In this note, I want to discuss the work of Weil as presented in Milne’s papers (in particuar this one). Given a finite Galois extension $L/k$ of fields, many readers will be familiar with the usual “Galois descent” procedure giving an equivalence between (for example) affine varieties over $k$ and affine varieties over $L$ with a suitable $Gal(L/k)$ action. Indeed, such extensions are finite etale, and the result is immediate from the general theory of etale descent and the “Galois isomorphism” $L \otimes_k L \cong \prod_{g \in Gal(L/k)} L$, which one can unravel and check implies that to give a $Gal(L/k)$-action is to give a descent datum.

If $L/k$ is no longer finite, this method breaks down. It will still be an fpqc cover, so descent theory works fine, but the Galois isomorphism breaks down. This can be seen using infinite Galois theory: for example by considering $\bar{Q}/Q$ and noting that only the open index 2 subgroups give rise to degree 2 etale algebras. In the case where $L/k$ is algebraic and ind-Galois, $Aut(L/k)$ is profinite and imposing continuity conditions like the one we saw above, one is able to recover Galois descent theory from the finite case. However, if the extension is transcendental, it is less clear how to proceed. For example, a key obviously interesting example is when one has an object over $\mathbb{C}$ together with an action of $Aut(\mathbb{C}/\mathbb{Q})$. Here $\mathbb{C}$ has uncountable transcendence degree.

Before we state our theorem, let us first (following Milne) make the question at hand a little more precise. Let $V$ be a variety (geometrically reduced separated scheme of finite type) over a field $\Omega$ which we assume is algebraically closed (to avoid having to define “Galois”). Let $k \subset \Omega$ be a subfield. A descent system for $V$ is a collection of isomorphisms $\theta_\sigma: \sigma V \stackrel{\cong}{\rightarrow} V$ one for each $\sigma \in Aut(\Omega/k)$ satisfying the cocycle condition $\theta_{\sigma_1 \sigma_2} = \theta_{\sigma_1} \sigma_1(\theta_{\sigma_2})$. We emphasise that such a thing is generally not the same as a descent datum. If there is a model $V_0/k$ giving rise to $(V,\theta)$ by base change, we say $(V,\theta)$ is effective.

Theorem (Weil): Let $\Omega/k$ be an algebraically closed extension of infinite transcendence degree, and suppose we are given $(V,\theta)$ a quasiprojective variety over $\Omega$ with descent system to $k$ as above. Then for $(V,\theta)$ to be effective it suffices for there to be a finitely generated subextension $L/k$ such that the restriction of $\theta$ to $\Omega/L$ is effective.

Milne notes the following consequence. Suppose your variety over $\Omega$ has the following properties.

1. There is a finite set of points $S$ such that any automorphism of $V$ which fixes each point of $S$ must be the identity.
2. There is a finitely generated extension $L/k$ with the property that $\theta_\sigma (\sigma P) = P$ for all $P \in S, \sigma \in Aut(\Omega/L)$.

Then the given pair is effective for descent $\Omega/k$.

Indeed since $V$ is quasiprojective we may always fix any model for it and enlarge $L$ to contain the coefficients of the polynomials describing the variety and the points of $S$. Thus we obtain a model $V_1/L$ and the condition (2) implies that the isomorphisms coming from the model and the given $\theta$ agree on where they send the points $P$. They therefore agree by condition (1), so we have established effectivity of the restriction of $\theta$ to $Aut(\Omega/L)$. The desired result now follows from the theorem.

As far as I can see, this seems to give a very practical tool for performing descent along transcendental extensions, and is necessary for Milne to show that Langlands’ conjecture on conjugation of Shimura varieties actually really implies the existence of canonical models.

In his note A Note of Shimura’s Paper “Discontinuous Groups and Abelian Varieties” Mumford constructs a Hodge type Shimura variety (a family of polarised abelian fourfolds with level structures) that is not of PEL type. He makes the necessary group from a quaternion algebra over a totally real field which is engineered to admit a symplectic representation which is absolutely irreducible (so the resulting abelian varieties cannot admit extra endomorphisms).

