Landau-Siegel Zeros and Cusp Forms

Dinakar Ramakrishnan

IAS Lecture

November 4, 1999

∮ 0. L-functions. For every m ≥ 1, let D_{m} denote the class of Dirichlet series L(s) = ∑_{n≥1} \frac{a_{n}}{n^{s}}, absolutely convergent in Re(s) > 1 with an Euler product Π_{p} P_{p}(p^{−s})^{−1} of degree m there, extending to whole s − plane as a meromorphic function of bounded order, in fact with no poles anywhere except at s = 1, and satisfying (relative to another Dirichlet series L^{∨}(s) in D_{m}) a functional equation of the form

L_{∞}(s) L(s) = W N^{\frac{1}{2} −s}L_{∞}^{∨}(1 − s)L^{∨}(1 − s),

where W ϵ ℂ^{∗}, Nϵℤ_{>0}, with the “archimedean factor” L_{∞}(s) being (π^{\frac{−sm}{2}} ) Π_{j=1}^{m} Γ(\frac{s+bj}{2} ), for sure (bj) ϵ ℂ^{m}.

Put D = ⋃_{m≥1} D_{m}. One says that L(s) is self-dual if L(s) = L^{∨}(s), in which case W ϵ {±1}. Examples:

(0) ζ(s), Dirichlet and Hecke L-functions.

(1) g a holomorphic newform of weight k ≥ 1 and level N,

L(s) = L(s + \frac{k − 1}{2}, g), L_{∞}(s) = π^{−s}Γ(\frac{s + (k − 1)/2}{2})Γ(\frac{s + (k + 1)/2}{2}).

(2) ϕ = Maass form of level N, an eigenfunction of Hecke operators.

L(s) = L(s, ϕ), L_{∞}(s) = π^{−s}Γ(\frac{s + δ + w}{2} )Γ(\frac{s + δ − w}{2} ), δϵ\{0, 1\}.

(3) L(s) = L(s, π_{f}), π = π_{∞} ⊗ π_{f} unitary cusp form on GL_{m}/F, [F : Q] < ∞, N = D_{F}^{m}Norm_{F/Q}(cond(π)); generalizes (0) − (2).

(3) (Rankin - Selberg) L(s) = L(s, π_{1,f} × π_{2,f}), π_{j} cusp form on GL(m_{j}). If

L(s, π_{1,p}) = Π_{i=1}^{m_{1}}(1 − α_{i}p^{−s})^{−1} & L(s, π_{2,p}) = Π_{j=1}^{m_{2}}(1 − β_{j}p^{−s})^{−1},

then

L(s, π_{1,p} × π_{2,p}) = Π_{i=1}^{m_{1}} Π_{j=1}^{m_{2}}(1 − α_{i}β_{j}p^{−s})^{−1}

(4) A typical member in the Selberg class. (Selberg requires Ramanujan, which we don’t, while he requires Euler product only in Re(s) >> 1)

(5) (Conjectural) L(s) = L(s, ρ) (Artin L-function)

where ρ : Gal(ℚ/ℚ) → GL_{m}(ℂ) is a continous homomorphism.

L(s, ρ) = Π_{p} det(1 − F r_{p}t|V_{ρ}^{I_{p}})|_{t=p^{−s}}^{−1}, F r_{p} : F robenius, I_{p} : Inertia

Everything known except for holomorphy at s ≠ 1, which is a big open problem.

(6) (Highly conjectural) M is a motive ℚ with coefficients in E ↪ ℂ of weight w, L(s) = L(s + \frac{w}{2} , M). (Known cases: CM ab. vars.: Shimura-Taniyama, Elliptic curve ℚ: Wiles, Breuil-Conrad-Diamond-Taylor).

Hope/Conjecture I: Every L(s) ϵ D_{m} is quasi-automorphic, ie, there exists an automorphic form π on GL_{m}/ℚ s.t. L_{p}(s)_{=}L(s, π_{p}) for almost all p. Moreover, L(s) is primitive iff π is cuspidal.

