Families of L-Functions and Their Symmetry

Abstract

A few years ago the first-named author proposed a working definition of a family of automorphic L-functions. Then the work by the second and third-named authors on the Sato–Tate equidistribution for families made it possible to give a conjectural answer for the universality class introduced by Katz and the first-named author for the distribution of the zeros near s  = 1/2. In this article we develop these ideas fully after introducing some structural invariants associated to the arithmetic statistics of a family.

Appendices

Appendix 1. Hasse–Weil L-Functions

Here we recall the definition of the Hasse–Weil L-function (2) and the modularity conjecture. The modularity conjecture (Conjecture 4 below) states that the L-functions arising from algebraic varieties over $\mathbb{Q}$ should be automorphic L-functions. In fact we will explain how L-functions are attached to l-adic Galois representations, in particular the étale cohomology space appearing in (2). To do so we recall the local Langlands correspondence for general linear groups in order to be precise about the matching of L-functions at ramified places. We also reformulate the modularity conjecture as a bijective correspondence between certain l-adic Galois representations and automorphic representations preserving L-functions, incorporating observations by Clozel and Fontaine-Mazur. The reader is referred to [Tay04] for an excellent survey of many topics discussed in this Appendix.

Let p be a prime and K a finite extension of $\mathbb{Q}_{p}$ with residue field cardinality q _K . Write W _K for the Weil group of K. For an algebraically closed field $\Omega$ of characteristic 0, denote by $\mathrm{Rep}_{n}(W_{K})_{\Omega }$ (resp. $\mathrm{Rep}(\mathop{\mathrm{GL}}\nolimits _{n}(K))_{\Omega }$) the set of isomorphism classes of n-dimensional Frobenius-semisimple Weil-Deligne representations of W _K (resp. irreducible smooth representations of $\mathop{\mathrm{GL}}\nolimits _{n}(K)$) on k-vector spaces. For simplicity an element of Rep _n (W _K ) will be called an (n-dimensional) WD-representation of W _K . Recall that such a representation is represented by (V, ρ, N) where V is an n-dimensional space over $\Omega$, $\rho: W_{K} \rightarrow \mathop{\mathrm{GL}}\nolimits _{\Omega }(V )$ is a representation such that ρ(I _K ) is finite and ρ(w) is semisimple for every w ∈ W _K , and $N \in \mathrm{ End}_{\Omega }(V )$ is a nilpotent operator such that wNw ⁻¹ =  | w | N where $\vert \cdot \vert: W_{K} \rightarrow \mathbb{R}_{>0}^{\times }$ is the transport of the modulus character on K ^× via class field theory. The local Langlands reciprocity map is a bijection

$$\displaystyle{\mathrm{rec}_{K}:\mathrm{ Rep}(\mathop{\mathrm{GL}}\nolimits _{n}(K))_{\mathbb{C}} \rightarrow \mathrm{ Rep}_{n}(W_{K})_{\mathbb{C}}}$$

uniquely characterized by a list of properties, cf. [HT01]. In particular L(s, π) = L(s, rec(π)), ɛ(s, π, ψ) = ɛ(rec(π), ψ) for any nontrivial additive character $\psi: F \rightarrow \mathbb{C}^{\times }$ (and a fixed Haar measure on F), and we also have an equality of conductors f(π) = f(rec _K (π)). Here the local L and ɛ factors as well as conductors are independently defined on the left and right-hand sides. Here we will only recall the definition of the conductor and L-factor for WD-representations, which is due to Grothendieck, leaving the rest of definitions and further references to [Tate] and [Tay04]. For $(V,\rho,N) \in \mathrm{ Rep}_{n}(W_{K})_{\Omega }$ the conductor is given by

$$\displaystyle{f(V ):=\dim (V/V ^{I_{K} } \cap \ker N) +\int _{ 0}^{\infty }\dim V/V ^{I_{K}^{u} }du,}$$