More specifically, let $K/\mathbb{Q}$ be a cubic totally real field. We wish to carefully choose a quaternion algebra $D/K$ such that $H:= D^{N=1}$ admits a symplectic representation.

The trick is to note that corestriction on Brauer groups associates to $D/K$ a quaternion algebra $Cor(D)/\mathbb{Q}$, equipped with a natural map $Nm:D \rightarrow Cor(D)$. In particular, if we can arrange for $Cor(D)$ to be split, we will certainly get a natural (8-dimensional) representation of $D^*$. By then imposing some conditions at infinity we can produce a symplectic structure on this representation, realising the resulting Shimura variety as one of Hodge type.

These conditions can be realised: just take $D$ to be ramified at any even number of places above each place of $\mathbb{Q}$, and at two of the three places at infinity.

Then $H(\mathbb{R}) = SU(2) \times SU(2) \times SL_2(\mathbb{R})$, and unravelling the construction one sees that the map $Nm: H(\mathbb{R}) \rightarrow SL_8(\mathbb{R})$ factors $SU(2) \times SU(2) \times SL(2) \rightarrow SO(4) \times SL(2) \rightarrow SL(8)$ where the first map is the standard degree 2 isogeny $SU(2) \times SU(2) \rightarrow SO(4)$ and the second is given by the natural action on $\mathbb{R}^4 \otimes \mathbb{R}^2$, and visibly preserves the natural symplectic pairing coming from the product of orthogonal and symplectic pairings on these two spaces. Symplectic structures descend, giving the required map

$H \rightarrow Sp_8$  over $\mathbb{Q}$.

One extends this game to a Hodge type Shimura datum by taking

$G=Nm(D^*) \subset GL_{8,\mathbb{Q}}$, which is an extension of the image of $H$ by $\mathbb{G}_m$ landing in $GSp_8$.

One takes the Hodge structure $h$ to be the composite of $Nm$ with the map taking $a+ib \in \mathbb{S}$ to $1$ on the $GU(2)\times GU(2)$ factor of $D^*_\mathbb{R}$ and the standard 2-dimensional embedding into the $GL_2(\mathbb{R})$ factor. It is obvious this extends to give the Siegel Shimura datum on $GSp_8$, so we have a Shimura datum of Hodge type, as required.

In this short post I want to note Deligne’s classic result on vector bundles with flat connections (let’s call them “differential equations”: terminology justified by the cyclic vector theorem”) and give a stupid example illustrating the basic shape of things. I should thank Ananth Shankar for helping unconfuse me faster than I could unconfuse myself.

Let $X$ be a smooth algebraic variety over $\mathbb{C}$. A vector bundle on $X$ is a locally free coherent (Zariski) sheaf $\mathcal{F}$, and a connection is an additive map of sheaves

$\nabla: \mathcal{F} \rightarrow \mathcal{F} \otimes_{\mathcal{O}_X} \Omega^1_{X/\mathbb{C}}$

satisfying the Leibniz law. There is the usual notion of flatness of a connection. A morphism between two such objects is an $\mathcal{O}_X$-linear map between the underlying sheaves making the obvious square commute.

One can also make the exactly parallel construction over the complex manifold $X^{an}$. Given an algebraic differential equation, we can analytify it and obtain an analytic differential equation: $(-)^{an}: DE(X) \rightarrow DE(X^{an}).$

Deligne’s result gives a functor in the other direction (let’s call it RH for “Riemann-Hilbert”)

$RH: DE(X^{an}) \rightarrow DE(X)$.

Theorem (Deligne): The functor RH (exists and) is fully faithful, and the essential image $RSDE(X)$ is characterised by taking a good compactification of $X$ and restricting to those differential equations with regular singularities at the boundary.

