Harmonic Analysis and Statistics of the First Galois Cohomology Group
aa r X i v : . [ m a t h . N T ] F e b Harmonic Analysis and Statistics of the FirstGalois Cohomology Group
Brandon Alberts and Evan O’DorneyFebruary 23, 2021
Abstract
We utilize harmonic analytic tools to count the number of elementsof the Galois cohomology group f P H p K, T q with discriminant-like in-variant inv p f q ď X as X Ñ 8 . Specifically, Poisson summation producesa canonical decomposition for the corresponding generating series as asum of Euler products for a very general counting problem. This type ofdecomposition is exactly what is needed to compute asymptotic growthrates using a Tauberian theorem. These new techniques allow for theremoval of certain obstructions to known results and answer some out-standing questions on the generalized version of Malle’s conjecture for thefirst Galois cohomology group.
Let K be a number field. Extensions of K with certain Galois groups andfixed resolvent can be parametrized by coclasses in the first cohomology group H p K, T q of a Galois module T . It is natural to count such coclasses asymp-totically by an invariant such as the norm of the discriminant or the product ofthe ramified primes.The first author [Alb20] proved a result counting coclasses by discriminant-like invariants which are Frobenian; that is, they are determined at almostall places p of K by the Frobenius element ´ F { Kp ¯ for some finite extension F { K (we call the finitely many places where this is not satisfied irregularplaces ). Moreover, the result also applies if we impose a local condition L p Ď H p K p , T q at each place p , provided that these conditions are again Frobenianin the appropriate sense. Explicitly, this result can be interpreted as givingthe asymptotic size of an “infinite Selmer group”, where for any family of localconditions L “ p L p q we define H L p K, T q “ t f P H p K, T q : res p p f q P L p u . If we allow L p to include non-trivial ramification at infinitely many places,this set may be infinite (for example, let L p “ H p K p , T q for all p so that1 L p K, T q “ H p K, T q ). We can still say something about the “size” by talkingabout how the 1-coclasses are distributed with respect to some ordering. Givenan admissible ordering inv : H p K, T q Ñ I K valued in the group of fractionalideals (as in [Alb20]), we can define H L , inv p K, T ; X q “ t f P H L p K, T q : N K { Q p inv p f qq ă X u , which is a finite set, and we can ask how this set grows as X Ñ 8 . We willoften omit the “inv” from the subscript when it is clear from context.We state the Asymptotic Wiles result proven in [Alb20] for comparison:
Theorem (Asymptotic Wiles Theorem; Theorem 4.7 [Alb20]) . Let T be a fi-nite Galois K -module, and L “ p L p q p and inv a family of local conditions andadmissible ordering respectively. Suppose(a) L and inv are Frobenian;(b) L p ď H p K p , T q is a subgroup and, for almost all places p , L p con-tains H ur p K p , T q . For infinitely many places, L p is strictly larger than H ur p K p , T q ;(c) For all places p , if f, f P H p K p , T q such that x f | I p y “ x f | I p y ď H p I p , T q then ν p p inv p f qq “ ν p p inv p f qq .Then | H L p K, T ; X q| „ c inv p K, L q X { a inv p L q p log X q b inv p L q´ for explicit positive constants a , b , and c . The method of proof in [Alb20] consists of using the Greenberg-Wiles identityto rewrite the Dirichlet series counting such coclasses as a finite sum of FrobenianEuler products, and then applying a Tauberian theorem to extract informationabout the average size of the coefficients. Due to the close relationship to theGreenberg-Wiles identity, and viewing | H L p K, T ; X q| as the asymptotic size ofan “infinite Selmer group”, this result is dubbed the Asymptotic Wiles Theorem.Since Greenberg-Wiles counts only groups of global coclasses in a compactbox (i.e., that L p are required to be subgroups ), some adjusting is necessaryto get information about infinitely many coclasses at once; this adjusting forcesone to impose some technical conditions on the local conditions L p and invariantinv.In this paper, we use harmonic analysis on adelic cohomology as a replace-ment for the Greenberg-Wiles identity, modeled after the celebrated use of har-monic analysis on the adeles in Tate’s thesis [Tat67]. Harmonic analysis onadelic cohomology has been used by Frei-Loughran-Newton [FLN18, FLN19] asan alternate approach to counting abelian extensions of number fields with pre-scribed local conditions, and by the second author (work in progress) to givea streamlined proof of the Ohno-Nakagawa reflection theorem and its gener-alizations. Poisson summation, in this setting, is a suitable generalization ofGreenberg-Wiles that will allow us to lift some of the hypotheses in the maintheorem to prove the following extended version of the Asymptotic Wiles theo-rem: 2 heorem 1.1. Let T be a finite Galois K -module, and L “ p L p q p and inv afamily of local conditions and admissible ordering respectively. Suppose(a) L and inv are Frobenian.