Nonvanishing of motivic L-functions

Andrzej Da̧browski and Jacek Pomykała

Introduction

Let E be a modular elliptic curve over ℚ and let L(E, s) denote the associated L-function. The Birch and Swinnerton-Dyer conjecture then predicts that L(E, s) has a zero at s = 1 of order precisely equal to the rank of the Mordell-Weil group E(ℚ). According to Waldspurger’s theorem [Wa] we know that there exists a real quadratic character χ such that the twisted L-function L(E, χ, s) does not vanish at s = 1. Recently V.A. Kolyvagin [Ko] has proved that E(ℚ) is finite if L(E, 1) ≠ 0 and there exists a suitable real quadratic character χ_{t} such that L(E, χ_{t}, s) has a simple zero at s = 1. The latter condition was proved to be true for infinitely many t [BFH][MuMu]. H. Iwaniec [Iw] and A. Perelli and J. Pomykała [PP] have proved a quantitative results on this condition; J. Pomykała [P] has generalized it to n-th derivative of L(E, χ, s).

Variants and generalizations are possible. For instance, S. Friedberg and H. Hoffstein [FH] established a non-vanishing theorem for quadratic twists of the Lseries of an arbitrary cuspidal automorphic form on GL(2) over any number field. In some cases such a result can be applied to the construction of l-adic representations associated to modular forms over imaginary quadratic fields [Ta]. Further examples as well as some perspective concerning the higher rank case is discussed in an excellent survey article [BFH2].

We should also mention that there are examples of irreducible cuspidal automorphic representations of GL(2) over a number field such that the corresponding twisted L-function vanishes at the center of the critical strip for all quadratic characters [Ro].

Here is our contribution to the above picture.

Let M be a pure motive over ℚ, and let L(M, s) denote the corresponding Lfunction. In this paper we prove, under certain assumptions, (a quantitative version of) a non-vanishing theorem for n-th derivatives of quadratic twists of L(M, s) at the center of the critical strip (Theorem 3). Also, assuming the Birch and Swinnerton-Dyer conjecture for abelian varieties over ℚ, we obtain a bound for the order of the (twisted) Shafarevich-Tate group (Theorem 4).

All the results of this paper except section 5 could be stated (and proved) without using any motivic language. Almost all we need to know about motives concerns the conjectural description of their L-functions (see section 1).

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Acknowlegements. Part of the work done by the first named author was on his visit to the Max-Planck-Institut für Mathematik in Bonn (Germany) in November 1996, and he thanks the Institute for the hospitality and support. He also thanks John Coates and Don Zagier for their comments on this article.

0. Statement of the results

Let M be a pure motive over ℚ of weight w = w(M) and of the global conductor c(M). For any Dirichlet character χ one can define the twisted motive M_{χ}; let L(M_{χ}, s) denote the corresponding L-function. Also, let L^{(n)}(M_{χ}, s) denote the n-th derivative of motivic L-function. Assume that L^{(n)}(M_{χ}, s) have good analytic properties (see section 1.1). Write

L(M, s) = ∑_{m=1}^{∞} a_{m}m^{−s}, Re(s) ≫ 0.

Let

D(M, s) = ∑_{m=kl^{2}} b_{m}m^{−s}, Re(s) ≫ 0,

where k has prime factors in 4c(M), (l, 4c(M)) = 1 and

b_{m} = a_{m} ∏_{p|m, (p,4c(M))=1} (1 + \frac{1}{p})^{−1} .

Such a series arises naturally when one try to follow Iwaniec’s work [Iw]. In a case of the motive attached to a newform f of weight k the series D(M(f), s) has meromorphic continuation to all the complex plane and does not vanish at s = \frac{k}{2} (see [Iw] for k = 2; in a case of arbitrary weight one can use the same trick). In general we do not expect for D(M, s) to have any ”nice” analytical properties. All we need (and assume) is that D(M, s) has analytic continuation to Re(s) > \frac{w}{2} − ϵ (ϵ > 0). Also, for our purposes, it is important to have an information about the behaviour at s = \frac{w+1}{2}. The next non-trivial example (communicated to us by D. Zagier) is M = Sym^{2}M(Δ), where Δ is the unique normalized cusp form of weight 12 and level 1: in this case D(M, s) can be analytically prolonged to (at least) Re(s) > 11 − ϵ and does not vanish at s = \frac{23}{2}. Note however that, in general, D(M, s) may have a pole at s = \frac{w+1}{2} (see Appendix).