– Compatible with Langlands philosophy ([La]) and with the conjecture of Cogdell-Piatetski-Shapiro. ([CoPS])

– This cannot be correctly formulated over number fields. Also, there exists an example of Patterson over function fields 𝔽_{q} satisfying analogous conditions, but with zeros on the lines Re(s) = \frac{1}{4} and Re(s) = \frac{3}{4}.

Hope/Conjecture II: For any L(s) ϵ D, if it has a pole of order r at s = 1, then ζ(s)^{r}|L(s), ie., L(s) = ζ(s)^{r}L_{1}(s), with L_{1}(s) ϵ D.

This is compatible with the conjectures of Selberg, Tate and Langlands.

∮ 1. Landau-Siegel Zeros Let L(s) ϵ D_{m} with L_{∞}(s) = π^{−ms} Π_{j=1}^{m} Γ(\frac{s+bj}{2} ). Define its thickened conductor to be

Ñ = N(1 + ∑_{j=1}^{m} |bj|).

Definition. Let c > 0. Then we say that L(s) has a Landau-Siegel zero relative to c if L(β) = 0 for some β ϵ (1 − \frac{c}{log Ñ} , 1).

Definition. Let F be a family, ie., a subclass of L-functions in D with Ñ → ∞ in F. We say that F admits no Landau-Siegel Zero if there exists an effective constant c > 0 such that no L(s) in F has a zero in (1 − \frac{c}{log Ñ} , 1).

Conjecture: Let F be a family in D. Then F admits no Landau-Siegel zero.

One reason for interest in this is that the lack of such a zero implies a good lower bound for L(s) at s = 1.

For example, consider the family F = {L(s, χ_{D})|D negative, fund. discriminant}.

Dirichlet : L(1, χ_{D}) = \frac{2πh_{D}}{w_{D}\sqrt{|d|}}, where w_{D} = # roots of 1 in Q(\sqrt{D})

and h_{D} = class number of Q(\sqrt{D}). If F has no L−S zero then one can show: (∀ε > 0) L(1, χ_{D}| ≥ C|D|^{ε}, for an effective C > 0. Consequently, h_{D} ≥ C_{1}|D|^{\frac{1}{2} −ε}. (∗)

This will solve Gauss’s class # problem for imaginary quadratic fields.

Siegel proved (∗) with an ineffective constant C_{2}. He was influenced by Landau’s earlier work on the problem giving |D| < C′h_{D}^{8} log^{3}(3h_{D}) ⇒ h_{D} > C_{ε}|D|^{\frac{1}{8} −ε}, for any ε > 0. The natural limit of Landau’s method seems to be |D|^{\frac{1}{4}} (mod logarithmic terms). Gross-Zagier-Goldfeld: h_{D} ≥ C log |D| for an effective C > 0.

Suppose we consider the family {L(s, χ_{D})} attached to the set of positive fundamental discriminants D. Then the expression for L(1, χ_{D}) involves h_{D}R_{D}, where R_{D} = log |u_{D}| is the regulator, with u_{D} denoting a fundamental unit of Q(\sqrt{D}). So we get a lower bound for h_{D}R_{D} and not for h_{D} itself. This is related to Gauss’s conjecture that F infinitely many D > 0 such that h_{D} = 1.

Very generally, let L(s) (= L(s + \frac{w}{2} , M), for a motive M/Q of even weight w. Call s = 1 (Deligne) critical for L(s) if neither L_{∞}(s) nor L_{∞}^{∨}(1 − s) has a pole at s = 1.

• L(s, χ_{D}) is critical at s = 1 iff D < 0.

• If g is a holomorphic new form of wt. 2k + 1, then s = 1 is critical for L(s) = L(s + \frac{k−1}{2}, g)

• If g is a holomorphic new form of wt. 2, then s = 1 is critical for sym^{2r}(g), for any r ≥ 1.

• if ϕ is a Maass form of weight o associated to an even Galois representation ρ; λ will be \frac{1}{4}. When M is critical with coefficients in Q, Deligne’s conjecture predicts

L(1, M) ϵ (period of M) ℚ^{∗}.

The transendental part of the period of M should be constant in the family {M ⊗ χ_{D}|D > 0}. Bloch-Kato conjectures: describe the rational #’s; they invlove a generalized class #’s, order

of Sha(M).