(b) For all but finitely many places p , L p is a union of cosets of H ur p K p , T q containing the identity coset.Then | H L p K, T ; X q | „ c inv p K, L q X { a inv p L q p log X q b inv p L q´ This is a significant improvement, in particular completely removing hy-pothesis (c) from the original result. We might call (c) the property of being constant on divisions , where a division of a group G is a minimal subset C Ă G closed under conjugation (i.e. g P C implies g h P C ) and under invert-ible powers (i.e. if n is coprime to the order of G and g P C , then g n P C ).Previously studied methods for counting number fields ordered by an arbitraryinvariant (such as Wright’s work on abelian extensions [Wri89]) do not requirethis assumption, suggesting that it is merely an artifact of the methods beingused in [Alb20]. By using harmonic analysis to bypass this condition, we verifythis intuition.As a particular example, if G Ď S n is a faithful transitive representation of G ,the discriminant disc p Σ q of the ´etale algebra attached to a coclass Σ P H p K, T q is a natural invariant to order by. (A more in depth discussion of discriminantsof ´etale algebras is found in [Alb20, Definition 3.2].) It satisfies property (c)at all but finitely many places, but some issues occur at wildly ramified placesor places ramified in the Galois action on T . The first author uses this toprove upper and lower bounds on | H L p K, T ; X q| of the same order of magnitudein [Alb20], and remarks that (c) is the obstruction to producing the asymptoticmain term. Theorem 1.1 is a sufficient generalization to prove the followingcorollary, which counts nonabelian extensions with specified nonabelian Galoisgroup G and specified resolvent by an abelian normal subgroup. In the languageof [Alb20], this counts p T IJ G q -towers ordered by discriminant, where T is anabelian normal subgroup of G . Corollary 1.2.
Let T IJ G be an abelian normal subgroup of a finite group, and π : Gal p K { K q Ñ G a homomorphism with T π p Gal p K { K qq “ G such that theGalois action of T factors through the conjugation action x.t “ π p x q tπ p x q ´ .(i) If L and inv satisfy the hypotheses of Theorem 1.1 and S is the set ofirregular places, then the limit lim X Ñ8 t f P H L p K, T ; X q : f ˚ π surjective u| H L p K, T ; X q| converges, where p f ˚ π qp x q “ f p x q π p x q is understood to apply to a repre-sentative of f in Z p K, T q . Moreover, the limit is(a) positive if π is surjective, b) positive if T “ x f p p I p q : f p P L p , p R S y , and(c) equal to if T “ x f p p I p q : f p P L r a inv p L qs p , p R S y where L r m s p “ t f P L p : ν p p inv p f qq “ m u . (ii) If G Ă S n is a transitive representation of G , then the invariant disc π p f q “ disc p f ˚ π q satisfies the hypotheses of Theorem 1.1, where disc : Hom p Gal p K { K q , S n q Ñ I K is the discriminant on the associated ´etale algebra of degree n . Corollary 1.2 is a generalization of [Alb20, Theorem 5.3], and for the mostpart the proof is the same after replacing the Asymptotic Wiles’ Theoremin [Alb20] with the more powerful Theorem 1.1. In particular, as describedin [Alb20], disc π satisfies condition p i qp b q so that the number of towers withbounded discriminant N p L { K, T IJ G ; X q “ t f P H π p K, T ; X q : f ˚ π surjective u is asymptotic to cX { a p T q p log X q b p K,T p π qq´ for some positive constant c as X Ñ 8 . This proves the generalization ofMalle’s conjecture to towers for T abelian described in [Alb20] with an exactasymptotic.In addition to allowing us to relax the hypotheses and prove the asymptoticmain terms, the harmonic analytic method has noteworthy elegance. The prooffound in this note is many times shorter than the original proof in [Alb20], andit naturally motivates the resulting finite sum of Euler products as the sum overthe dual arising in a Poisson summation. Our goal is to study the average value of certain functions over the global coho-mology classes H p K, T q . Given such a function w , we are looking to understandthe summation ÿ f P H p K,T q w p f q . This can appear as an average of an arithmetic function if w p f q “ w X p f q is zerowhen inv p f q ě X , so that we can consider the limiting behavior as X Ñ 8 . Itcould also appear as a Dirichlet series, such as w p f q “ inv p f q ´ s , on which weare interested in studying the meromorphic continuation. (Note that in manycases these two examples are intimately related, and results for one can implyresults for the other.)The idea is to realize H p K, T q as a discrete subgroup of some locally com-pact group, so that ř w p f q appears to be one side of a Poisson summation. Werecall the Poisson formula for locally compact abelian groups (see, for instance,Hewitt and Ross [HR70], p. 246): 4 heorem 2.1. Let G be a locally compact abelian group and w : G Ñ C acontinuous function of class L . Suppose that G Ď G is a subgroup such that p a q For each x P G , the integral w p x q “ ż y P G w p x ` y q dy is absolutely convergent; p b q The resulting function w : G { G Ñ C is L ; p c q Its Fourier transform x w is L on p G { G q _ – G _ { G K (where G K is theannihilator of G ).Then, with respect to an appropriate scaling of the measures on G and G K , ż x P G w p x q dx “ ż y P G K ˆ w p y q dy. In our case, we will realize G “ H p K, T q as a discrete group, so that ż f P H p K,T q w p f q df “ ÿ f P H p K,T q w p f q (up to scaling the measure).The goal of this section is to set up the other features of Poisson summationin this setting, including the definition of G and the corresponding groups G _ and G K . Definition 2.2.
