irreducible.

  2. Let $n>1$ and let $F=x_1x_2-nx_3^2$. It is irreducible over $\bar{\mathbb Q}$, but reducible mod any prime $p\mid n$. So if you want some bound, you need to know more properties of $F$.

  3. (=1bis) If $F$ is irreducible but not geometrically irreducible, then the . answer is no for general base ring. For example, if $R=\mathbb C[t]$ and $F=x_1^2-tx_2^2$, then $F$ is irreducible over $\mathbb C(t)$, but only one closed fiber is irreducible. However, over $\mathbb Z$, there is chance that the answer is yes using Chebotarev.

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  • $\begingroup$ Thanks! I will read your answer in detail later, but there's one thing I want to clarify. In the book I am reading (Browning's book on qualitative arithmetic of varieties), Lang-Weil estimate is only stated for irreducible affine varieties. I have two questions: 1. Does the estimate hold for any variety? In particular, do I need anything like quasi-compactness? 2. Is the bound uniform in some variables, if I consider only the geometrically irreducible varieties? $\endgroup$
    – user27126
    Feb 6, 2013 at 1:31
  • $\begingroup$ @Sanchez: you should read Lang-Weil's original paper. The estimate is true for any scheme of finite type. For (2), it depends on what do you call uniform. I guess the answer would be no. $\endgroup$
    – user18119
    Feb 6, 2013 at 9:23
  • $\begingroup$ Thanks again for your answer! Some more questions: 1. Is there any elementary way to see 1 in the special case here, without resolving to the result in EGA? Is the result in EGA difficult? 2. What do I need to know to read Lang-Weil paper? I will look it up tomorrow I think, but would like to know if I am up to it first. Thanks! $\endgroup$
    – user27126
    Feb 7, 2013 at 6:36
  • $\begingroup$ 1. I don't know ! 2. If I remember correct, the proof is by induction using Bertini and Weil's estimate for curves. $\endgroup$
    – user18119
    Feb 7, 2013 at 16:22
  • $\begingroup$ 1. Thanks anyway. The statement looks easy until I try to prove it so I thought maybe I'm missing something.. 2. That sounds quite okay, thanks again! $\endgroup$
    – user27126
    Feb 7, 2013 at 18:05

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