Upshot: If {M ⊗ χ_{D}} has no Landau-Siegel zeros, and if the B-K conjectures hold , then one should have a good lower bound for the order of Sha(M_{D}), as D → ∞.

Another reason for being interested in L(1) is the following:

Let π be a cusp form on GL_{n}/Q. Then L(s, π × π^{∨}) has a simple pole at s = 1 and its residue there is essentially given by (φ, φ) for a new vector φϵV_{π}. On the other hand, L(s, π × π^{∨}) = ζ(s)L(s, π, Ad); so one gets control of ∥φ∥ by bounding L(1, π, Ad) from below. For n = 2 and π self-dual, L (1, π, Ad) is simply L (1, sym^{2}(π)). ∮ 3. Known Cases

(0) (Siegel, –) Fix R > 0, and consider F_{R} = {L(S, χ_{D})| D < 0, |D| ≤ R}. Then there exists an effective C > 0 such that there is at most one χ_{D} ϵ F_{R} which has a zero in (1 − \frac{c}{log_{R}} , 1).

(1) (Stark) Let F be a finite Galois extension of ℚ. Then the Dedekind zeta function ζ_{F} (s) has no Landau-Siegel zero unless F contains a quadratic extension of ℚ. This holds for non-normal extensions as well if one assumes the Artin conjecture.

(2) (Goldfeld-Hoffstein-Lockhart-Lieman) Let F = {L(s, sym^{2}(π_{f})}|π non-dihedral cusp form on GL_{2}ℚ}. Then F admits no Landau-Siegel zero.

(3) (Hoffstein-Ramakrishnan)

(i) Let F = {L(s, π_{f})|π cusp form on GL_{2}/F }. Then F admits no L − S zero.

(ii) Analog of (0) holds for {L(s, π)|π cusp form on GL_{m}, all m.}

(iii) Assume the Rankin-Selberg L-functions are modular. Then the analog of (i) holds for GL_{m}, for any m ≥ 1. Remark: For GL(3), the paper [HR] gave a reduction, which was established by W. Banks to give a non-existence result for L − S zeros (for GL(3)) ([Ba]). ∮ 4. Idea of proof of the results of [HR]:

The starting point is the following general principle, well known to experts in the area:

LEMMA Let L(s) ϵ D_{m} is a positive Dirichlet series having a pole of order r ≥ 1 at s = 1, with L′(s)/L(s) < 0 for sϵ(1, ∞). Then there exists an effective constant C > 0, depending only on m and r, such that L(s) has most r real zeros in (1 − \frac{C}{log Ñ} , 1).

So, given L_{1}(s) ϵ D, one can rule out L-S zeros for L_{1}(s) if we can find a positive L(s) with order of pole r ≥ 1 such that L(s) = L_{1}(s)^{k}L_{2}(s), with k > r and L_{2}(s) holo in (t, 1) for a fixed t.

LEMMA L(s, π × π^{∨}) is a positive Dirichlet series, ∀ auto form π, with L′(s)/L(s) < 0 for real s ϵ (0, ∞).

[GHLL] case: π self-dual, non-dihedral cusp form on GL(2), S^{2} = sym^{2};

L(s) = ζ(s)L(s, S^{2}(π))^{3}L(s, S^{2}(S^{2}(π))), r = 2.

[HR] case: π self-dual, non-dihedral cusp form on GL(2);

L(s) = ζ(s)^{2}L(s, π)^{4}L(s, S^{2}/π))^{3}L(s, S^{3}(π))L(s, S^{2}(π) × S^{2}(π)), r = 3.

Crucial fact: L(s, sym^{3}(π)) is holomorphic in (\frac{3}{4}, 1) which had been known at the time by Bump-Ginzburg-Hoffstein ([BGH]). Now one can also appeal to Kim-Shahidi who prove that L(s, sym^{3}(π)) is even entire ([KSh]).

General principle: (in terms of Galois representations of Artin type)

Let ρ ϵ Hom_{cont}(Gal, GL_{n}(ℂ) be irreducible. We will assume ρ is self-dual; the general case can be reduced to this.

Elementary, but key point: If n > 1, ∃ irreducible τ ≠ 1 of Gal occuring in ρ ⊗ ρ.