where I _K ^u is the upper numbering filtration on the inertia group I _K . Now let Frob _K denote the geometric Frobenius in W _K ∕I _K . The local L-factor is defined to be

$$\displaystyle{L(s,V ):=\det (1 -\mathrm{ Frob}_{K}q_{K}^{-s}\vert V ^{I_{K} } \cap \ker N)^{-1}}$$

so that we have the equality $L(s,\pi ) =\det (1 -\mathrm{ Frob}_{K}q_{K}^{-s}\vert \mathrm{rec}(\pi )^{I_{K}} \cap \ker N)^{-1}$ for $\pi \in \mathrm{ Rep}(\mathop{\mathrm{GL}}\nolimits _{n}(K))_{\mathbb{C}}$.

Now fix a field isomorphism $\iota: \overline{\mathbb{Q}}_{l} \simeq \mathbb{C}$ and let $\rho:\mathop{ \mathrm{Gal}}\nolimits (\overline{F}/F) \rightarrow \mathop{\mathrm{GL}}\nolimits _{n}(\overline{\mathbb{Q}}_{l})$ be a continuous semisimple Galois representation which is unramified at almost all primes and potentially semistable (equivalently de Rham) at places of F above l. Such a ρ is to be called algebraic. At each finite place v of F, there is a functor WD _v from continuous representations of $\mathop{\mathrm{Gal}}\nolimits (\overline{F}_{v}/F_{v}) \rightarrow \mathop{\mathrm{GL}}\nolimits _{n}(\overline{\mathbb{Q}}_{l})$ (assumed potentially semistable if v | l) to WD-representations of $W_{F_{v}}$. The construction of WD _v relies on Grothendieck’s monodromy theorem when $v \nmid l$ and Fontaine’s work in l-adic Hodge theory if v | l.

The (global) conductor for ρ is $\prod _{v}\mathfrak{p}_{v}^{f_{v}}$ where $\mathfrak{p}_{v}$ is the prime ideal of $\mathcal{O}_{F}$ corresponding to v, and $f_{v} = f(\rho \vert _{\mathop{\mathrm{Gal}}\nolimits (\overline{F}_{ v}/F_{v})})$. With ρ is associated a product function in a complex variable s, which is a priori formal infinite product:

$$\displaystyle{L(s,\rho ):=\prod _{v:\text{finite}}L_{v}(s,\rho ),\qquad L_{v}(s,\rho ):= L(s,\iota \mathrm{WD}_{v}(\rho \vert _{\mathop{\mathrm{Gal}}\nolimits (\overline{F}_{ v}/F_{v})})).}$$

When ρ arises as a subquotient in the l-adic cohomology of an algebraic variety over F, one can apply Deligne’s purity theorem to show that L(s, ρ) converges absolutely for Re(s) ≫ 1 (with often explicit lower bound). Further there is a recipe for the archimedean factor L _∞ (s, ρ) in terms of Hodge-Tate weights of ρ at places above l. (See the definition of $\Gamma (R,s)$ in [Tay04, § 2], taking R to be the induced representation of ρ from $\mathop{\mathrm{Gal}}\nolimits (\overline{F}/F)$ to $\mathop{\mathrm{Gal}}\nolimits (\overline{F}/\mathbb{Q})$.) This leads to a completed L-function

$$\displaystyle{\Lambda (s,\rho ):= L(s,\rho )L_{\infty }(s,\rho ).}$$

In the main body of the paper we were interested in the L-functions for Galois representations arising from varieties. Let X be a smooth projective variety over $\mathbb{Q}$, so X has good reduction modulo p for all but finitely many primes p. Then a reciprocity law for X on a concrete level would be a description of the number of points of X in $\mathbb{F}_{p}$ (and its finite extensions) in terms of automorphic data at p (i.e., local invariants at p of several automorphic representations of general linear groups) as p runs over the set of primes with good reduction, cf. [Lan76]. This may be thought of as a non-abelian reciprocity law generalizing the Artin reciprocity law in class field theory as well as an observation about elliptic modular curves by Eichler-Shimura. Now we say that $\rho:\mathop{ \mathrm{Gal}}\nolimits (\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \mathop{\mathrm{GL}}\nolimits _{n}(\overline{\mathbb{Q}}_{l})$ comes from geometry if