Let C be the set of primitive quadratic characters χ of conductors c(χ) which are relatively prime to c(M). Let

C(T) = {χ ∈ C : c(χ) ≤ T }.

In a case χ = χ_{t}, χ_{t}(·) = (\frac{−t}{·}) (the Kronecker symbol) we sometimes write M_{t} = M_{χ_{t}}, etc.

Let ϵ(M) be the rooth number in the functional equation (1.1) for L(M, s). Our next restriction on M is that M = M^{∨}(−w). Then of course ϵ(M) = ±1, and more precisely

ϵ(M_{χ}) = (−1)^{ord_{s= \frac{w+1}{2}} L(M_{χ},s)}

for all quadratic characters χ of conductors relatively prime to c(M). It is certainly the case for modular forms of trivial characters, abelian varieties, etc.

Let T be defined as follows

T := {t square free : 0 < t ≡ −ν^{2}mod(4c(M)) for some ν prime to 4c(M)}.

The proof of the following crucial result depends on Iwaniec’s fundamental work [Iw].

Theorem 1. Assume that n ≠ 0 or ϵ(M) = −1. Then we have the following asymptotic equality (T → ∞)

∑_{χ_{t}∈C(T), t∈T} L^{(n)} (M_{t}, \frac{w + 1}{2}) F (\frac{t}{T} ) ∼ C(M, n) ∫ F (u)du T (log T)^{n},

where C(M, n) ≠ 0 and F is a smooth function compactly supported in ℝ^{+} with positive mean value.

We carry over the method of the paper [HB] to prove the following estimate, which also have important implications in our work on distribution of zeros [DP].

Theorem 2. Let ϵ > 0 and assume that M is of rank ≤ 2. Then, for T > T_{0}(M, ϵ) and |s − \frac{w+1}{2}| ≤ 2(log T)^{−1}, it holds

(0.1)  ∑_{χ∈C(T)} |L(M_{χ}, s)|^{2} ≪ T^{1+ϵ}.

We expect the above estimate to hold in general. The case of rank ≥ 2 presents the difficulties analogous to those for higher moments of classical Dirichlet L-series.

Let

N_{n}(T, M) = ♯\{χ ∈ C(T) : L^{(n)}(M_{χ}, \frac{w + 1}{2}) ≠ 0\}.

Assuming (0.1) for all (pure) motives we derive from Theorem 1 the following estimate.

Theorem 3. We have

N_{n}(T, M) ≫ T^{1−ϵ}.

It is perhaps time to recall the empirical distribution of the ranks of elliptic curves given by D. Zagier and G. Kramarz [ZK] (for the family x^{3} + y^{3} = m: about 23.3% have rank 2 or greater) and by A. Brumer and O. McGuinness [BM] (for elliptic curves of prime conductor: about 20% have even rank ≥ 2). It may suggest that the expectation that half of all elliptic curves have rank 0 and half rank 1, with higher ranks occuring asymptotically for only 0% of all curves (with respect to any natural ordering) is wrong. (We refer to section 6 for further comments.)

Our application concerns the bound for the order of the twisted Shafarevich-Tate groups. We generalize the main result of [MaMu] (where the case of an elliptic curve was treated) as follows.

Theorem 4. Let A be an abelian variety over ℚ. Assume that the Birch and Swinnerton-Dyer conjecture holds true for the quadratic twists of A. Assuming (5.1), there exist infinitely many t such that A_{t} has analytic rank 0 and

♯III(A_{t}) ≫ c(A_{t})^{\frac{1}{4} −ϵ}.

