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The handle
http://hdl.handle.net/1887/67539
holds various files of this Leiden University
dissertation.
Author: Pagano, C.
Summary
JUMP SETS IN LOCAL FIELDS 119
with βK2p d pvQppdqiq “ βK1piq ` vQppdq for each iP IK1´ te˚K1u. References
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Symposium on Symbolic and Algebraic Computation, 68 (2017), 923–934.
[2] L. Capuano, I. del Corso, Upper ramification jumps in abelian extensions of exponent p, Rivista di
Matematica della Universita’ di Parma, (2014), 317–329.
[3] I. Fesenko, S. Vostokov, Local fields and their extensions, Second edition, (2002).
[4] S. Pauli, C. Greve, Galois groups of Eisenstein polynomials whose Ramification Polygon has one side. [5] E. Maus, Die gruppentheoretische Struktur der Verzweigungsgruppenreihen, Journal f¨ur die reine und
angewandte Mathematik 230 (1968), 1–28.
[6] H. Miki, On the ramification numbers of cyclic p-extensions over local fields, Journal f¨ur die reine und angewandte Mathematik 328 (1981), 99–115.
[7] S. Mochizuki, A Version of the Grothendieck Conjecture for p-adic Local Fields, The International
Journal of Math. 8 (1997), 499–506.
[8] D. S. Romano, Galois groups of strongly Eisenstein polynomials, Dissertation, UC Berkeley, 2007. [9] J.P. Serre, Local fields, Springer-Verlag, (1995).
[10] J.P. Serre, Une “formule de masse” pour les extensions totalement ramifi´ees de degr´e donn´e d’un corps local, C.R. Acad. Sc. Paris S´erie A 286 (1978), 1031–1036.
[11] Y. Sueyoshi, Ramification numbers in cyclic p-extensions over p-adic number fields, Memoirs of the
Faculty of Science, Kyushu University. Series A, Mathematics 38 (1984), 163–168.
[12] C. Pagano, E. Sofos, 4-ranks and the general model for statistics of ray class groups of imaginary quadratic number fields. arXiv:1710.07587, (2017).
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Summary
This thesis consists of three chapters. Each chapter is on a different subject. How-ever, all chapters address issues that arise in counting arithmetically interesting objects.
Chapter 1 is a joint paper with Peter Koymans about unit equations in positive characteristic. In this paper we establish the first upper bound that is uniform in the characteristic for the number of “solutions” to the unit equation. With this tool we settle a conjecture of F. Voloch. If p is a prime number, r a positive integer, K is a field with char(K) = p and Γ⊆ K∗× K∗a finitely generated subgroup of rank r, the unit equation is the equation
x + y = 1,
to be solved in (x, y)∈ Γ but (x, y) ∈ Γp. Denote by S(Γ) the set of solutions to the unit equation for Γ. Our main theorem establishes that
#S(Γ)≤ 31 · 19r.
Chapter 2 is a joint paper with Efthymios Sofos about statistical properties of ray class groups of fixed integral conductor of imaginary quadratic number fields. If c is a positive integer and K is a finite extension of Q, the ray class group of conductor c of K is the group
Cl(K, c) := I(K, c) Pr(K, c),
where I(K, c) is the subgroup of IK :={fractional ideals in K} that is generated by ideals of OK that are coprime to c and Pr(K, c) is the subgroup of IK that is generated by principal ideals (α) with α∈ OK− {0} and α congruent to 1 modulo c. When K varies among imaginary quadratic number fields whose discriminant is coprime to c and congruent to 1 modulo 4, we establish the asymptotic behavior of the natural map
(2Cl(K, c))[2]→ (2Cl(K))[2], obtaining as a corollary the joint distribution of
(#2(Cl(K, c))[2], #(2Cl(K))[2]).
Summary
This thesis consists of three chapters. Each chapter is on a different subject. How-ever, all chapters address issues that arise in counting arithmetically interesting objects.
Chapter 1 is a joint paper with Peter Koymans about unit equations in positive characteristic. In this paper we establish the first upper bound that is uniform in the characteristic for the number of “solutions” to the unit equation. With this tool we settle a conjecture of F. Voloch. If p is a prime number, r a positive integer, K is a field with char(K) = p and Γ⊆ K∗× K∗a finitely generated subgroup of rank r, the unit equation is the equation
x + y = 1,
to be solved in (x, y)∈ Γ but (x, y) ∈ Γp. Denote by S(Γ) the set of solutions to the unit equation for Γ. Our main theorem establishes that
#S(Γ)≤ 31 · 19r.
