We describe our online database of finite extensions of Q p , and how it can be used to facilitat... more We describe our online database of finite extensions of Q p , and how it can be used to facilitate local analysis of number fields.
International Journal of Number Theory, Mar 1, 2011
We present the first explicitly known polynomials in Z[x] with nonsolvable Galois group and field... more We present the first explicitly known polynomials in Z[x] with nonsolvable Galois group and field discriminant of the form ±p A for p ≤ 7 a prime. Our main polynomial has degree 25, Galois group of the form P SL 2 (5) 5 .10, and field discriminant 5 69 . A closely related polynomial has degree 120, Galois group of the form SL 2 (5) 5 .20, and field discriminant 5 311 . We completely describe 5-adic behavior, finding in particular that the root discriminant of both splitting fields is 125 • 5 -1/12500 ≈ 124.984 and the class number of the latter field is divisible by 5 4 .
In foundational papers, Gross, Zagier, and Kohnen established two formulas for arithmetic interse... more In foundational papers, Gross, Zagier, and Kohnen established two formulas for arithmetic intersection numbers of certain Heegner divisors on integral models of modular curves. In [GZ1], only one imaginary quadratic discriminant plays a role. In [GZ2] and [GKZ], two quadratic discriminants play a role. In this paper we generalize the two-discriminant formula from the modular curves X 0 (N ) to certain Shimura curves defined over Q. Our intersection formula was stated in [Ro], but the proof was only outlined there. Independently, the general formula was given, in a weaker and less explicit form, in [Ke2]; there it was proved completely. This paper is thus a synthesis of parts of [Ro] and [Ke2]. The intersection multiplicities computed here were used in [Ku] to derive a relation between height pairings and special values of the derivatives of certain Eisenstein series. Two more recent related works are [KR], which concentrates on computing local intersection multiplicities at ramified primes under quite general hypotheses, and [KRY], which relates intersection numbers on Shimura curves to coefficients of modular forms. We note also that Zhang [Zh] has generalized all of [GZ1] from ground field Q to general totally real ground fields F , working with general Shimura curves. We expect that all of [GKZ] should generalize similarly. We are happy to thank B. Gross and S. Kudla for their encouragement during the long period in which this work was carried out.
Integrals from Feynman diagrams with massive particles soon outgrow polylogarithms. We consider t... more Integrals from Feynman diagrams with massive particles soon outgrow polylogarithms. We consider the simplest situation in which this occurs, namely for diagrams with two vertices in two space-time dimensions, with scalar particles of unit mass. These comprise vacuum diagrams, on-shell sunrise diagrams and diagrams obtained from the latter by cutting internal lines. In all these cases, the Feynman integral is a moment of n = a + b Bessel functions, of the form The corresponding L-series are built from Kloosterman sums over finite fields. Prior to the Creswick conference, the first author obtained empirical relations between special values of L-series and Feynman integrals with up to n = 8 Bessel functions. At the conference, the second author indicated how to extend these. Working together we obtained empirical relations involving Feynman integrals with up to 24 Bessel functions, from sunrise diagrams with up to 22 loops. We have related results for moments that lie beyond quantum field theory.
We consider the set of classical newforms with rational coefficients and no complex multiplicatio... more We consider the set of classical newforms with rational coefficients and no complex multiplication. We study the distribution of quadratic-twist classes of these forms with respect to weight k and minimal level N . We conjecture that for each weight k ≥ 6, there are only finitely many classes. In large weights, we make this conjecture effective: in weights 18 ≤ k ≤ 24, all classes have N ≤ 30, in weights 26 ≤ k ≤ 50, all classes have N ∈ {2, 6}, and in weights k ≥ 52, there are no classes at all. We study some of the newforms appearing on our conjecturally complete list in more detail, especially in the cases N = 2, 3, 4, 6, and 8, where formulas can be kept nearly as simple as those for the classical case N = 1.
We present a method for computing complete lists of number fields in cases where the Galois group... more We present a method for computing complete lists of number fields in cases where the Galois group, as an abstract group, appears as a Galois group in smaller degree. We apply this method to find the twenty-five octic fields with Galois group PSL 2 (7) and smallest absolute discriminant. We carry out a number of related computations, including determining the octic field with Galois group 2 3 : GL 3 (2) of smallest absolute discriminant.
In this paper we give a conjectural refinement of the Davenport-Heilbronn theorem on the density ... more In this paper we give a conjectural refinement of the Davenport-Heilbronn theorem on the density of cubic field discriminants. We explain how this refinement is plausible theoretically and agrees very well with computational data.