1. Preliminaries on motivic L-functions

1.1. Motivic L-functions. We will define a pure motive M over ℚ of weight w = w(M) and rank d = d(M), by specifying Betti, de Rham and l-adic realisations (for each prime l)

H_{B}(M), H_{DR}(M), H_{l}(M)

which are, respectively, vector spaces over ℚ, ℚ and ℚ_{l} of dimension d, and which are endowed with the additional structures and comparison isomorphisms [BK][CP-R][D]. In particular H_{B}(M) admits an involution ρ_{B}, H_{l}(M) is Gal(\overline{ℚ}/ℚ)-module, and there is a Hodge decomposition into ℂ-vector spaces

H_{B}(M) ⊗ ℂ = ⊕_{i+j=w}H^{i,j}(M),

where, letting ρ_{B} act on the vector space on the left via the first factor in the tensor product, we have

ρ_{B}(H^{i,j}(M)) = H^{j,i}.

The L-function of M is defined for Re(s) ≫ 0 as the following Euler product

L(M, s) = ∏_{p} L_{p}(M, p^{−s}),

extended over all primes p and where

\begin{aligned}L_{p}(M, X)^{−1} & = det (1 − X · r_{l}(F r^{−1})|_{H_{l}(M)^{Ip}} ) \\ & = (1 − α_{p}^{(1)}X)...(1 − α_{p}^{(d)}X); \end{aligned}

here r_{l} is the representation giving Gal(\overline{ℚ}/ℚ)-module structure on H_{l}(M), and F r_{p} ∈ Gal(\overline{ℚ}/ℚ) is the Frobenius element at p, defined modulo conjugation and modulo the inertia subgroup I_{p} ⊂ G_{p} ⊂ Gal(\overline{ℚ}/ℚ) of the decomposition group G_{p}. We make the standard hypothesis that the coefficients of L_{p}(M, X)^{−1} do not depend on l ≠ p.

We next state the conjectural functional equation for L(M, s). First let’s recall the definition of the Euler factor L_{∞}(M, s) at infinity. We suppose that if w(M) is even and H^{\frac{w}{2} , \frac{w}{2}} ≠ 0, the involution ρ_{B} acts on it by a scalar ±1. Put

Γ_{ℝ}(s) = π^{− \frac{s}{2}} Γ(\frac{s}{2}), Γ_{ℂ}(s) = 2(2π)^{−s}Γ(s), h(i, j) = dim_{ℂ}H^{i,j}(M),

and, if w(M) is even and H^{\frac{w}{2} , \frac{w}{2}} (M) ≠ 0, we put

r^{α}(M) = dim_{ℂ}\{x ∈ H^{\frac{w}{2} , \frac{w}{2}} (M) : ρ_{B}(x) = α(−1)^{\frac{w}{2}} x\}, α = ±1.

We define

L_{∞}(M, s) = L_{∞,ℝ}(M, s)L_{∞,ℂ}(M, s),

where

L_{∞,ℝ}(M, s) = Γ_{ℝ}(s − \frac{w}{2} )^{r^{+}(M)}Γ_{ℝ}(s + 1 − \frac{w}{2} )^{r^{−}(M)},

and

L_{∞,ℂ}(M, s) = ∏_{i<j} Γ_{ℂ}(s − i)^{h(i,j)}.

We put

Λ(M, s) = L_{∞}(M, s)L(M, s).

We also assume the existence of a finite set of bad primes S such that, for each prime l and each r ∉ S ∪ {l}, the inertia group I_{r} acts trivially on H_{l}(M). For each r ∈ S, and each l ≠ r, we additionally assume that the conductor of H_{l}(M) as a representation of the local Galois group D_{r} does not depend on l. The global conductor c(M) of M is then defined as the product of local conductors at primes r ∈ S.

For each p ∉ S, we assume that the reciprocal complex roots of L_{p}(M, X)^{−1} have absolute value equal to p^{\frac{w}{2}} ; according to the monodromy conjecture, a more refined version of the purity should hold also at bad primes. It implies, that L(M, s) converges in the complex half-plane Re(s) > \frac{w}{2} + 1.

Conjecture [D][CP-R]. The function Λ(M, s) has a meromorphic continuation over the whole complex plane, and satisfies the functional equation

(1.1)  Λ(M, s) = ϵ(M)c(M)^{\frac{w+1}{2} −s}Λ(M^{∨}(−w), w + 1 − s),

where ϵ(M) is a complex number which is independent of s. Moreover, L(M, s) is holomorphic unless both w(M) is even and ℚ(−\frac{w}{2} ) is a direct summand of M, in which case L(M, s) may possibly have a pole at the point s = \frac{w}{2} + 1.

