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Reflection Theorems for Number Rings

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Reflection Theorems for Number Rings Reflection theorems for number rings Evan O’Dorney July 13, 2021 Contents I Introduction 4 1 Introduction 4 1.1 Historicalbackground . ............ 4 1.2 Methods......................................... ....... 5 1.3 Results......................................... ........ 6 1.4 Outlineofthepaper ............................... .......... 7 1.5 Acknowledgements ................................ .......... 7 2 Examples for the lay reader 7 2.1 Reflectionforquadraticequations. ................ 7 2.2 Reflectionforcubicequations . .............. 9 2.3 Reflection for 2 n n boxes.................................... 11 2.4 Reflectionforquarticequations× × . ............... 12 3 Notation 15 II Galois cohomology 15 4 Étale algebras and their Galois groups 16 4.1 Étalealgebras................................... .......... 16 4.2 TheGaloisgroupofanétalealgebra . .............. 17 4.3 Resolvents...................................... ......... 17 4.4 Subextensionsandautomorphisms . .............. 18 4.5 Torsors ......................................... ....... 18 arXiv:2107.04727v1 [math.NT] 10 Jul 2021 4.6 AfreshlookatGaloiscohomology . ............. 20 5 Extensions of Kummer theory to explicitize Galois cohomology 23 5.1 TheTatepairingandtheHilbertsymbol. ............... 27 6 Rings over a Dedekind domain 29 6.1 Indicesoflattices............................... ............ 29 6.2 Discriminants................................... .......... 30 6.3 Quadraticrings.................................. .......... 32 6.4 Cubicrings ...................................... ........ 33 6.5 Quartic rings and their cubic resolvent rings . ................... 35 1 7 Cohomology of cyclic modules over a local field 41 7.1 Discriminants of Kummer and affine extensions . ................ 42 7.2 TheShafarevichbasis ............................. ........... 44 7.3 ProofofTheorem7.1............................... .......... 49 7.4 Thetamecase ..................................... ....... 51 III Composed varieties 51 8 Composed varieties 52 8.1 Examples ........................................ ....... 53 8.2 Integral models; localization of orbit counts . .................... 54 8.3 Fourier analysis of the local and global Tate pairings . ..................... 56 IV Reflection theorems: first examples 60 9 Quadratic forms by superdiscriminant 61 10 Class groups: generalizations of the Scholz and Leopoldt reflection theorems 67 10.1Dualorders ..................................... ......... 67 10.2 Dual orders are plentiful for quadratic extensions . ...................... 68 10.3 Localdualgeneralizedorders . ............... 69 10.4 Relation to the Scholz reflection theorem . .................. 71 V Reflection theorems: cubic rings 71 11 Cubic Ohno-Nakagawa 71 11.1 Abijectiveproofofthetamecase. ............... 74 11.2 Acomputationalproof. ............ 77 11.3 Abijectiveproofofthewildcase . ............... 85 12 Non-natural weightings 90 12.1 Local weightings given by splitting types . ................... 90 12.2 Discriminantreduction. .............. 91 12.3 Subringzetafunctions . ............. 93 12.4 Invariant lattices at 2 ........................................ 96 12.5 Binary cubic forms over Z[1/N] ................................... 98 VI Reflection theorems: quartic rings and related objects 100 13 Reflection for 2-adic quartic orders, and applications 100 13.1 Resultsonbinaryquarticforms . ...............103 13.2 The conductor property of the resolvent ring . ..................107 14 Tame quartic rings with non-split resolvent, by multijection 109 14.1 Invertibilityofidealsinorders. ..................109 14.2 Self-duality of the count of quartic orders . ...................111 VII Counting quartic rings with prescribed resolvent 113 15 Introduction 113 2 16 The group H1 of quartic algebras with given resolvent 114 17 Reduced bases 115 17.1 The extender basis of a cubic resolventring . ..................119 18 Resolvent conditions 120 18.1 Transformation, and ring volumes in the white zone . ...................124 18.2 Fromringvolumestoringcounts . .............. 125 19 The conic over K 127 19.1Diagonalconics.................................O ........... 130 19.2 Thesolutionvolumeoftheconic . .............. 132 19.3TheBrauerclass ................................. .......... 135 19.4Thesquareness .................................. .......... 136 19.5 ..................................................138 N11 20 Boxgroups 141 20.1Signatures..................................... .......... 141 20.2 Boxgroups in unramified splitting type . .................142 20.3 Boxgroups in splitting type 13 ....................................145 20.4 Therecenteringlemma. ............ 149 20.5Charmedcosets.................................. .......... 