The adelic cohomology of a Galois K -module T is the re-stricted direct product H p A K , T q – ź p H p K p , T q H ur p K p ,T q , or more explicitly H p A K , T q – " p f p q P ź p H p K p , T q ˇˇˇˇ f p P H ur p K p , T q for all but finitely many p * . This is a locally compact abelian group. Its Pontryagin dual, by local Tateduality, is isomorphic to H p A K , T ˚ q where T ˚ “ Hom p T, µ q is the Tate dualof T .There is a map i T : H p K, T q Ñ H p A K , T q K ã Ñ ¯ K p . We gather the necessary informationon this map from the Poitou-Tate nine-term exact sequence [NSW08, Theorem8.6.10], which in the case that S is the set of all places is given by0 H p K, T q H p A K , T q H p K, T ˚ q _ H p K, T q H p A K , T q H p K, T ˚ q _ H p K, T q H p A K , T q H p K, T ˚ q _ , i T p i T ˚ q _ where each of the homomorphisms are continuous and the nine terms havethe following topologies respectively: finite, compact, compact, discrete, locallycompact, compact, discrete, discrete, finite.Although i T need not be injective, we see that the kernel is the continuousimage of a compact group into a discrete cocompact group, and so is necessarilyfinite. Moreover, the middle row shows that the images of i T and i T ˚ are duallattices in their respective cohomology groups. We thus derive the followingcorollary of Poisson summation: Theorem 2.3.
Normalize the measure on H p A K , T q so that the compact sub-group ź p finite H ur p K, T q ˆ ź p infinite H p K, T q has measure . Then for any function w : H p A K , T q Ñ C satisfying thehypotheses of Theorem 2.1 for G “ H p A K , T q and G “ H p K, T q , ÿ f P H p K,T q w p f q “ | H p K, T q || H p K, T ˚ q | ÿ g P H p K,T ˚ q ˆ w p g q . Proof.
The image im p ι T q is a discrete subgroup of H p A K , T q , and the Poitou-Tate nine-term exact sequence implies im p ι T ˚ q “ im p ι T q K . Thus, Poisson sum-mation (Theorem 2.1) implies that there exists a constant c such that ż f P im p ι T q w p f q df “ c ¨ ż g P im p ι T ˚ q ˆ w p g q dg. The constant c depends on how the measures df and dg are scaled. We specifieda scaling at the beginning, so we cannot assume the two integrals are equal onthe nose. We know that im p ι T q and im p ι T ˚ q are discrete, so that the above isequivalent to ÿ f P im p ι T q w p f q “ c ¨ ÿ g P im p ι T ˚ q ˆ w p g q . Recalling that w is defined on H p A K , T q and that the corresponding map w : H p K, T q Ñ C is defined to be the pullback along ι T , we see in particular6hat w p f h q “ w p f q for any h P ker p ι T q . That same statement is true for T replaced by T ˚ , so | ker p ι T q | ÿ f P H p K,T q w p f q “ c | ker p ι T ˚ q | ÿ g P H p K,T ˚ q ˆ w p g q . It now suffices to determine the constant coefficient c | ker p ι T ˚ q || ker p ι T q | , which is independent of w . We note that when w is the characteristic functionof a compact open box ś p L p , this identity is exactly Greenberg-Wiles as statedin [DDT95, Theorem 2.19]. This necessarily implies that the constant coefficientis equal to | H p K, T q || H p K, T ˚ q | , concluding the proof.The use of Theorem 2.3 separates into two cases:1. If w is periodic with respect to a compact open subgroup of the form ź p R S H ur p K p , T q for some finite set of places S , then ˆ w p g q has finite support. This impliesthat the sum over H p K, T ˚ q is secretly a finite sum. In this case, it willsuffice to understand the behavior of ˆ w p g q at individual values of g . (Wenote that this is similar to classical Poisson summation for real-valuedfunctions, where the Fourier transform of a periodic function has discretesupport. The extra structure of being periodic with respect to an open subgroup is what gives an even smaller support in this case.)2. If w is not periodic , then the sum over H p K, T ˚ q is truly an infinitesum. This situation is worth considering, as we can show that it is closelyrelated to other important questions in number theory and arithmeticstatistics. While dealing with the infinite sum requires more powerfulanalytic tools than we will employ in this paper, the behavior of eachindividual summand ˆ w p g q will give important insight into certain questionsabout nonabelian extensions.We will study several examples of both periodic and non-periodic w functionswhich are important to open questions in number theory, and in the periodiccase we will use these results to prove Theorem 1.1. w The examples we will consider come from multiplicative functions:7 efinition 3.1.