Remark. Let us recall that in a case M = M^{∨}(−w) the root number ϵ(M) can be given by the analytic formula

(1.2)  ϵ(M) = (−1)^{ord_{s=0}L(M,s)}.

Let χ be a real quadratic character of conductor c(χ). To such χ one can associate a motive [χ] over ℚ of weight 0 and rank 1 so that L([χ], s) = L(s, χ). Also given any (pure) motive M, we can define M_{χ} = M ⊗ [χ]; it is again a motive over ℚ of the same weight and rank so that we have

L(M_{χ}, s) = ∏_{p} L_{p}(M, χ(p)p^{−s}).

Lemma 1 [CP-R]. Assume that (c(χ), c(M)) = 1. Then

c(M_{χ}) = c(M)c(χ)^{d(M)}.

1.2. The conjecture of Birch and Swinnerton-Dyer. Let A be an abelian variety over ℚ, and let S be a set of rational primes where A has bad reduction. The L-function L(A, s) = L_{S}(A, s) is defined, for Re(s) > \frac{3}{2}, by

L(A, s) = ∏_{p∉S} L_{p}(H^{1}(A), p^{−s}).

Let III(A) be the Shafarevich-Tate group of A, and let A(ℚ), A′(ℚ) = Pic^{o}(A) (resp. A(ℚ_{p})) are the groups of global (resp. local) points. Let R denote the determinant of the height pairing

A(ℚ) × A′(ℚ) → ℝ.

Also put π_{∞}(A) = vol(A(ℝ)), π_{bad}(A) = vol(∏_{p∈S} A(ℚ_{p})).

Conjecture (Birch and Swinnerton-Dyer, Tate [T]). L(A, s) has a zero of order r = rankA(ℚ) at s = 1 and

lim_{s→1} L(A, s)(s − 1)^{−r} = \frac{♯III(A) · R}{♯(A(ℚ)_{tors})♯Pic^{o}(A)_{tors})} π_{∞}(A)π_{bad}(A).

S. Bloch [Bl] proved that, in a case r = 0, the Birch and Swinnerton-Dyer conjecture is equivalent to the Tamagawa number conjecture. Later, S. Bloch and K. Kato [BK] defined the notion of the Shafarevich-Tate group, the groups of local and global points etc., and formulated the Tamagawa number conjecture in a general motivic situation.

2. Proof of Theorem 1

Put i_{0} = min{i : i ≤ j, h(i, j) ≠ 0}. For n ∈ ℤ we introduce the generalized incomplete gamma function

V_{n}(M, X) = \frac{1}{2πi} ∫_{(3/4)} L_{∞}(M, s + i_{0} − \frac{w − 1}{2}) s^{−n}X^{−s}ds,

where we integrate over the line Re(s) = \frac{3}{4}.

Put d_{0} = \frac{1}{2}h(\frac{w}{2} , \frac{w}{2} ), d_{1} = ∑_{i<j} h(i, j). Consider the approximate functions

A_{n}(M_{χ}, X) = ∑_{m=1}^{∞} a_{m}χ(m)m^{− \frac{w+1}{2}} V_{n} (M, \frac{π^{d_{0}}(2π)^{d_{1}}m}{X}) .

By the functional equation (1.1) we obtain

\frac{1}{2πi} ∫_{(−3/4)} L(M_{χ}, s + \frac{w + 1}{2})\frac{L_{∞}(M, s + i_{0} − \frac{w−1}{2})}{s^{n}} (\frac{X}{π^{d_{0}}(2π)^{d_{1}}} )^{s} ds

= \frac{ϵ(M_{χ})}{2πi} ∫_{(−3/4)} L(M_{χ}, \frac{w + 1}{2} − s)\frac{L_{∞}(M, \frac{w+1}{2} − s)}{L_{∞}(M, s + \frac{w+1}{2})}

× \frac{L_{∞}(M, s + i_{0} − \frac{w−1}{2})}{s^{n}} (\frac{π^{d_{0}}(2π)^{d_{1}}X}{c(χ)^{d}c(M)} )^{s} ds