• ρ is unramified away from finitely many primes,

• there exists a finite collection of smooth projective varieties X _i and integers $d_{i},m_{i} \in \mathbb{Z}$ (indexed by i ∈ I) such that ρ appears as a subquotient of

  $$\displaystyle{\bigoplus _{i\in I}H_{\mathrm{et}}^{d_{i} }(X \times _{\mathbb{Q}}\overline{\mathbb{Q}},\overline{\mathbb{Q}}_{l})(m_{i}).}$$

As usual (m _i ) denotes the Tate twist. One can speak of the obvious analogue with $\mathbb{Q}$ replaced by any finite extension F over $\mathbb{Q}$. In the language of L-functions the following conjecture presents a precise form of the reciprocity law as above.

Conjecture 4.

Let $\iota: \overline{\mathbb{Q}}_{l} \simeq \mathbb{C}$ be an isomorphism. If $\rho:\mathop{ \mathrm{Gal}}\nolimits (\overline{F}/F) \rightarrow \mathop{\mathrm{GL}}\nolimits _{n}(\overline{\mathbb{Q}}_{l})$ comes from geometry, then L(s,ρ) is automorphic, namely there exists an isobaric automorphic representation $\Pi$ of $\mathop{\mathrm{GL}}\nolimits _{n}(\mathbb{A}_{F})$ such that $L_{v}(s,\Pi ) = L_{v}(s,\rho )$ at every finite place v and v = ∞ (so that $L(s,\Pi ) = L(s,\rho )$ and $\Lambda (s,\Pi ) = \Lambda (s,\rho )$ ).

The Hasse-Weil conjecture predicts that L(s, ρ) should have nice analytic properties such as analytic continuation, functional equation, and boundedness in vertical strips. If we believe in the Hasse-Weil conjecture, the converse theorem (discovered by Weil and then developed notably by Piatetskii-Shapiro and Cogdell) gives us a good reason to also believe that Conjecture 4 is true.

The conjecture begs two natural questions, namely a useful characterization of ρ coming geometry and a description of $\Pi$ that arise from such ρ. The conjectural answers have been provided by Fontaine-Mazur and Clozel, respectively. Indeed a conjecture by Fontaine-Mazur asserts that a continuous semisimple l-adic representation ρ comes from geometry if and only if it is algebraic. Following Clozel a cuspidal automorphic representation $\Pi$ of $\mathop{\mathrm{GL}}\nolimits _{n}(\mathbb{A}_{F})$ is said to be L-algebraic if, roughly speaking, the L-parameters for $\Pi$ at infinite places consist of algebraic characters in a suitable sense (see [BG11] for the definition; this differs from [Clozel] in that no adjustment by the $\frac{n-1} {2}$-th power is made, cf. comments below Conjecture 5). An isobaric sum of cuspidal representations $\boxplus _{i=1}^{r}\Pi _{i}$ is algebraic if every $\Pi _{i}$ is algebraic. Then we can reformulate Conjecture 4 as one about the existence of the global Langlands correspondence preserving L-functions:

Conjecture 5.

Fix ι as above. Then there exists a bijection $\Pi \leftrightarrow \rho$ between the set of L-algebraic isobaric automorphic representations of $\mathop{\mathrm{GL}}\nolimits _{n}(\mathbb{A}_{F})$ and the set of algebraic n-dimensional semisimple l-adic representations of $\mathop{\mathrm{Gal}}\nolimits (\overline{F}/F)$ (up to isomorphism) such that the local L-factors are the same, so that $L(s,\Pi ) = L(s,\rho )$ and $\Lambda (s,\Pi ) = \Lambda (s,\rho )$ .