We call a function w : H p A K , T q Ñ C multiplicative if w p f q “ ź p w p f | G Kp q . The discriminant is multiplicative, so the function w p f q “ disc p f q ´ s is nec-essarily multiplicative (as is the corresponding example for any admissible or-dering inv as defined in [Alb20, Definition 4.4]). Multiplicativity is preservedby the Fourier transform, so thatˆ w p g q “ ź p ˆ w p g | G Kp q“ ź p ¨˝ | H p K p , T q | ÿ f p P H p K p ,T q x f p , g | G Kp y w p f p q ˛‚ where x¨ , ¨y is the Tate pairing. Note that if g “ w p f q “ disc p f q ´ s then it follows thatˆ w p q “ ź p ¨˝ | H p K p , T q | ÿ f p P H p K p ,T q disc p f p q ´ s ˛‚ is exactly the local series given by Bhargava (in the case of the trivial action) andthe first author (in the case of nontrivial actions). This suggests a reformulationof the Malle-Bhargava principle in terms of Fourier transforms: Heuristic 3.2 (Malle-Bhargava principle on H p K, T q ) . Suppose w : H p A K , T q Ñ C is a “reasonable” multiplicative function satisfying the hypotheses of Poissonsummation (Theorem 2.1). Then there exists a positive constant c ą such that ÿ f P H p K,T q w p f q “ c ¨ ˆ w p q ` “lower-order term”. Following in the footsteps of Cohen-Lenstra [CL84], we do not give a defini-tion for a “reasonable” function; instead we will consider a series of examples inthis paper that should be considered “reasonable.” The notion of a “lower-orderterm” depends on the context as well, and we don’t give an explicit definitionin all cases. In the example w p f q “ disc p f q ´ s , ˆ w p q is a Dirichlet series whoserightmost pole is at s “ { a p T q of order b p K, T q [Alb20, Theorem 3.3]. We thentake “lower-order term” to mean a Dirichlet series whose rightmost pole is at s P t z P C : Re p z q ă { a p T quY t { a p T qu and if it is at s “ { a p T q then it is of anorder ă b p K, T q . By applying a Tauberian theorem, we see that this expressionidentifies the sum of coefficients of ˆ w p q as the main term of ÿ disc p f qă X , Remark:
In the discriminant ordering (or other similar orderings), thereare certain cases where the main term really is a constant multiple of ˆ w p q .This will happen when there exist other g P H p K, T ˚ q with ˆ w p g q is on thesame “order of magnitude” as ˆ w p q , and so must also contribute to the mainterm.One easily shows that | ˆ w p g q| ď | ˆ w p q| for all g P H p K, T ˚ q and all w for which the Fourier transform exists, which suggests that ˆ w p q should attainthe correct order of magnitude, if not the correct multiplicity. (Note: there iscertainly something to prove here, as we have done nothing to show that the infi-nite sum of ˆ w p g q does not attain a larger order of magnitude than the individualsummands. This is why we merely refer to this statement as a “heuristic”.) w Suppose w is periodic with respect to a compact open subgroup Y ď H p A K , T q and satisfies the hypotheses of Theorem 2.3. Then ˆ w is supported on the com-pact open subgroup Y K , and ÿ g P H p K,T ˚ q ˆ w p g q “ ÿ g P H p K,T ˚ qX Y K ˆ w p g q is a sum over the group H p K, T ˚ q X Y K . Discreteness of H p K, T ˚ q and com-pactness of Y K imply that H p K, T ˚ qX Y K is finite, which makes the summationeasier to evaluate.An important example of a periodic function is the absolute discriminantwhen T has the trivial action. If Y “ ź p ∤ Hom ur p Q p , T q , then | disc p f ` y q| “ | disc p f q| for every y P Y , as the discriminant only seesramification behavior. Y ď H p A Q , T q is necessarily a compact open subgroupin the adelic topology. The function w p f q “ | disc p f q| ´ s is shown to satisfy thehypotheses of Theorem 2.3 by [FLN18], which they then use to compute thelocation and orders of poles of the corresponding Dirichlet series.Our main example of a periodic function is a generalization of the absolutediscriminant to the largest class of functions for which the same ideas will work.This follows the ideas in [Alb20] by allowing nontrivial actions on T , allowingrestricted local conditions L “ p L p q , and allowing an admissible ordering to beused instead of the usual discriminant: Proposition 4.1.