= (−1)^{n+1} \frac{ϵ(M_{χ})}{2πi} ∫_{(3/4)} L(M_{χ}, s + \frac{w + 1}{2})\frac{L_{∞}(M, s + i_{0} − \frac{w−1}{2})}{s^{n}}

(\frac{π^{d_{0}}(2π)^{d_{1}}}{c(χ)^{d}c(M)/X} )^{−s} ds

= (−1)^{n+1}ϵ(M_{χ})A_{n}(M_{χ}, c(χ)^{d}c(M)/X). Put

G_{n}(M_{χ}, s, X) = L(M_{χ}, s + \frac{w + 1}{2})\frac{L_{∞}(M, s + i_{0} − \frac{w−1}{2})}{s^{n}} (\frac{X}{π^{d_{0}}(2π)^{d_{1}}} )^{s} .

We obtain by the Cauchy theorem

(2.0) \begin{aligned}res_{s=0}G_{n}(M_{χ}, s, \sqrt{c(χ)^{d}c(M)}) & = A_{n}(M_{χ}, \sqrt{c(χ)^{d}c(M)}) \\ & + (−1)^{n}ϵ(M_{χ})A_{n}(M_{χ}, \sqrt{c(χ)^{d}c(M)}). \end{aligned}

Consider the Laurent expansions

L_{∞}(M, s + i_{0} − \frac{w − 1}{2}) = ∑_{m=−r}^{∞} α_{m}(M)s^{m}, α_{−r}(M) ≠ 0,

where r = r(M) = h(\frac{w}{2} , \frac{w}{2} ) + ∑_{i<j} h(i, j), and

(\frac{X}{π^{d_{0}}(2π)^{d_{1}}} )^{s} = ∑_{k=0}^{∞} b_{k}(M, X)s^{k}.

It is plain to see that

\begin{aligned}res_{s=0}G_{n}(M_{χ}, s, X) & = ∑_{k=1}^{r} d_{−k}(M, X)\frac{L^{(n+k−1)}(M_{χ}, \frac{w+1}{2})}{(n + k − 1)!} \\ & + ∑_{l=0}^{n−1} c_{l}(M, X)\frac{L^{(n−1−l)}(M_{χ}, \frac{w+1}{2})}{(n − 1 − l)!}, \end{aligned} (2.1)

where

d_{−r+m}(M, X) = ∑_{t=0}^{m} α_{−r+t}(M)b_{m−t}(M, X),

and

c_{l}(M, X) = ∑_{t=0}^{l} α_{l−t}(M)b_{t}(M, X) + ∑_{m=0}^{r−1} α_{−r+m}(M)b_{r−m+l}(M, X).

In particular, since

b_{t}(M, X) = \frac{1}{t!} (log \frac{X}{π^{d_{0}}(2π)^{d_{1}}} )^{t} ,

we obtain (2.2)

res_{s=0}G_{n}(M_{χ}, s, X) ∼ α_{−r}(M) ∑_{l=0}^{n+r−1} b_{l}(M, X)\frac{L^{(n+r−1−l)}(M_{χ}, \frac{w+1}{2})}{(n + r − 1 − l)!}, X → ∞.

Now the problem reduces to the question about the asymptotics of the sum

2 ∑_{t∈T} ′A_{n} (M_{t}, \sqrt{c(χ_{t})^{d}c(M)}) F (\frac{t}{T} ) .

Following Iwaniec’s method [Iw] we obtain that the main contribution is equal to

\begin{aligned}S_{0} & = \frac{γ(M)T}{2c(M)} ∑_{m=kl^{2}} \frac{a_{m}ϕ(l)}{l}m^{− \frac{w+1}{2}} \\ & × ∑_{a≤T^{3/14},(a,4lc(M))=1} μ(a)a^{−2} ∫ F (u)V_{n} (M, \frac{π^{d_{0}}(2π)^{d_{1}}m}{uT \sqrt{c(M)}}) du, \end{aligned}

where

γ(M) := ♯{d(mod 4c(M)) : d ≡ −ν^{2}(mod 4c(M)), (ν, 4c(M)) = 1},

k has prime factors in 4c(M) and (l, 4c(M)) = 1.