Remark.

The strong multiplicity one theorem and the Chebotarev density theorem imply that if there is a correspondence $\Pi \leftrightarrow \rho$ as above then it should be a bijective correspondence and unique (but it does depend on the choice of ι). It is expected that the set of cuspidal $\Pi$ maps onto the set of irreducible ρ. A stronger property, often referred to as the local-global compatibility, is believed to be true at finite places v: it says that $\mathrm{rec}_{F_{v}}(\Pi _{v}) =\iota \mathrm{ WD}(\rho \vert _{\mathop{\mathrm{Gal}}\nolimits (\overline{F}_{ v}/F_{v})})$. (This is stronger only at ramified places.) In particular it should be true that ρ and $\Pi$ have the same conductor (at finite places). Since we are concerned with unitary duals, we have adopted the unitary normalization for the Langlands correspondence and algebraicity. For arithmetic considerations it is customary to twist $\Pi$ by the $\frac{1-n} {2}$-th power of the modulus character in the conjecture. If so, one should replace “L-algebraic” by “C-algebraic,” cf. [BG11].

It is worth noting that Conjecture 4 suffices for our purpose in discussing geometric families. An important part of the Langlands program has been to confirm Conjecture 4 when ρ is the l-adic cohomology of a Shimura variety (in any degree), which in turn led to many instances of the map $\Pi \mapsto \rho$ in Conjecture 5. Another remarkable result toward the conjectures is the modularity of elliptic curves over $\mathbb{Q}$ due to Wiles and Breuil-Conrad-Diamond-Taylor, who identified L(s, ρ) with the L-function of a weight 2 modular form when ρ is the étale H ¹ of an elliptic curve over $\mathbb{Q}$. Recent developments include modularity lifting and potential modularity theorems. As we have no capacity to make a long list of all known cases of either Conjecture 4 or 5, we mention survey articles [Tay04] and [Harr10] for the reader to begin reading about progress until 2009.

We close the discussion with a comment on the unitarity of local components and the issue of correct twist, cf. Remark (iv) below the definition of geometric families in Sect. 1. Consider the automorphic representation $\Pi$ corresponding via the above conjectures to $\rho = H_{\mathrm{et}}^{d}(X \times _{\mathbb{Q}}\overline{\mathbb{Q}},\overline{\mathbb{Q}}_{l})$ for a smooth proper variety X over $\mathbb{Q}$ (which is not necessarily geometrically connected). Set $\Pi ':= \Pi \otimes \vert \det \vert ^{d/2}$. If X has good reduction modulo a prime p, then the geometric Frobenius acts on the H ^d-cohomology with absolute values p ^d∕2 under any choice of ι. (This is Deligne’s theorem on the Weil Conjectures if p ≠ l. The argument extends to p = l by work of Katz-Messing.) Hence the twist the Satake parameters of $\Pi '_{p}$ have absolute value 1, so $\Pi '_{p}$ is unitary. In general when X has bad reduction modulo p, the unitarity of $\Pi '_{p}$ can be deduced from the weight-monodromy conjecture in mixed characteristic (as stated in [Saito]). Despite recent progress, cf. [Sch12], the latter conjecture is still open. What we said of ρ should remain true when ρ is a subquotient of $H_{\mathrm{et}}^{d}(X \times _{\mathbb{Q}}\overline{\mathbb{Q}},\overline{\mathbb{Q}}_{l})$.