Let S be a finite set of primes and let G be a finite group. Let N F (S, G) be the set of Galois ... more Let S be a finite set of primes and let G be a finite group. Let N F (S, G) be the set of Galois number fields K ⊂ C with Gal(K/Q) ∼ = G and discriminant divisible only by primes in S. The sets N F (S, G) are finite, by a classical theorem of Hermite. We are interested in the following inverse Galois problem. For given S and G, find defining polynomials f (x) ∈ Q[x] for as many fields in N F (S, G) as possible. This problem is most interesting when |N F (S, G)| can be expected to be small, so that one can reasonably aim for complete lists. Thus we are interested in "collecting number fields", the most valuable specimens being those which are least ramified, for a given Galois group G. In this paper we consider only cases with S = {2, 3}. If G has a faithful permutation representation of degree ≤ 7, we have completely identified N F (S, G) previously with Jones [JR1], [Jon]. Here we supplement these complete lists with incomplete lists of fields with larger groups G, involving the simple groups P SL 2 (9), SL 2 (8), SL 3 (3), A 8 , P Sp 4 (3), A 9 , Sp 6 (2), and A 32 . We construct all our fields by the standard technique of specializing three point covers. However our requirement that the final field discriminant be of the form ±2 a 3 b is extremely restrictive. In the main, it restricts consideration to three point covers with bad reduction at 2 and 3 only. To keep ramification within {2, 3}, the specialization point τ then has to be chosen judiciously as well. Throughout we used Mathematica, supplemented by the discf, factorpadic, and polred functions of Pari. Of the three point covers which we use, for ten it was hard to find a defining polynomial. The polynomials f 9a , f 9b , f 13a , and f 27a were found by Elkies, Matzat, Malle, and Häfner, respectively, [Elk1], [Mat], [Mal1], [Häf]. The polynomials f 13b , f 13c , f 27b , f 27c , f 27d , and f 28 are new here. We do not go into much detail about how we found the new polynomials, our methods being standard. Rather, we emphasize various aspects of the specialization step. Section 1 provides an overview of the fields constructed in this paper. Section 2 and 3 discuss three point covers and especially specialization. Sections 4-10 discuss the individual constructions, sorted approximately by |G|; each of these sections also informally discusses a more theoretical issue, as indicated by the section titles. Section 11 indicates a direction for further research we feel is important and promising. We conclude in Section 12 with two speculative conjectures on the general nature of the numbers |N F (S, G)|. Throughout we work as much as possible in a purely number-theoretic setting. However the reader should realize that a systematic study of the sets N F (S, G) would fundamentally involve both the theory of motives and the theory of automorphic representations:
The study of the moduli of covers of the projective line leads to the theory of Hurwitz varieties... more The study of the moduli of covers of the projective line leads to the theory of Hurwitz varieties covering configuration varieties. Certain onedimensional slices of these coverings are particularly interesting Belyi maps. We present systematic examples of such "Hurwitz-Belyi maps." Our examples illustrate a wide variety of theoretical phenomena and computational techniques. Contents 1. Introduction 1 2. Two Belyi maps unexpectedly defined over Q 5 3. Hurwitz maps, Belyi pencils, and Hurwitz-Belyi maps 9 4. The two rational Belyi maps as Hurwitz-Belyi maps 12 5. The semicubical clan 16 6. The braid-triple method 27 7. Hurwitz-Belyi maps exhibiting spin separation 30 8. Hurwitz-Belyi maps with |G| = 2 a 3 b p and ν = (3, 1) 34 9. Hurwitz-Belyi maps with |G| = 2 a 3 b 5 and ν = (3, 1) 37 10. Hurwitz-Belyi maps with |G| = 2 a 3 b 7 and ν = (3, 1) 42 11. Some 5-and 6-point Hurwitz-Belyi maps 47 12. Expectations in large degree 52 References 54
We give conditions for the monodromy group of a Hurwitz space over the configuration space of bra... more We give conditions for the monodromy group of a Hurwitz space over the configuration space of branch points to be the full alternating or symmetric group on the degree. Specializing the resulting coverings suggests the existence of many number fields with surprisingly little ramification -for example, the existence of infinitely many A m or S m number fields unramified away from f2; 3; 5g. 1. Introduction 511 2. Hurwitz covers 513 3. Braid groups 515 4. Lifting invariants 517 5. The full-monodromy theorem 523 6. Proof of I D ) II 529 7. Proof of II D ) I 536 8. Full number fields 541 Acknowledgements 544 References 544 MSC2010: primary 14D05; secondary 20F36, 11R21.