This is a little counterintuitive at first. One imagines there being “more” objects in $DE(X^{an})$ than in $DE(X)$ because one can write down more connections, but actually the extra freedom causes more stuff to become isomorphic, resulting in a “smaller” analytic category. Of course, one has $\mathcal{F} \cong RH(\mathcal{F})^{an}$ but not (unless $X$ is proper) an identification in the other direction.

Let’s see what happens in a stupid example. Let $X=\mathbb{A}^1$ and $\mathcal{F}$ the trivial line bundle. Let’s take our favourite non-vanishing algebraic section $v$, and observe that any connection is automatically flat and determined by $\nabla(v)$

(since any other section can be written as $f(t)v$ which by Leibniz is given by

$\nabla(f(t)v) = v f'(t) \otimes dt + f(t) \nabla(v)$).

If $g(t) \in \mathbb{C}[t]$ let us denote by $\nabla_g$ the connection taking $v \mapsto g(t) v \otimes dt$.

Let’s suppose we have a map $\alpha: \nabla_g \rightarrow \nabla_h$ between two such differential equations. Let $\alpha(v) = u(t) v$, and the compatibility of $\alpha$ with the connections gives the relation

$u'(t) + u(t)h(t) = u(t)g(t)$.

In other words, $u$ must satisfy a first order differential equation, and in fact we see that up to a constant factor

$u(t) = e^{\int (g(t)-h(t))dt}$

which is never algebraic unless $g(t)-h(t)=0$.

The upshot is that in the algebraic category each $\nabla_g$ gives a distinct differential equation, while in the analytic category one can use the above recipe to construct isomorphisms between each of the $\nabla_g$ and in particular they are all equal to $\nabla_0$, which is the unique algebraic connection with regular singularities at $\infty$.

I’m about to give a talk on how to conclude lots of cases of Fontaine-Mazur for 2-dimensional $Gal(\bar{\mathbb{Q}}/\mathbb{Q})$-representations from Emerton’s study of completed cohomology. For this to work, one needs Bockle’s arguments after Gouvea-Mazur and Coleman to deduce “big” R=T theorems from “small” ones which are not too difficult given the success of the original Taylor-Wiles-(Diamond?)-Kisin approach. I attach a set of notes saying roughly how this works and giving references to the literature.

Today I’ve found myself mainly thinking about the interaction between Frobenius and the filtration on crystalline cohomology, referring to Mazur’s classic paper http://projecteuclid.org/download/pdf_1/euclid.bams/1183533965 as well as Ogus’ paper on “Griffiths Transversality in Crystalline Cohomology” and thought I would record a summary here before I forget everything. Of course, both of these papers are from the 1970s, so if there has been any significant advance or later examples of interest that any readers know of I’d be extremely interested to hear.

Firstly the setup. Let $X_0/k$ be a smooth projective variety over a perfect field of characteristic $p$, and let us also suppose we have a smooth proper lift $X/W(k)$. We can form the crystalline cohomology and have comparisons $M:= H^i_{crys}(X_0/W(k)) =H^i_{crys}(X/W(k)) = H^i_{dR}(X/W(k))$. Let us assume these are free modules and in fact also that the Hodge cohomology groups $H^q(X, \Omega^p_{X/W(k)})$ are free. Then we have the further relation that $M/pM = H^i_{dR}(X_0/k)$.

What structures are in play? Algebraic de Rham cohomology comes with a Hodge filtration $F^k \subset M$ (which depends on the lift $X$, though of course mod $p$ it does not), and crystalline cohomology is equipped with a semilinear Frobenius $\Phi: \sigma^*M \rightarrow M$ (which does not depend on the lift).

The fundamental relationship between these two structures is given by a theorem of Mazur, which implies under the freeness assumptions we have made that the Frobenius determines the mod $p$ Hodge filtration.

Theorem (Mazur): The reduction mod $p$ of $\Phi^{-1}(p^jM)$ is precisely the reduction of $F^j$.

(Note for the statement that Frobenius induces an isomorphism $\Phi^*M/p\Phi^*M = H^i_{dR}(X_0^{(p)}/k) \rightarrow H^i_{dR}(X_0/k)=M/pM$ preserving the Hodge filtration.)