Let K be a number field, S a finite set of places of K , and T a Galois module. If(a) L “ p L p q is a family of subsets L p Ă H p K p , T q such that for all p R S , L p is closed under translation by H ur p K p , T q and P L p ; b) inv is an admissible ordering as defined in [Alb20, Definition 4.4],then the function w : H p A K , T q Ñ C defined by w p f q “ N K { Q p inv p f qq ´ s f P ś p L p f R ś p L p satisfies the hypotheses of Theorem 2.3 and is periodic with respect to the com-pact open subgroup Y “ Y S “ ź p R S H ur p K p , T q . Proof. w is periodic with respect to Y by definition, so it suffices to check thehypotheses of Poisson summation.We first prove that w is L on H p A K , T q for ℜ p s q sufficiently large. Define a p w q “ lim inf N K { Q p Ñ8 min f P H p K p ,T q ν p p inv p f qq‰ ν p p inv p f qq . Then ż H p A K ,T q | w p f q| df ! ź p ´ ` O p N K { Q p p q ´ a p w q ℜ p s q q ¯ , which converges absolutely for ℜ p s q ą { a p w q . We use this to prove each ofthe hypotheses of Poisson summation (Theorem 2.1) with G “ H p A K , T q and G “ im p ι T q :(a) Since w is Y -periodic, we have | w p x q| “ µ p Y q ż Y | w p x ` y q| dy, where, since Y is open, the Haar measures on Y and G can be taken tocoincide. Hence ż G | w p x ` z q| dz “ µ p Y q ż G ż Y | w p x ` y ` z q| dy dz “ p G X Y q µ p Y q ż Y ` G | w p x ` z q| dz ď p G X Y q µ p Y q ż G | w p z q| dz ă 8 , where we used that, since Y is compact and G is discrete, their intersec-tion is finite. Thus the integral defining w is absolutely convergent.10b) We note that ż G { G | w p x q| dx “ ż G { G (cid:12)(cid:12)(cid:12)(cid:12) ż G w p x ` y q dy (cid:12)(cid:12)(cid:12)(cid:12) dx ď ż G { G ż G | w p x ` y q| dy dx “ ż G | w p x q| dx , which converges by w L , thus w is L for ℜ p s q ą { a p w q .(c) Since w is L , its Fourier transform x w is bounded and hence L on thecompact space G _ { G K .This is almost precisely the setup for Theorem 1.1, where the only conditionmissing is that L and inv are Frobenian. Indeed, Proposition 4.1 implies that ÿ f P H p K,T q w p f q is a Dirichlet series which can be expressed as a finite sum of Euler products.Knowing that L and inv are Frobenian implies that the Euler products are ofFrobenian factors, which is sufficient for calculating the locations of their polesas is done in [Alb20, Proposition 2.2]. A Tauberian theorem is then used toconvert the analytic information at the poles into asymptotic information. Wesketch the argument here, but refer to [Alb20] for the details. Proof of Theorem 1.1.
Let w be as in Proposition 4.1. Then ÿ f P H p K,T q w p f q “ | H p K, T q|| H p K, T ˚ q| ÿ g P H p K,T ˚ q ˆ w p g q is a Dirichlet series. w is periodic with respect to Y , which implies ˆ w has finitesupport in H p K, T ˚ q . The rate of growth of | H L p K, T ; X q| is determined by thelocation and order of the rightmost pole of this series via a Tauberian theorem.It then suffices to show that the rightmost pole occurs at 1 { a inv p L q of order b inv p L q .First considerˆ w p q “ ź p ¨˝ | H p K p , T q| ÿ f P L p N K { Q p inv p f qq ´ s ˛‚ . This is precisely the Frobenian Euler product studied in [Alb20], denoted Q p , s q in Propositions 4.14 and 4.16 of that paper, which was shown to have rightmost11ole at 1 { a inv p L q of order b inv p L q . For each other term, note that | ˆ w p g q| “ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ź p ¨˝ | H p K p , T q| ÿ f P L p x f, g | G Kp y N K { Q p inv p f qq ´ s ˛‚ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ď ź p ¨˝ | H p K p , T q| ÿ f P L p N K { Q p inv p f qq ´ ℜ p s q ˛‚ , which agrees with ˆ w p q at ℜ p s q . Taking a limit as s Ñ { a inv p L q implies thatif ˆ w p g q has a pole (including a branch singularity like p s ´ { a inv p L qq ´ { forexample) at s “ { a inv p L q , it is of order at most b inv p L q . Thus, after adding thefinitely many other terms ˆ w p g q to ˆ w p q , it suffices to show that this rightmostpole of ˆ w p q