## GALOIS EXTENSIONS RAMIFIED AT ONE PRIME (2007)

### BibTeX

@MISC{Hoelscher07galoisextensions,

author = {Jing Long Hoelscher and David Harbater and Ching-li Chai and Jing Long Hoelscher and David Harbater Advisor},

title = {GALOIS EXTENSIONS RAMIFIED AT ONE PRIME},

year = {2007}

}

### OpenURL

### Abstract

I want to use this page to express my appreciation to all the many people who

### Citations

134 |
Revetements etales et groupe fondamental
- Grothendieck
- 1971
(Show Context)
Citation Context ...ure of k. The algebraic (or étale) fundamental group π1(U) is defined to be the inverse limit of the inverse system of groups occurring as Galois groups of pointed étale Galois covers of U (see SGA I =-=[Gro]-=-). Define πA(U) to be the set of finite quotients of the algebraic fundamental group π1(U), i.e. the set of the finite groups that can occur as Galois group of étale Galois covers of U. In the situati... |

49 |
On certain functions connected with polynomials in Galoisfield
- Carlitz
- 1935
(Show Context)
Citation Context ... still an odd permutation, which cannot be 1, contradiction. So Sn /∈ π t A (Uf) when p ̸= 2 and f is of odd degree. 3.2 Cyclotomic function fields Analogously to cyclotomic number fields, L. Carlitz =-=[Ca1]-=- defined a “cyclotomic function field” Fq(t)(λf) for a polynomial f ∈ Fq[t], where λf is a primitive root of the equation Cf(z) = 0 for the Carlitz module C for Fq[t]. Later D. Hayes [Hay1] published ... |

22 |
Coverings of algebraic curves
- Abhyankar
(Show Context)
Citation Context ...ields. On the other hand, over an algebraically closed field k of characteristic p, the situation is better understood. There, a precise form of the above question was posed in Abhyankar’s Conjecture =-=[Ab]-=-, which was proved by M. Raynaud [Ra] and D. Harbater [Ha2]. Namely, given a smooth connected projective k-curve X with genus g and a finite set S of n > 0 closed points of X, we may consider the set ... |

22 |
Formes modulaires et représentations l-adiques. Séminaire Bourbaki. 1968/69: Exposés 347–363, Exp
- Deligne
- 1971
(Show Context)
Citation Context ...ides p k and is of degree p, so Rk+1 = Rk(ζ p k+1) for k >> 0. 2.5 Modular forms Another tool we can use to construct a Galois extension over Q with given ramification is modular forms. By P. Deligne =-=[De]-=-, we can attach a semisimple continuous representation ρf : GQ → GL2( ¯ Fp) to every eigenform, whose image corresponds to a finite Galois extension over Q with given ramification. If we restrict Galo... |

20 |
A set of polynomials
- Carlitz
- 1940
(Show Context)
Citation Context ...tained in a constant field extension of a cyclotomic function field for some polynomial f ∈ Fq[t]. Let k = Fq(t), let A = Fq[t] and let k ac be the algebraic closure of k. Carlitz showed in [Ca1] and =-=[Ca2]-=- that the additive group of k ac becomes a right module over A under the following action: For any u ∈ k ac and any polynomial f ∈ A, define u f = f(ϕ + µ)(u), where ϕ : k ac → k ac is the Frobenius a... |

12 |
The nonexistence of certain Galois extensions unramified outside 5
- Brueggeman
- 1999
(Show Context)
Citation Context ...e p = 3 was also proved by Serre in [Se1]. In 1997, Sharon Brueggeman showed, in accordance with Serre’s conjecture, the nonexistence of certain Galois extensions unramified outside 5. Theorem 1.5.2 (=-=[Br1]-=-, Theorem 1.1). Assume GRH. Let G be the Galois group of a finite Galois extension K/Q which is unramified outside 5. Let ρ : G ↩→ GL2( ¯ F5) be a faithful semisimple odd representation. Then ρ = χa 5... |

9 |
Elliptic modules, Math. Sbornik 94
- Drinfeld
- 1974
(Show Context)
Citation Context ...ublished an exposition of Carlitz’s idea and showed that it provided an explicit class field theory for rational function fields. Later developments, due independently to Hayes [Hay2] and V. Drinfeld =-=[Dr]-=-, showed that Carlitz’s ideas can be generalized to provide an explicit class field theory for any global function field. Consider extensions over function fields Fq(t) where Fq is a finite field of o... |

6 |
Septic number fields which are ramified only at one small prime
- Brueggeman
(Show Context)
Citation Context ...nd Doud using improved discriminant bounding and polynomial searching, in determining the non-existence of various Galois extensions of Q which are unramified outside a single prime p. Theorem 1.5.4 (=-=[Br2]-=-, Theorem 4.1). Let K be a number field which is ramified only at a single prime p and p ≤ 7. Then its Galois group is not isomorphic to SL(3, 2), A7, or S7. Corollary 1.5.5 ( [Jo2], Theorem 2.2). If ... |

2 |
On geometric Zp-extensions of function fields
- Gold, Kisilevsky
- 1988
(Show Context)
Citation Context ...〉 for all i ≥ 0 and denote the F∞ Γi Fi F Γ/Γi Let pen be the highest power of p dividing the class number of Fn. In the case of m = 1, i.e. geometric cyclotomic Zp-extensions, R. Gold, H. Kisilevsky =-=[GK]-=- and A. Aiba [Ai] have some results about the Iwasawa modules and Iwasawa invariants. Much less is known if m > 1. When m = 2, the author believes it is possible to prove the following conjecture. Con... |

1 |
On the vanishing of Iwasawa invariants of geometric cyclotomic
- Aiba
(Show Context)
Citation Context ...nd denote the F∞ Γi Fi F Γ/Γi Let pen be the highest power of p dividing the class number of Fn. In the case of m = 1, i.e. geometric cyclotomic Zp-extensions, R. Gold, H. Kisilevsky [GK] and A. Aiba =-=[Ai]-=- have some results about the Iwasawa modules and Iwasawa invariants. Much less is known if m > 1. When m = 2, the author believes it is possible to prove the following conjecture. Conjecture 3.4.3. Gi... |

1 | Class number of cyclotomic function fields, Transactions of the American Mathematical Society 351
- Guo, Shu
- 1999
(Show Context)
Citation Context ...omposed into a product G ′ n × G ′′ n, where G ′ n ∼ = (Fq[t]/(f)) ∗ is a cyclic group of order qd − 1 and G ′′ n is a product of cyclic p-groups G (i,j) n with orders g (j) n (see Proposition 3.2 in =-=[GS]-=-), i.e. Gn ∼ = G ′ n × r∏ i=1 (G (i,1) n × ... × G (i,m) n ), where g (1) n ≥ ... ≥ g (m) n . The order of G ′′ n is qd(n−1) . In particular, when n = 1, the order of G ′′ n is 1. Let pdn be the exact... |