We describe our online database of finite extensions of Q p , and how it can be used to facilitat... more We describe our online database of finite extensions of Q p , and how it can be used to facilitate local analysis of number fields.
International Journal of Number Theory, Mar 1, 2011
We present the first explicitly known polynomials in Z[x] with nonsolvable Galois group and field... more We present the first explicitly known polynomials in Z[x] with nonsolvable Galois group and field discriminant of the form ±p A for p ≤ 7 a prime. Our main polynomial has degree 25, Galois group of the form P SL 2 (5) 5 .10, and field discriminant 5 69 . A closely related polynomial has degree 120, Galois group of the form SL 2 (5) 5 .20, and field discriminant 5 311 . We completely describe 5-adic behavior, finding in particular that the root discriminant of both splitting fields is 125 • 5 -1/12500 ≈ 124.984 and the class number of the latter field is divisible by 5 4 .
In foundational papers, Gross, Zagier, and Kohnen established two formulas for arithmetic interse... more In foundational papers, Gross, Zagier, and Kohnen established two formulas for arithmetic intersection numbers of certain Heegner divisors on integral models of modular curves. In [GZ1], only one imaginary quadratic discriminant plays a role. In [GZ2] and [GKZ], two quadratic discriminants play a role. In this paper we generalize the two-discriminant formula from the modular curves X 0 (N ) to certain Shimura curves defined over Q. Our intersection formula was stated in [Ro], but the proof was only outlined there. Independently, the general formula was given, in a weaker and less explicit form, in [Ke2]; there it was proved completely. This paper is thus a synthesis of parts of [Ro] and [Ke2]. The intersection multiplicities computed here were used in [Ku] to derive a relation between height pairings and special values of the derivatives of certain Eisenstein series. Two more recent related works are [KR], which concentrates on computing local intersection multiplicities at ramified primes under quite general hypotheses, and [KRY], which relates intersection numbers on Shimura curves to coefficients of modular forms. We note also that Zhang [Zh] has generalized all of [GZ1] from ground field Q to general totally real ground fields F , working with general Shimura curves. We expect that all of [GKZ] should generalize similarly. We are happy to thank B. Gross and S. Kudla for their encouragement during the long period in which this work was carried out.
Integrals from Feynman diagrams with massive particles soon outgrow polylogarithms. We consider t... more Integrals from Feynman diagrams with massive particles soon outgrow polylogarithms. We consider the simplest situation in which this occurs, namely for diagrams with two vertices in two space-time dimensions, with scalar particles of unit mass. These comprise vacuum diagrams, on-shell sunrise diagrams and diagrams obtained from the latter by cutting internal lines. In all these cases, the Feynman integral is a moment of n = a + b Bessel functions, of the form The corresponding L-series are built from Kloosterman sums over finite fields. Prior to the Creswick conference, the first author obtained empirical relations between special values of L-series and Feynman integrals with up to n = 8 Bessel functions. At the conference, the second author indicated how to extend these. Working together we obtained empirical relations involving Feynman integrals with up to 24 Bessel functions, from sunrise diagrams with up to 22 loops. We have related results for moments that lie beyond quantum field theory.
We consider the set of classical newforms with rational coefficients and no complex multiplicatio... more We consider the set of classical newforms with rational coefficients and no complex multiplication. We study the distribution of quadratic-twist classes of these forms with respect to weight k and minimal level N . We conjecture that for each weight k ≥ 6, there are only finitely many classes. In large weights, we make this conjecture effective: in weights 18 ≤ k ≤ 24, all classes have N ≤ 30, in weights 26 ≤ k ≤ 50, all classes have N ∈ {2, 6}, and in weights k ≥ 52, there are no classes at all. We study some of the newforms appearing on our conjecturally complete list in more detail, especially in the cases N = 2, 3, 4, 6, and 8, where formulas can be kept nearly as simple as those for the classical case N = 1.
We present a method for computing complete lists of number fields in cases where the Galois group... more We present a method for computing complete lists of number fields in cases where the Galois group, as an abstract group, appears as a Galois group in smaller degree. We apply this method to find the twenty-five octic fields with Galois group PSL 2 (7) and smallest absolute discriminant. We carry out a number of related computations, including determining the octic field with Galois group 2 3 : GL 3 (2) of smallest absolute discriminant.
In this paper we give a conjectural refinement of the Davenport-Heilbronn theorem on the density ... more In this paper we give a conjectural refinement of the Davenport-Heilbronn theorem on the density of cubic field discriminants. We explain how this refinement is plausible theoretically and agrees very well with computational data.