The sum over a > T^{3/14} contributes to T^{11/14+ϵ}.

Now

\begin{matrix}\frac{a_{m}}{m^{\frac{w+1}{2}}} \frac{ϕ(l)}{l} ∑_{(a,4c(M)l)=1} \frac{μ(a)}{a^{2}} = \\ = \frac{a_{m}}{m^{\frac{w+1}{2}}} ∏_{p} (1 − \frac{1}{p^{2}} ) ∏_{p|4c(M)l} (1 − \frac{1}{p^{2}} )^{−1} ∏_{p|l} (1 − \frac{1}{p}) . \end{matrix}

Since (l, 4c(M)) = 1 we conclude the above is equal to

\begin{matrix}\frac{6}{π^{2}} \frac{a_{m}}{m^{\frac{w+1}{2}}} ∏_{p|4c(M)} (1 − \frac{1}{p^{2}} )^{−1} ∏_{p|l, (p,4c(M))=1} (1 − \frac{1}{p^{2}} )^{−1} ∏_{p|l, (p,4c(M))=1} (1 − \frac{1}{p}) = \\ = \frac{6}{π^{2}} \frac{a_{m}}{m^{\frac{w+1}{2}}} ∏_{p|4c(M)} (1 − \frac{1}{p^{2}} )^{−1} ∏_{p|l, (p,4c(M))=1} (1 + \frac{1}{p})^{−1} . \end{matrix}

Hence

\begin{aligned}S_{0} & = \frac{3γ(M)T}{π^{2}c(M)} ∏_{p|4c(M)} (1 − \frac{1}{p^{2}} )^{−1} ∑_{m=kl^{2}} \frac{b_{m}}{m^{\frac{w+1}{2}}} ∫ F (u)V_{n} (M, \frac{π^{d_{0}}(2π)^{d_{1}}m}{uT \sqrt{c(M)}}) du \\ & = C_{M}T ∫ F (u)B_{n}(M, uT \sqrt{c(M)})du + O(T^{11/14+ϵ}); \end{aligned} (2.3)

here

B_{n}(M, X) = ∑_{m=kl^{2}} b_{m}m^{− \frac{w+1}{2}} V_{n} (M, \frac{π^{d_{0}}(2π)^{d_{1}}m}{X}) ,

with

b_{m} = a_{m} ∏_{p|m, (p,4c(M))=1} (1 + \frac{1}{p})^{−1} ,

where k has prime factors in 4c(M) and (l, 4c(M)) = 1.

Assume that D(M, s) has analytic continuation to Re > \frac{w}{2} − ϵ. First, applying the Mellin transformation we see that (2.4)

B_{n}(M, X) = \frac{1}{2πi} ∫_{(3/4)} D(M, s + \frac{w + 1}{2})\frac{L_{∞}(M, s + i_{0} − \frac{w−1}{2})}{s^{n}} (\frac{X}{π^{d_{0}}(2π)^{d_{1}}} )^{s} .

Consider

H_{n}(M, s, X) := res_{s=0}D(M, s + \frac{w + 1}{2})\frac{L_{∞}(M, s + i_{0} − \frac{w+1}{2})}{s^{n}} (\frac{X}{π^{d_{0}}(2π)^{d_{1}}} )^{s} .

Assume for the moment that D(M, s) has no pole and does not vanish at s = \frac{w+1}{2}. Then by the analogue of (2.2) with L(M_{χ}, s) replaced by D(M, s) we conclude

(2.5)  B_{n}(M, X) ∼ α_{−r}(M)b_{n−1+r}(M, X) D(M, \frac{w + 1}{2}), X → ∞,

and combinying this with (2.3) we obtain the asymptotic equality (T → ∞)

\begin{matrix}\frac{C_{M}}{(n − 1 + r)!}D(M, \frac{w + 1}{2}) ∫ F (u)du T (log T)^{n−1+r} \\ ∼ ∑_{t≤T} ∑_{l=0}^{n−1+r} b_{l}(M, t\sqrt{c(M)})\frac{L^{(n−1+r−l)}(M_{t}, \frac{w+1}{2})}{(n − 1 + r − l)!}F (\frac{t}{T} ) , \end{matrix} (2.6)

with C_{M} = \frac{3γ(M)}{π^{2}c(M)} ∏_{p|4c(M)}(1 − p^{−2})^{−1}. To complete the proof of Theorem 1 in this case one takes C(M, 0) = 0 if ϵ(M) = 1, C(M, 0) = C_{M} if ϵ(M) = −1 and C(M, n) = (−1)^{ϵ(M)}(−2)^{n−1}C_{M} if n ≠ 0, and uses the induction in n.