One immediate consequence of this is the conjecture of Katz relating the slopes of Frobenius to the shape of the Hodge filtration.

Corollary: For the crystalline cohomology of $X_0/W(k)$, the Newton polygon lies above the Hodge polygon.

We do not explain what this means here, except to remark that Mazur’s theorem allows you to find bases in which the matrix for $\Phi$ has columns divisible by powers of $p$ of widths given by the Hodge numbers. The statement, which is about relating these numbers to valuations of eigenvalues, is then just an easy result in linear algebra.

The next obvious question is what can we say about the Hodge filtration $F^k$ coming from our smooth lift? The above result tells us that

$\Phi(\sigma^*F^k) \subset p^kM + p\Phi(\sigma^*M)$.

Mazur was also able to prove that $\Phi(\sigma^*F^k) \subset p^{v(k)} M$, where $v(k) := \text{sup}_{l \geq k} v_p(p^l/l!)$.

Question (Mazur): Is it in fact the case that $\Phi(\sigma^*F^k) \subset p^k M$? If so, we say $M$ is strongly divisible, and this statement is equivalent to Frobenius inducing an isomorphism $\sum_i p^{-i} \sigma^*F^i \rightarrow M$.

Note that whenever the Hodge filtration has length shorter than $p$, this is immediate from Mazur’s second inequality, since $v(k)=k$ in this case for all nonempty pieces of filtration. For example, for the cohomology of a curve or $H^1$ of an abelian variety we do get something strongly divisible.

The reason for the distance between Mazur’s estimate and the notion of strong divisibility is the phenomenon of Griffiths transversality, which was investigated by Ogus in the crystalline context following Griffiths’ work in classical Hodge theory.

Classically suppose you have a variety $X/\mathbb{C}[[t]]$ and you want to study its de Rham cohomology. Using the Gauss-Manin connection, one can identify the cohomology group itself with the cohomology of the constant family $X'$ defined by the fibre at $t=0$. However, the Hodge filtrations will not agree, but are allowed to vary within the confines imposed by Griffiths transversality. Explicitly, one can show that

$F_{X}^k \subset \sum_{i \geq 0} t^i/i! F_{X'}^{k-i}$.

In the crystalline situation, given two smooth lifts $X,X'$, one obtains a similar formula,

$F_{X}^k \subset \sum_{i \geq 0} p^i/i! F_{X'}^{k-i}$.

However, note that the denominators start to cancel off the powers of $p$, once $i \geq p$, and this is exactly why Mazur can only get the estimate involving $v(k)$. For a more general statement in the same vein see Ogus’ “corollary 2.5.”

These are all inequalities, and it is natural to ask if they are “strict”. In other words, does Griffiths transversality really happen (do these filtrations vary), and to what extent? One obvious example to bear in mind is abelian varieties, where every possible lift of the Hodge filtration corresponds to a lift (and determines it uniquely: this is Grothendieck-Messing theory). If, as the filtration lengths increase, there is enough freedom for the lifts to vary widely within the constraints imposed by transversality, then one would expect a counterexample to strong divisibility.

Ogus manages precisely this, and his example is as follows. Suppose $p>2$, and consider the hypersurface $X'/W(k)$ given by

$X': X_0^{p+2} + \dots X_{p+1}^{p+2} + pX_0\dots X_{p+1} = 0$.

Then $\Phi(\sigma^*(F^p_{X'}(H^p_{crys}(X_0/W(k)))))$ fails to lie in $p^p H^p_{crys}(X_0/W(k))$.

This is achieved by comparing to the Fermat hypersurface which also visibly lifts the special fibre. One can show that in fact the cohomology of this is strongly divisible, exploiting the fact the group actions in play give an explicit decomposition of the cohomology, making it easy to control. Comparing the filtration coming from this to the filtration coming from $X'$, using a careful deformation-theoretic study of these differences, Ogus deduces that the latter fails to be strongly divisible.