does not cancel out.The proof that the poles do not cancel out is the same as the proof of theAsymptotic Wiles Theorem in [Alb20], so we only summarize the main points.Write L r m s p “ t f P L p : ν p p inv p f qq “ m u . Let T ď T be the subgroup definedby T “ x f p p I p q : p R S, f p P L r a inv p L qs p y , and let i : T Ñ T denote the inclusion. The restrictions L p T q p “ p i ˚ q ´ p L p q Ď H p K p , T q form an admissible family L p T q of local conditions on H p K, T q ,and i ˚ : H L p T q p K, T ; X q Ñ H L p K, T ; X q has finite kernel. The set L r a inv p L qs p Ď L p of f p of minimal weight a inv p L q is also included in i ˚ p L p T q p q , which impliesthat a inv p L q “ a inv p L p T qq . The fact that L p and L p T q p are closed undertranslation by unramified coclasses implies that, for each place p unramified inthe action on T , | L r a inv p L qs p || H p K p , T q| “ I p ´ L r a inv p L qs p ¯ “ I p ´ i ˚ L p T q r a inv p L qs p ¯ “ I p ´ L p T q r a inv p L qs p ¯ “ | L p T q r a inv p L qs p || H p K p , T q| , noting that H p K p , A q “ H ur p K p , A q for p ∤ and that I p acting trivially on T implies the restriction of i ˚ to H p I p , T q Ñ H p I p , T q is injective. The b -invariant is the average value of these ratios for p ď x as x Ñ 8 (see [Alb20,Definition 4.6]) so that b inv p L q “ b inv p L p T qq . Note that for any nonzero g P ` p i ˚ q ´ Y S ˘ K Ď H p K, p T q ˚ q , we have res p p g q ‰ p by the Chebotarev density theorem. The coefficient of p ´ a inv p L p T qq s in y i ˚ w p g q is given by ÿ f P L r a inv p L qs p x f, res p p g qy p ´ a inv p L q s , p ´ a inv p L p T qq s in y i ˚ w p q for a positive proportion of p by the fact that the Tate pairing isa perfect pairing. [Alb20, Proposition 2.2] then shows that the singularity at s “ { a inv p L p T qq of y i ˚ w p g q , if it is a singularity, is of strictly lower orderthan the rightmost pole of y i ˚ w p q . Thus we know the rightmost pole for thegenerating function of H L p T q p K, T ; X q does not cancel out and a Tauberiantheorem implies that | H L p T q p K, T ; X q| „ y i ˚ w p q „ cX { a inv p L q p log X q b inv p L q´ . As this is a lower bound for | H L p K, T ; X q| (up to the size of the finite kernel of i ˚ ), we know the rightmost poles must not cancel out in this case either.Corollary 1.2 is then a direct consequence of Theorem 1.1. We summarizethe proof here, but we refer to [Alb20] for the details. Proof of Corollary 1.2.
We remark that if f „ f under the coboundary rela-tion, then there exists a t P T such that p f ˚ π qp G K q “ t p f ˚ π qp G K q t ´ . Thereforethe T -conjugacy class of the image is preserved under the coboundary relation,so in particular surjectivity is preserved under the coboundary relation and thenumerator in the corollary statement is well-defined.That the limit exists follows from the following inclusion-exclusion argument.For a subgroup H ď G , let L p H X T q “ p L p X H p K p , H X T qq p . (Here, andin what follows, we suppress pullbacks and pushforwards along the inclusion H X T ã Ñ T .) Then, as long as there exists at least one f H P H L p K, T q suchthat p f H ˚ π qp G q ď H , it follows that t f P H L p K, T ; X q : p f ˚ π qp G K q ď H u “ f H ˚ H L p H X T q , inv fH p K, H X T ; X q where inv f H p f q “ inv p f H ˚ f q .It is clear that L p H X T q and inv f H inherit the hypotheses of Theorem 1.1from L and “inv” away from the places ramified in f H , so Theorem 1.1 impliesthat t f P H L p K, T ; X q : p f ˚ π qp G K q ď H u„ c inv fH p K, L p H X T qq X { a inv fH p L p H X T qq p log X q b inv fH p L p H X T qq´ . Let µ G be the M¨obius function on the poset of T -conjugacy classes of sub-groups H ď G , ordered by inclusion up to conjugation, so that t f P H L p K, T ; X q : f ˚ π surjective u| H L p K, T ; X q|„ ÿ H µ G p H q c inv fH p K, L p H X T qq X { a inv fH p L p H X T qq p log X q b inv fH p L p H X T qq´ c inv p K, L q X { a inv p L q p log X q b inv p L q´ .