Let S be a finite set of primes and let G be a finite group. Let N F (S, G) be the set of Galois ... more Let S be a finite set of primes and let G be a finite group. Let N F (S, G) be the set of Galois number fields K ⊂ C with Gal(K/Q) ∼ = G and discriminant divisible only by primes in S. The sets N F (S, G) are finite, by a classical theorem of Hermite. We are interested in the following inverse Galois problem. For given S and G, find defining polynomials f (x) ∈ Q[x] for as many fields in N F (S, G) as possible. This problem is most interesting when |N F (S, G)| can be expected to be small, so that one can reasonably aim for complete lists. Thus we are interested in "collecting number fields", the most valuable specimens being those which are least ramified, for a given Galois group G. In this paper we consider only cases with S = {2, 3}. If G has a faithful permutation representation of degree ≤ 7, we have completely identified N F (S, G) previously with Jones [JR1], [Jon]. Here we supplement these complete lists with incomplete lists of fields with larger groups G, involving the simple groups P SL 2 (9), SL 2 (8), SL 3 (3), A 8 , P Sp 4 (3), A 9 , Sp 6 (2), and A 32 . We construct all our fields by the standard technique of specializing three point covers. However our requirement that the final field discriminant be of the form ±2 a 3 b is extremely restrictive. In the main, it restricts consideration to three point covers with bad reduction at 2 and 3 only. To keep ramification within {2, 3}, the specialization point τ then has to be chosen judiciously as well. Throughout we used Mathematica, supplemented by the discf, factorpadic, and polred functions of Pari. Of the three point covers which we use, for ten it was hard to find a defining polynomial. The polynomials f 9a , f 9b , f 13a , and f 27a were found by Elkies, Matzat, Malle, and Häfner, respectively, [Elk1], [Mat], [Mal1], [Häf]. The polynomials f 13b , f 13c , f 27b , f 27c , f 27d , and f 28 are new here. We do not go into much detail about how we found the new polynomials, our methods being standard. Rather, we emphasize various aspects of the specialization step. Section 1 provides an overview of the fields constructed in this paper. Section 2 and 3 discuss three point covers and especially specialization. Sections 4-10 discuss the individual constructions, sorted approximately by |G|; each of these sections also informally discusses a more theoretical issue, as indicated by the section titles. Section 11 indicates a direction for further research we feel is important and promising. We conclude in Section 12 with two speculative conjectures on the general nature of the numbers |N F (S, G)|. Throughout we work as much as possible in a purely number-theoretic setting. However the reader should realize that a systematic study of the sets N F (S, G) would fundamentally involve both the theory of motives and the theory of automorphic representations:
The study of the moduli of covers of the projective line leads to the theory of Hurwitz varieties... more The study of the moduli of covers of the projective line leads to the theory of Hurwitz varieties covering configuration varieties. Certain onedimensional slices of these coverings are particularly interesting Belyi maps. We present systematic examples of such "Hurwitz-Belyi maps." Our examples illustrate a wide variety of theoretical phenomena and computational techniques. Contents 1. Introduction 1 2. Two Belyi maps unexpectedly defined over Q 5 3. Hurwitz maps, Belyi pencils, and Hurwitz-Belyi maps 9 4. The two rational Belyi maps as Hurwitz-Belyi maps 12 5. The semicubical clan 16 6. The braid-triple method 27 7. Hurwitz-Belyi maps exhibiting spin separation 30 8. Hurwitz-Belyi maps with |G| = 2 a 3 b p and ν = (3, 1) 34 9. Hurwitz-Belyi maps with |G| = 2 a 3 b 5 and ν = (3, 1) 37 10. Hurwitz-Belyi maps with |G| = 2 a 3 b 7 and ν = (3, 1) 42 11. Some 5-and 6-point Hurwitz-Belyi maps 47 12. Expectations in large degree 52 References 54
We give conditions for the monodromy group of a Hurwitz space over the configuration space of bra... more We give conditions for the monodromy group of a Hurwitz space over the configuration space of branch points to be the full alternating or symmetric group on the degree. Specializing the resulting coverings suggests the existence of many number fields with surprisingly little ramification -for example, the existence of infinitely many A m or S m number fields unramified away from f2; 3; 5g. 1. Introduction 511 2. Hurwitz covers 513 3. Braid groups 515 4. Lifting invariants 517 5. The full-monodromy theorem 523 6. Proof of I D ) II 529 7. Proof of II D ) I 536 8. Full number fields 541 Acknowledgements 544 References 544 MSC2010: primary 14D05; secondary 20F36, 11R21.
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