This blog post is a place to collect information about the Harvard number theory learning seminar spring 2014. We meet Mondays 4:15-6pm (actual times: if you run on Harvard time, we meet at 4:08) in SC 507.

After a brief intense seminar on Scholze’s thesis and the MSRI conference, we are currently reading the paper of Matt Emerton on Local-Global compatibility for GL_2(Q) in the p-adic Langlands correspondence.

While nominally being organised by myself, this is now really being run by Erick Knight, to whom all angry complaints and difficult questions about the mathematical content and organisation should be addressed. I am still prepared to accept compliments, lavish gifts, etc. and still exercise the right to write dates in the format DD/MM.

Schedule of Talks

(2 introductory talks by Erick on the general p-adic Langlands program)

24/03 – Introduction, Completed cohomology and promodular representations (Rong)

07/04 – p-adic Langlands done correctly (Yihang)

14/04 – Completed Hecke Algebras 1 (Koji)

21/04 – Completed Hecke Algebras 2 (Bao?)

28/04 – Proof of Main Theorem (Yunqing)

04/05 – Application to the Fontaine-Mazur Conjecture (Tom)

In a short article appearing on the arxiv today Cuntz and Deninger seem to give a new simple construction of the ring of Witt vectors. Seems like good news for people like me who use them on a daily basis but would probably need to go looking for a copy of Serre’s “Local Fields” if asked for the details on how to actually construct them. This post represents my processing the definition appearing in their paper.

Firstly, recall that for any perfect $\mathbb{F}_p$-algebra $R$, a strict $p$-ring for $R$ is a ring $A$ in which $p$ is not a zero-divisor, Hausdorff and complete for the $p$-adic topology, and with $A/p \cong R$. The key fact about these rings is the following.

Proposition: For every perfect $\mathbb{F}_p$-algebra $R$, a strict $p$-ring $W(R)$ exists, and is unique up to unique isomorphism as a ring over $R$ (in fact $R \mapsto W(R)$ is a functor).

This is of course well-known. The new construction goes as follows. View $R$ as a monoid (remember multiplication and the identity element), and form the free ring $\mathbb{Z}R$ thereon. This has elements $\sum_i n_i[r_i]$ with addition as a “free $\mathbb{Z}$-module” and multiplication coming from the monoid structure. For example, this means that weirdly $[0]\not= 0$ yet (but $[1]=1$). There is also (induced from the identity on $R$ as a monoid) a natural ring map $\mathbb{Z}R \rightarrow R$. Let $I$ be its kernel, and form the $I$-adic completion $A$. It turns out (and the proof in their paper is pretty short) that this is a strict $p$-ring and so canonically isomorphic to $W(R)$.

One interesting feature of this construction is that of course you can run it for $R$ not perfect, where the Witt construction also gives you something but which is non-canonical (a Cohen ring’). Apparently this construction in general gives a different one from the Witt construction (so it is genuinely different: not a clever repackaging).

In this post I want to advertise Serre’s lovely note, following the opening section which gives a simple definition of p-adic modular forms (via q-expansions) and uses this and some elementary congruences to construct the classical p-adic L-functions. Serre’s writing is excellent so I give only a brief account and let the interested reader consult the master.

The definition goes as follows. Fix a level (in fact, let’s just take level 1), and a prime $p$ (which we will take to be odd, but the case $p=2$ appears no harder). By taking q-expansions, one can consider the algebra of classical modular forms of level 1 (all weights and indeed mixed weights) as a subalgebra $\mathcal{M} \subseteq \mathbb{C}[[q]].$ In fact, flat base change and the construction of models for modular curves implies one can actually do this on the integral and rational levels in such a way as loses no information: one is naturally led to study $\mathcal{M}_\mathbb{Z} \subseteq \mathbb{Z}[[q]]$ and $\mathcal{M}_\mathbb{Q} = \mathcal{M}_\mathbb{Z} \otimes \mathbb{Q}$. But now note that $\mathbb{Z}[[q]]\otimes \mathbb{Q} \subset \mathbb{Z}_p[[q]][1/p]$, a subring of $\mathbb{Q}_p[[q]]$ which can be equipped with the structure of a Banach algebra via the ‘sup norm’ (the distance between $\sum a_n q^n$ and $\sum b_n q^n$ is the largest value of $|a_n-b_n|$).