Chapter 2 is a joint paper with Efthymios Sofos about statistical properties of ray class groups of fixed integral conductor of imaginary quadratic number fields. If c is a positive integer and K is a finite extension ofQ, the ray class group of conductor c of K is the group
Cl(K, c) := I(K, c) Pr(K, c),
where I(K, c) is the subgroup of IK :={fractional ideals in K} that is generated by ideals of OK that are coprime to c and Pr(K, c) is the subgroup of IK that is generated by principal ideals (α) with α∈ OK− {0} and α congruent to 1 modulo c. When K varies among imaginary quadratic number fields whose discriminant is coprime to c and congruent to 1 modulo 4, we establish the asymptotic behavior of the natural map
(2Cl(K, c))[2]→ (2Cl(K))[2], obtaining as a corollary the joint distribution of
(#2(Cl(K, c))[2], #(2Cl(K))[2]).
Even though there is a surjective natural map 2Cl(K, c) 2Cl(K), the surjectiv-ity of the induced map (2Cl(K, c))[2]→ (2Cl(K))[2] encounters a cohomological obstruction. In a refined version of our main theorem, we show the equidistribu-tion of this obstrucequidistribu-tion in the full obstrucequidistribu-tion group (viewed as a probability space with the counting measure).
These results extend the only previously known case, which is c = 1, where there is only the ordinary class group. This was due E. Fouvry and J. Kl¨uners.
Next, we extend the Cohen–Lenstra and the Gerth heuristics from class groups to general ray class groups. The Cohen–Lenstra heuristic is a probabilistic model designed by H. Cohen and H. Lenstra, which predicts conjecturally the exact asymptotic outcome of most statistical questions about theZ[12]-module Cl(K)⊗Z
Z[1
2] as K varies among imaginary quadratic number fields. Later F. Gerth
for-mulated a heuristic about Cl(K)[2∞]. We formulate a more general probabilistic model aimed at predicting the exact asymptotic outcome of most statistical ques-tions about ray class groups, viewed as exact sequences of Galois modules. This statistical model agrees with our result on 4-ranks, yielding a heuristic interpre-tation of the equidistribution of the above mentioned cohomological obstructions. Moreover, our model explains the precise constants given by a theorem of I. Varma about the average 3-torsion of ray class groups. With this statistical model for ray class groups, both our results on 4-ranks and Varma’s result on the 3-torsion obtain a precise heuristical explanation and are placed within a broad conjectural framework.
Chapter 3 is about the arithmetic of local fields and it mostly focuses on the sub-class of p-adic fields for some prime number p. If p is a prime number, a p-adic field is a finite field extension K/Qp. The multiplicative group K∗carries a natural filtration
K∗⊇ O∗
K⊇ 1 + mK⊇ ... ⊇ 1 + miK⊇ ...,
where OK denotes the ring of integers of K and mK is its unique maximal ideal. One can show that the sequence
1 + mK⊇ ... ⊇ 1 + mi K⊇ ...
is a filtration ofZp-modules. In this work I give a parametrization of the set of sequences ofZp-modules
M1⊇ ... ⊇ Mi⊇ ...
that are isomorphic to 1 + mK ⊇ ... ⊇ 1 + mi
K⊇ ... for some local field K. This means that there exists an isomorphism ofZp-modules
ϕ : 1 + mK→ M1
such that ϕ(1 + mi
K) = Mi. In case such a K exists, we say that the sequence M1⊇ ... ⊇ Mi ⊇ ... is admissible. I parametrize admissible sequences in terms of
certain combinatorial objects called jump sets. One of the main theorems in this study is the remarkable property that this parametrization is weight preserving, in the following sense. It turns out that there is a natural way to attach to each jump set a weight. One can give the weight of a jump set also a natural interpretation in terms of the Haar measure. On the other hand, Serre introduced a natural probability measure on the set of totally ramified extensions of given degree of a given local field. In this chapter I show that the total mass of the set of local fields whose filtration of subgroups is isomorphic to a given admissible sequence equals the combinatorial weight of the corresponding jump set. Finally I use my identification between the set of jump sets and the set of admissible sequences to give a simpler and more conceptual proof of a classification, due to H. Miki, of the possible sets of upper jumps of a cyclic totally ramified p-power degree extension of a fixed p-adic field K.