150 20.6Theprojectors .................................. .......... 152 20.7Notation....................................... ......... 153 21 Ring volumes for ξ1′ 154 21.1Thesmearinglemma............................... .......... 154 21.2 The zones when is strongly active (black, plum, purple, blue, green, and red)....... 155 N11 21.3 The zones when 11 isweaklyactive(brownandyellow) . 166 21.4Thebeigezone ...................................N ......... 171 21.5Orthogonality.................................. ........... 173 21.6Smearedanswers ................................. .......... 176 21.7 The average value of a quadratic character on a box . ..................178 22 Ring volumes for ξ2′ 180 22.1 12 ..................................................180 22.2 M ..................................................181 M22 23 Further remarks on the code 184 VIII Unanswered questions 184 24 Doubly traced quartic rings 185 25 Reflection for 2 n n boxes 187 × × IX Appendices 187 A The Grothendieck-Witt ring and the proof of Lemma 19.11 188 3 B Examples of zone totals 189 Abstract The Ohno-Nakagawa reflection theorem is an unexpectedly simple identity relating the number of GL2Z-classes of binary cubic forms (equivalently, cubic rings) of two different discriminants D, −27D; it generalizes cubic reciprocity and the Scholz reflection theorem. In this paper, we provide a framework for generalizing this theorem using a global and local step. The global step uses Fourier analysis on the 1 adelic cohomology H (AK , M) of a finite Galois module, modeled after the celebrated Fourier analysis on AK used in Tate’s thesis. The local step is combinatorial, more elementary but much more mysterious. We establish reflection theorems for binary quadratic forms over number fields of class number 1, and for cubic and quartic rings over arbitrary number fields, as well as binary quartic forms over Z; the quartic results are conditional on some computational algebraic identities that are probabilistically true. Along the way, we find elegant new results on Igusa zeta functions of conics and the average value of a quadratic character over a box in a local field. Part I Introduction 1 Introduction 1.1 Historical background In 1932, using the then-new machinery of class field theory, Scholz [49] proved that the class groups of the quadratic fields Q(√D) and Q(√ 3D), whose discriminants are in the ratio 3, have 3-ranks differing by at most 1. This is a remarkable− early example of a reflection theorem. A generalization− due to Leopoldt [31] relates different components of the p-torsion of the class group of a number field containing µp when decomposed under the Galois group of that field. Applications of such reflection theorems are far-ranging: for instance, Ellenberg and Venkatesh [21] use reflection theorems of Scholz type to prove upper bounds on ℓ-torsion in class groups of number fields, while Mihăilescu [34] uses Leopoldt’s generalization to simplify a step of his monumental proof of the Catalan conjecture that 8 and 9 are the only consecutive perfect powers. Through the years, numerous reflection principles for different generalizations of ideal class groups have come into print. A very general reflection theorem for Arakelov class groups is due by Gras [23]. A quite different direction of generalization was discovered by accident in 1997: The following relation was conjectured by Ohno [44] on the basis of numerical data and proved by Nakagawa [38], for which reason we will call it the Ohno-Nakagawa (O-N) reflection theorem: Theorem 1.1 (Ohno–Nakagawa). For a nonzero integer D, let h(D) be the number of GL2(Z)-orbits of binary cubic forms f(x, y)= ax3 + bx2y + cxy2 + dy3 of discriminant D, each orbit weighted by the reciprocal of its number of symmetries (i.e. stabilizer in GL2(Z)). Let h3(D) be the number of such orbits f(x, y) such that the middle two coefficients b,c are multiples of 3, weighted in the same way. Then for every nonzero integer D, we have the exact identity 3h(D), D> 0 h3( 27D)= (1) − ( h(D), D< 0. By the well-known index-form parametrization (see 6.9 below), h(D) also counts the cubic rings of discriminant D over Z, weighted by the reciprocal of the order of the automorphism group. It turns out that h (D) counts those rings C for which 3 tr ξ for every ξ C. When D is a fundamental discriminant, the 3 | C/Z ∈ corresponding cubic extensions are closely related, via class field theory, to the 3-class group of Q(√D) and we get back Scholz’s reflection theorem, as Nakagawa points out ([38], Remark 0.9). 4 Theorem 1.1 was quite unexpected, because GL2(Z)-orbits of binary cubics have been tabulated since Eisenstein without unearthing any striking patterns. Even the exact normalizations h(D), h3(D) had been in use for over two decades. They appear in the Shintani zeta functions
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