It thus makes sense to consider the closure of $\mathcal{M}_\mathbb{Q}$ with respect to this topology, and it is this which Serre defines to be the algebra $\mathcal{M}_p$ of $p$-adic modular forms. In other words, a p-adic modular form is just a power series with p-adic coefficients which can be approximated uniformly coefficient-wise by the q-expansions of classical modular forms.

Pure classical modular forms have a weight $k$. What about p-adic modular forms? One crucial computation in Serre’s paper that makes the theory work says that if two modular forms are congruent modulo a large power of $p$ then their weights are also congruent modulo a large power of $p$ and also modulo $p-1$. From this one can easily attach a weight in the group $X = \mathbb{Z}/(p-1)\mathbb{Z} \times \mathbb{Z}_p$ to a p-adic modular form (and since we are working at level 1, all weights arising will be even (so the subgroup $2\mathbb{Z}/(p-1)\mathbb{Z}$ on the first factor is all that is hit). We should remark that this computation has as an input the classical theorem of Clausen-von Stadt on the denominators of Bernoulli numbers.

Another interesting consequence of this which Serre notes is the following. Suppose we have a power series $f= a_0+a_1q+a_2q^2+...$ and we would like to show it’s a p-adic modular form. Maybe we can write it as a putative limit of some modular forms $f^i=a^i_0+a^i_1q+a^i_2q^2+...$, except we only know that for all $n \geq 1$ the coefficients $a_n^i \rightarrow a_n$ converge (but know nothing about the constant coefficients). Suppose further we also know the weights $k^i$ converge to some nonzero $k \in X$.

Well then the weight zero modular form $a_0$ and $f$ cannot be too close p-adically, where the length scale implicit in the phrase ‘too close’ is given with reference to one of the modular forms in question, so one gets a bound of the form $sup_{n \geq 1} |a_n| \geq C |a_0|$, where $C$ is some constant depending on how far $k$ is from zero. Applying this to our sequence $f^i$ we learn in particular that the $a^i_0$ lie in a closed bounded subset of $\mathbb{Q}_p$, so there is a convergent subsequence, and passing to the corresponding subsequence of modular forms we deduce that $f$ is a p-adic modular form after all.

One could complain that this is a silly theorem: when will we have that all but the constant coefficient is known to converge? Recall the classical weight $k$ Eisenstein series, whose q-expansions are given in terms of the Bernoulli numbers $B_k = -k \zeta(1-k)$ and the “sum of $(k-1)$th powers of all divisors of $n$” function $\sigma_{k-1}(n)$:

$G_k = -\frac{B_k}{2k} + \sum_{n\geq 1} \sigma_{k-1}(n)q^n$.

If we take a p-adically convergent sequence $k_1,k_2,... \rightarrow k \in X$ of weights that also tends to infinity in the archimedean metric, then one sees explicitly that p-adically:

$\sigma_{k_i-1}(n) = \sum_{d|n} d^{k_i-1} \rightarrow \sum_{d|n, p\not|d} d^{k-1} =: \sigma^*_{k-1}(n)$.

Therefore the sequence $G_{k_i}$ of Eisenstein series fit exactly in the situation of the above theorem, and we deduce that there is a p-adic Eisenstein series of any weight $k \in X$, which is well-defined because $\sigma^*_{k-1}(n)$ depends only on $k$. In particular, we deduce the existence of a well-defined continuous p-adic function $\zeta_p:X-\{1\} \rightarrow \mathbb{Q}_p$ such that $\zeta_p(k) = \frac{1}{2} \zeta(1-k)$ for all $k \in \mathbb{Z}_{\geq 2}$. One can check that this is exactly the p-adic zeta function constructed classically by Kubota-Leopoldt and featuring on the analytic side of the main conjecture of Iwasawa theory.