In a general case we still can write down an analogue of (2.5) and, consequently of (2.6) (with a different constant term on the left).

3. Proof of Theorem 2

We slightly modify Heath-Brown’s arguments ([HB], section 10). The main tool is the following large sieve type estimate ([HB], Corollary 3)

(3.1)  ∑_{χ∈C(D)} | ∑_{n≤N} a_{n}χ(n)|^{2} ≪ (DN)^{ϵ}N(N + D) max_{n} |a_{n}|^{2}.

Let A denote the number of Γ_{ℝ}’s plus twice number of Γ_{ℂ}’s in L_{∞}(M, s). Theorem 2 follows from the following result. (Here we do not assume that M = M^{∨}(−w).)

Theorem 2′. Let ϵ > 0 and assume that d = d(M) ≤ 2. Then we have for \frac{w+1}{2} ≤ σ ≤ \frac{w+2}{2} and D > D_{0}(M, ϵ)

∑_{χ∈C(D)} |L(M_{χ}, σ + it)|^{2} ≪ (D + (D(|t| + 1)^{A})^{w+2−2σ}) (D|t| + 1)^{ϵ}.

Proof. Let s = σ + it,

S^{+}(D, s) = ∑_{χ∈C(D)} |L(M_{χ}, s)|^{2},

S^{−}(D, s) = ∑_{χ∈C(D)} |L(M_{χ}^{∨}(−w), s)|^{2}

and ν^{±}(σ) be the infimum of those exponents ν^{±} respectively for which

S^{±}(D, σ + it) ≪ (D + (DT^{A})^{w+1−2σ}) (DT^{A})^{ν^{±}}

uniformly in D and t, where T = |t| + 1. We will show that ν^{±} ≤ 0. Similarly as in [HB] p.268 we have for any X > 0 and \frac{w+1}{2} ≤ α ≤ σ ≤ \frac{w+2}{2} (3.2)

S^{+}(D, s) ≪ ∑_{χ∈C(D)} | ∑_{n} a_{n}χ(n)n^{−s}e^{− \frac{n}{X}} |^{2} + X^{2(α−σ)} ∫_{−∞}^{∞} S^{+}(D, α + iu)e^{−|u−t|}du.

Using the functional equation (1.1) we obtain

S^{+}(D, α + iu) = ∑_{χ∈C(D)} |U(D, α + iu)|^{2}|L(M_{χ}^{∨}(−w), w + 1 − α − iu)|^{2}

where (3.3) U = U(D, α + iu) = ϵ(M_{χ})c(M_{χ})^{\frac{w+1}{2} −(α+iu)}L_{∞}(M_{χ}^{∨}(−w), w + 1 − s)L_{∞}(M_{χ}, s)^{−1}.

The last quotient contains the Γ-factors of type

Γ (\frac{1}{2}(w + 1 − s + a)) (Γ(\frac{1}{2}(s + a))^{−1}

or the similar ones with double arguments. In the second case we use the duplication formula

Γ(s) = Γ(\frac{s}{2})Γ(\frac{s + 1}{2})2^{s−1}π^{− \frac{1}{2}}

to have \frac{s}{2} in all arguments of Γ-factors. If A denotes the total number of them, then using the equality

Γ(σ + it) = \sqrt{2π}|t|^{σ− \frac{1}{2}} e^{− \frac{π}{2} |t|} (1 + O(|t|^{−1})

valid for σ_{0} ≤ σ ≤ σ_{1} and t > t_{0}(σ_{1}, σ_{2}), we deduce that the ratio of the L_{∞}-terms in (3.3) is ≪_{σ} |t|^{A(\frac{w+1}{2} −σ)}. Hence by Lemma 1 (ϵ(M_{χ}) does not depend on D) we have

\begin{aligned}|U|^{2} & ≪ c(χ)^{d(w+1−2α)}|t|^{A(w+1−2α)} \\ & ≪ (D^{d}|t|^{A})^{w+1−2α} . \end{aligned}