Appendix 2. Non-Criticality of the Central Value for Orthogonal Representations

Deligne ([Del79]) made a conjecture on special values of motivic L-functions. For a given L-function there is a set of the so-called critical values of s to which his conjecture applies. For our purpose we take on faith a motivic version of Conjecture 5 (cf. [Lan12, § 6] and Remark 3.5 above) on the existence of a bijection between absolutely irreducible pure motives M of rank n over $\mathbb{Q}$ and cuspidal C-algebraic automorphic representations π of $\mathop{\mathrm{GL}}\nolimits _{n}(\mathbb{A})$ such that

$$\displaystyle{L(s + \frac{n - 1} {2},M) = L(s,\pi ).}$$

Thereby Deligne’s conjecture translates to a conjecture on automorphic L-functions. We copy the definition of s being critical from the motivic side to the automorphic side in the obvious way. We are particularly interested in the question of whether the central value s = 1∕2 is critical for a cuspidal automorphic L-function which is unitarily normalized (for this a twist by a suitable power of the modulus character may be needed). The goal of appendix is to show

Proposition 6.

Suppose that a cuspidal automorphic representation π of $\mathop{\mathrm{GL}}\nolimits _{n}(\mathbb{A})$  is

• (1) orthogonal (i.e., π is self-dual and $L(s,\pi,\mathop{\mathrm{Sym}}\nolimits ^{2})$ has a pole) and

• (2) regular and C-algebraic.

Then s = 1∕2 is not critical for L(s,π).

The statement, in particular the definition of criticality, is unconditional in that no unproven assertions need to be assumed. However the proof is conditional on Conjecture 5 as well as various conjectures around motives that are supposed to be true (see Sect. 1 of [Del79] for the latter). We freely assume them below.

Proof.

There should be a pure irreducible rank n motive M over $\mathbb{Q}$ corresponding to π. We follow the conventional normalization so that the weight of M is w = n − 1. (Note that the second assumption on π implies that M has Hodge numbers 0 or 1. In the Hodge realization the dimension of M ^p, q is at most one, and zero if p + q ≠ n − 1.) Since π is self-dual, M is self-dual up to twist. More precisely there is a perfect pairing

$$\displaystyle{M \otimes M \rightarrow \mathbb{Q}(1 - n)}$$

where $\mathbb{Q}(1 - n)$ is the (1 − n)-th power of the Tate motive.

The center of symmetry for L(s, M), the L-function associated with M, is at s = (1 + w)∕2 = n∕2. The necessary condition (which may not be sufficient) for it to be critical is that $n/2 \in \mathbb{Z}$, namely that n is even (so w is odd). Hence we may and will assume that n is even. Now consider the l-adic realization

$$\displaystyle{M_{l} \otimes M_{l} \rightarrow \mathbb{Q}_{l}(1 - n),}$$

where M _l is now an irreducible l-adic representation of $\mathop{\mathrm{Gal}}\nolimits (\overline{\mathbb{Q}}/\mathbb{Q})$. By a result of Bellaiche-Chenevier’s ([BC11]) the sign of M _l is equal to (−1)ⁿ⁻¹ = −1, meaning that the above pairing on M _l is symplectic. (To apply their result we need both assumptions (1) and (2) on π.) Translating back to the automorphic side we deduce that π is also symplectic. We have shown that if s = 1∕2 is critical then π is symplectic, completing the proof. □ 

Example.

When n = 1 and π corresponds to a Dirichlet character χ, it is well known that the central value s = 1∕2 for L(s, χ) is not critical. In this case π is clearly orthogonal and the proposition applies.

Example.

Consider the case of n = 2 where π corresponds to weight k cuspforms ($k\geqslant 1$). Since we are concerned with self-dual representations, we normalize the correspondence such that π is self-dual. Then π is regular algebraic if and only if k is even. (To deal with odd weight forms, one could twist π by a half-power of the modulus character, but then π would be self-dual only up to a twist.) In case k is even, we associate with π a pure motive M of rank 2 and weight 1 such that dimM ^{1−k∕2, k∕2} = dimM ^{1−k∕2, k∕2} = 1. It is equipped with a symplectic pairing $M \times M \rightarrow \mathbb{Q}(-1)$.