In this post I want to briefly record the following nice recipe for predicting the dimension of a Galois representation you expect to extract from a Shimura variety (and the local factors you expect to see). More precisely, let’s suppose I have a cohomological cuspidal representation $\pi$ of a reductive group $G/F$. Then Langlands would conjecture the existence of a map

$Gal(\bar{F}/F) \rightarrow ^LG(\bar{\mathbb{Q}_l})$.

However, it’s not clear the L-group necessarily has a preferred representation, and ultimately this is what we will construct in the cohomology of our Shimura variety. To see where this extra information comes from, recall that the Shimura variety is defined by both the group $G$ and a map $h:\mathbb{S} \rightarrow G_\mathbb{R}$. This latter datum can be made into a Hodge cocharacter $\mu: \mathbb{G}_m \rightarrow G$. But a cocharacter of $G$ is a character of its Langlands dual, i.e. a weight, and it is possible to take the irreducible algebraic representation with this as highest weight, and extend it to the whole L-group.

Composing the map predicted by Langlands reciprocity with this representation gives the Galois representation we might hope to find in the cohomology of our Shimura variety. As a nice bonus, the cocharacters floating around are miniscule, so computing the dimensions of these irreducible reps is really easy (the weight spaces are multiplicity free and just Weyl-translates of the highest weight).

This is a quick note to record some thoughts following from Toby Gee’s first lecture of his course at the Arizona Winter School, where he observes that quadratic reciprocity is a completely immediate consequence of basic algebraic number theory. I feel rather silly for never having noticed this before, and hope I don’t insult the reader by providing a post on it.

That quadratic reciprocity follows immediately from class field theory is standard, and for the rational numbers class field theory can be decomposed into the irreducibility of cyclotomic polynomials (reciprocity laws for cyclotomic extensions) and the Kronecker-Weber theorem (cyclotomic extensions fill out all the abelian extensions). Of these, I would only consider the second to be `hard’.

The key point that makes quadratic reciprocity strictly easier than class field theory is that for $p$ an odd prime, the quadratic extensions $K_p := \mathbb{Q}(\sqrt{(-1)^{(p-1)/2} p})/\mathbb{Q}$ are the unique quadratic extensions ramified only at $p$. They therefore obviously satisfy the Kronecker-Weber theorem, since $\mathbb{Q}(\zeta_p)/\mathbb{Q}$ has a degree 2 subextension which is ramified only at $p$ and thus equal to $K_p$.

We can easily make this more explicit. Consider the quadratic character

$\chi: G_\mathbb{Q} \rightarrow Gal(K_p/\mathbb{Q}) = \{ \pm 1 \}$.

By definition (more or less), for $q \not= p$ odd, this character is unramified and $\chi(Frob_q) = \left(\frac{(-1)^{(p-1)/2} p}{q}\right)$.

On the other hand, by our previous observation (“Kronecker Weber” in this special case), and the standard isomorphism between $Gal(\mathbb{Q}(\zeta_p)/\mathbb{Q})$ and $(\mathbb{Z}/p\mathbb{Z})^*$ (“irreducibility of cyclotomic poly”: here just Eisenstein’s theorem), we obtain the factorisation:

$\chi: G_\mathbb{Q} \rightarrow (\mathbb{Z}/p\mathbb{Z})^* \rightarrow \{\pm 1 \}$,

where $Frob_q$ is mapped to the class of $q$ modulo $p$. Equipped with this description (and recalling that $(\mathbb{Z}/p\mathbb{Z})^*$ is cyclic), it is clear that

$\chi(Frob_q) = \left(\frac{q}{p}\right)$.

Comparing the two expressions obtained, we recover the classical quadratic reciprocity law. One can also handle $q=2$ the same way with a small amount of care (over the correct way to interpret the first expression: Hensel’s lemma doesn’t give a direct comparison with a Legendre symbol in this case).