13s the summation is a finite sum of functions of the form x a p log x q b ´ , the limitcertainly exists (we know the limit is not infinite because the original ratio isbounded between 0 and 1).Part (c) of Corollary 1.2 then follows from a more in-depth study of the a -and b -invariants to show that only the term H “ G of the summation contributessomething nonzero to the limit, while parts (a) and (b) follow by bounding itbelow by a ratio of functions of the form | H L p K, T ; X q| with the same a - and b -invariants. The proof-method for these steps is exactly the same as for [Alb20,Theorem 5.3], so we direct the reader to that paper for the details.In order to prove part (ii) it suffices to show that disc π is admissible andFrobenian. This proven in [Alb20, Lemma 5.1]. w In all of the following examples, w p f q will be a function of a complex variable s P C so that ÿ f P H p K,T q w p f q is a Dirichlet series, similar to the periodic example in Proposition 4.1. How-ever, these examples will not be periodic and we will find that Theorem 2.3decomposes this Dirichlet series as an infinite sum of Euler products.We can still reference the Malle-Bhargava principle, which suggests thatsum should have the same “order of magnitude” ˆ w p q . Although this is notnecessarily true for arbitrary infinite sums (even sums of Euler products), weare able to give evidence that this close to the truth. Proposition 5.1.
Let K be a number field, S a finite set of places of K , and T a Galois module. Define the function w : H p A K , T q Ñ C by w p f q “ N K { Q p inv p f qq ´ s f P ś p L p f R ś p L p , where(a) L “ p L p q is a family of subsets L p Ă H p K p , T q such that for all p R S , H ur p K p , T q Ă L p ,(b) inv is an admissible ordering as defined in [Alb20].Then w satisfies the hypotheses of Theorem 2.3.Proof. Simply note that | w p f q| is bounded above by ¯ w p f q “ | N K { Q p inv p f qq| ´ s ,the case that occurs when L p “ H p K p , T q is as large as possible. Note that¯ w p f q is Y S -periodic and satisfies the hypotheses of Theorem 2.3 by Proposition4.1. The remainder of the proof proceeds exactly like the proof of Proposition4.1. 14he difference between Propositions 5.1 and 4.1 is the removal of the trans-lation invariance requirement on the family of allowable local conditions L . Theonly condition we impose is that all but finitely many places allow for any un-ramified behavior, which we need to assume in order to avoid specifying thesplitting types of infinitely many unramified primes (if we did specify these, itmight happen that no coclass satisfies all the local conditions!)In this setting, Theorem 2.3 tells us that ÿ f P H p K,T q w p f q “ | H p K, T q|| H p K, T ˚ q| ÿ g P H p K,T ˚ q ˆ w p g q is an infinite sum of Euler products. In order to apply a Tauberian theoremto get asymptotic information with these kinds of local restrictions, some morework needs to be done to uniformly control the order of ˆ w p g q .We present the simplest example of this phenomenon below: Example 5.2.
Let K “ Q , T “ C have the trivial action, and for each prime p ∤ , choose a subset L p with H ur p Q p , C q Ĺ L p Ĺ H p Q p , C q ;necessarily | L p | “
3. This amounts to choosing one of the two ramified quadraticextensions of Q p ; the easiest choice, which we will make though it is not essential,is L p “ H ur p Q p , C q Y t h p u , h p corresponds to Q p p? p q . Complete the family p L p q of local conditions by setting L “ H ur p Q , C q , L “ H p R , C q . Then counting cohomology classes satisfying the local conditions L p , with re-spect to discriminant, amounts to counting squarefree integers D ” p | D , D { p is a square modulo p : a cute question in arithmeticstatistics, which we have not seen before.If we define w : H p Q , C q Ñ C as in Proposition 5.1 with L as above andthe usual discriminant disc, then it follows (since C is its own Tate dual) that ÿ f P H p Q ,C q w p f q “ ÿ g P H p Q ,C q ˆ w p g q . For an individual g , we see thatˆ w p g q “ ź p ∤ ¨˝ | H p Q p , C q| ÿ f p P L p x f p , g | G Q p y p ´ ν p p disc p f p qq s ˛‚ “ ź p | disc p g q ˆ x h p , g | Q p y p ´ s ˙ ź p ∤ p g q8 ˆ ` x h p , g | Q p y p ´ s ˙ . ˘ ´ ω p disc p g qq | disc p g q| ´ s , while the second prod-uct is like an L -function. If χ g is the character of g , then ź p ∤ p g q8 ˆ ` x h p , g | Q p y p ´ s ˙ “ ÿ ∤ n χ g p n q ´ ω p n q µ p n q n ´ s , which can be shown to be (conditionally) convergent and nonzero at s “ g “
0. If g “
0, it has a pole of order 1 { s “