Case i Can take τ = ρ. (Notation: \underline{1} = trivial representation)

Write

ρ^{⊗2} = \underline{1} ⊕ kρ ⊕ ρ′, ρ ⊄ ρ′, k ≥ 1.

Then

η : = (1 + ρ)^{⊗2} = 2.\underline{1} ⊕ (2 + k)ρ ⊕ ρ′, k ≥ 1.

⇒ L(s, η) = ζ(s)^{2}L(s, ρ)^{2+k} L(s, ρ′).

Done if L(s, ρ′) is holomorphic to the left of s = 1. (This corresponds to the [GHLL] -case).

Case ii τ ≠ ρ.

Write

ρ^{⊗2} = \{ \begin{matrix}1 ⊕ k τ ⊕ k τ^{∨} ⊕ τ′, if τ ≠ τ^{∨} \\ 1 ⊕ l τ ⊕ τ′, if τ ≠ τ^{∨}, \end{matrix}

with k, l ≥ 1, τ, τ^{∨} ⊄ τ′.

Then

η : = (1 ⊕ ρ ⊕ τ)^{⊗2} = 3.\underline{1} ⊕ 2ρ ⊕ τ′′ ⊕ (ρ ⊗ τ^{∨}) ⊕ (ρ ⊗ τ),

for some τ′′. Note that ρ occurs in ρ ⊗ τ^{∨}.

Hence L(s, η) = ζ(s)^{3}L(s, ρ)^{4}L(s, ρ′).

Done if L(s, ρ′) is holo. near 1.

(This corresponds to Hoffstein-Ramakrishnan case.)

Suppose we assume the functoriality principle of Langlands ([La]). More specifically, suppose we assume that the Rankin-Selberg L-functions L(s, π × π′) on GL(n) × GL(m) are modular, ie., given as the standard L-series L(s, π ⊠ π′) for automorphic representations π ⊠ π′ of GL(nm). For n = m = 2, this is carried out in [Ra]. Then we can transport this Galois argument over to the automorphic side and conclude that for any cusp form π on GL(n), n > 1, L(s, π) admits no Landau-Siegel zero.

But such a result can be proved without any hypothesis for L(s, π) ([HR]) and L(s, S^{2}(π)) ([GHLL]) for any cusp form π on GL(2). We refer to [HR] for further details.

Selected References

[Ba] W. Banks, Twisted symmetric square L-functions and the non-existence of Siegel zeros on GL(3), Duke Math Journal, 87 (1997), no. 2, 343–353.

[BGH] D. Bump, D. Ginzburg and J. Hoffstein, The symmetric cube, Invent. Math. 125 (1996), no. 3, 413–449.

[CoPS] J. Cogdell and I. Piatetski-Shapiro, Converse Theorems for GL_{n}, Publications Math. IMES 79 (1994), 157–214.

[GHLL] D. Goldfeld, J. Hoffstein and D. Liemann, An effective zero free region, appendix to: J. Hoffstein, P. Lockhart, Coefficients of Maass forms and the Siegel zero, Ann. Math. (2) 140 (1994), 161–181.

[HR] J. Hoffstein and D. Ramakrishnan, Siegel zeros and cusp forms, IMRN (1995), no. 6, 279–308.

[KSh] H. Kim and F. Shahidi, Symmetric cube L-functions of GL_{2} are entire, Annals of Math. (to appear)

[La] R. P. Langlands, Automorphic Representations, Shimura varieties and motives. Ein Marchen, Proc. Symp, Pure Math. 33 (1979), part 2, 205–246.

[L] E. Landau, Bemer Rungen Zum Heilbronnschen Saltz, Acta Arithmetica 1 (1936), 1–18.

[Ra] D. Ramakrishnan, Modularity of the Rankin-Selberg L-series, and Multiplicity one for SL(2), to appear in Annals of Math.

[S] C. L. Siegel, Über die Classenzahl quadratischer Zahlkörper, Acta Arithmetica 1, (1936), 83–86.

[St] H. M. Stark, Some effective cases of the Brauer-Siegel theorem, Invent. math. 23 (1974), 135–152.