1. In order to concludethat ˆ w p q really is the main term, we would need to show that (up to the signsof the summands) ÿ g P H p Q ,C q ´ ω p disc p g qq | disc p g q | ´ s ÿ ∤ n χ g p n q ´ ω p n q µ p n q n ´ s is of the same or smaller “order of magnitude”. This amounts to computing themoments of L -function look-alikes, and while in principle this is an accessiblequestion we will not address it in this paper. It is known that central embedding problems have a solution if and only if theyhave a solution locally [Ser08]. Suppose B and T are abelian groups which fitinto a short exact sequence1 T G B ι π for which T is a central subgroup of G . Additionally, put a Galois action on G which acts trivially on the subgroup T . If we then define the family of localconditions L “ p L p q such that L p “ π ˚ p H p K p , G qq Ď H p K p , B q , it follows that t f P H p K, B q : f P π ˚ p H p K, G qqu “ H L p K, B q is a generalized Selmer group. Moreover, the fact that the extension is cen-tral implies that H ur p K p , B q Ă π ˚ p H p K p , B qq for all p so that L satisfies thehypotheses of Proposition 5.1! This is a major step in the right direction, andindicates that studying non-periodic functions can be used to apply this methodto certain nonabelian groups G that fit into such a short exact sequence.To complete the process, for every f P H L p K, B q associate an r f P H p K, G q such that π ˚ r f “ f . Then all other lifts of f are given by twists of r f by someelement of H p K, T q , by the inflation restriction sequence. We decompose thegenerating function for H p K, G q ordering by discriminant as follows: ÿ F P H p K,G q N K { Q p disc p F qq ´ s “ ÿ z P H p K,T q ÿ f P H L p K,B q N K { Q p disc p r f ˚ z qq ´ s r f and then in the variable z , to dualizethis expression; or we can equivalently apply Poisson summation to the productspace H p K, T q ˆ H p K, B q . If we define w p f, z q “ N K { Q p disc p r f ˚ z qq ´ s f P ś p L p , we find that w is multiplicative and satisfies all the conditions of Theorem 2.3by a similar proof to that of Proposition 4.1. Therefore ÿ F P H p K,G q N K { Q p disc p F qq ´ s “ | H p K, T ˚ q|| H p K, B ˚ q|| H p K, T q|| H p K, B q| ÿ x P H p K,T ˚ q ÿ g P H p K,B ˚ q ˆ w p g, x q is an infinite sum of Euler products which each converge for sufficiently large s (in this case, ℜ p s q ą { a p G q ). (Note that if each Euler product converges, thenso does their sum, by Poisson summation.) We remark that w is periodic in the z variable, but not in the f variable due to the fact that L p is not necessarilyclosed under translation by H ur p K, T q at any place p .This process generalizes to repeatedly taking central extensions, so that wecan state the following: Corollary 5.3.
Let G be a finite nilpotent group with a Galois action, i.e. aniterated central extension of a Galois K -module by groups with trivial Galoisactions. Then ÿ f P H p K,G q N K { Q p disc p f qq ´ s is equal to an infinite sum of Euler products which each converge absolutely for ℜ p s q ą { a p G q , where the sum is over coclasses with coefficients in the duals ofthe factors of the upper central series of G . We intend this discussion to be a “look to the future” for this method,which is why we left the statement of Corollary 5.3 rather vague. Indeed, thisdecomposition is not unique but depends on the choices of lifts f ÞÑ r f . It maybe the case that some choices of lifts are nicer than others, and these choicesshould be informed by the analytic tools used to study non-periodic functions. References [Alb20] Brandon Alberts. Statistics of the first Galois cohomology group: Arefinement of Malle’s conjecture, December 2020. Preprint availableat https://arxiv.org/abs/1907.06289 .[Bha10] Manjul Bhargava. Mass formulae for extensions of local fields, andconjectures on the density of number field discriminants.
InternationalMathematics Research Notices , July 2010.17CL84] H. Cohen and H. W. Lenstra. Heuristics on class groups of numberfields.
Lecture Notes in Mathematics Number Theory Noordwijkerhout1983 , pages 33–62, 1984.[DDT95] H. Darmon, F. Diamond, and R. Taylor. Fermat’s last theorem.
Cur-rent Developments in Mathematics , 1995(1):1–154, 1995.[FLN18] Christopher Frei, Daniel Loughran, and Rachel Newton. The Hassenorm principle for abelian extensions.
American Journal of Mathe-matics , 140(6):1639–1685, 2018.[FLN19] Christopher Frei, Daniel Loughran, and Rachel Newton. Numberfields with prescribed norms, 2019.[HR70] Edwin Hewitt and Kenneth A. Ross.
Abstract harmonic analysis.Vol. II: Structure and analysis for compact groups. Analysis on lo-cally compact Abelian groups . Die Grundlehren der mathematischenWissenschaften, Band 152. Springer-Verlag, New York-Berlin, 1970.[NSW08] J¨urgen Neukirch, Alexander Schmidt, and Kay Winberg.
Cohomologyof Number Fields , volume 323. Springer, 2nd edition, 2008.[Ser08] Jean-Pierre Serre.
Topics in Galois Theory . A K Peters, 2008.[Tat67] John Tate. Fourier analysis in number fields and Hecke’s zeta-functions. In J. W. S. Cassels and A. Frohlich, editors,
AlgebraicNumber Theory . Thompson, Washington, DC, 1950/1967.[Wri89] David J. Wright. Distribution of discriminants of abelian exten-sions.