The summation over n > X(log DT)^{2} in (3.2) is negligible due to the decay of exp (−\frac{n}{X} ). We split the remaining part of the sum into the intervals of type (N, 2N] and apply (3.1) to obtain

∑_{χ∈C(D)} | ∑_{N≤n≤2N} a_{n}χ(n)n^{−s}e^{− \frac{n}{X}} |^{2} ≪ (D + X^{w+2−2σ})(DX)^{ϵ}

for \frac{w+1}{2} ≤ σ ≤ \frac{w+2}{2}. Let Δ = σ − \frac{w+1}{2} and η > 0 be sufficiently small and X = (DT^{A})^{1+η}. By the definition of ν^{±} and the above estimate we obtain

S^{+}(D, s) ≪

≪ ((DT^{A})^{w+2−2σ} + D) (DT^{A})^{η+3ϵ}

+ (DT^{A})^{2(α−σ)(1+η)}(D^{d}T^{A})^{w+1−2α}(D + (DT^{A})^{2α−w})(DT^{A})^{ν^{−}(w+1−α)+ϵ}.

If \frac{w+1}{2} + η ≤ σ ≤ \frac{w+2}{2} then we choose α = w + 1 − σ getting

\begin{aligned}S^{+}(D, s) & ≪ ((DT^{A})^{w+2−2σ} + D) (DT^{A})^{η+3ϵ} \\ & + (DT^{A})^{−4Δ(1+η)}(D^{d}T^{A})^{2Δ}D(DT^{A})^{ν^{−}(w+1−α)+ϵ} \end{aligned} (3.4)

and by the inequality d ≤ 2 we obtain, by comparying the exponents that ν^{+}(σ) ≤ max(η + 3ϵ, −4η^{2} + ϵ + ν^{−}(σ)). Writting the same for the dual motive we conclude by the symmetry that ν^{−}(σ) ≤ max(η + 3ϵ, −4η^{2} + ϵ + ν^{+}(σ)). Choosing 2η > \sqrt{ϵ} we conclude that ν^{+}(σ) ≤ 0 in this case.

If \frac{w+1}{2} ≤ σ ≤ \frac{w+1}{2} + η then choosing α = \frac{w+1}{2} − η we have by the above that ν^{+}(w + 1 − α) ≤ 0 and by (3.4) we obtain

\begin{aligned}S^{+}(D, s) & ≪ ((DT^{A})^{w+2−2σ} + D) (DT^{A})^{η+3ϵ} + (D^{d}T^{A})^{w+1−2α}(D + (DT^{A})^{2α−w}) \\ & ≪ ((DT^{A})^{w+2−2σ} + D) (DT^{A})^{η+3ϵ} + D(DT^{A})^{4η+ϵ}, \end{aligned}

hence in this case also ν^{+}(σ) ≤ 0. Analogously we prove that ν^{−}(σ) ≤ 0. This completes the proof of Theorem 2.

4. Proof of Theorem 3

By (0.1) we see that

∑_{χ∈C(D)} |L(M_{χ}, s)|^{2} ≪ D^{1+ϵ}

uniformly for |s − s_{0}| ≤ 2(log D)^{−1}, where s_{0} = \frac{w+1}{2}.

Also we have, by the Cauchy integral formula, that

L^{(n)}(M_{χ}, s) = \frac{n!}{2πi} ∫_{|s−ξ|=r} \frac{L(M_{χ}, ξ)}{(ξ − s)^{n+1}} dξ

with r = (log D)^{−1}. Hence

∑_{χ∈C(D)} |L^{(n)}(M_{χ}, s_{0})|^{2}

≪ (log D)^{2n+2} ∑_{χ∈C(D)} (∫_{|ξ−s_{0}|=r} |L^{(n)}(M_{χ}, ξ)|^{2}dξ) (∫_{|ξ−s_{0}|=r} dξ)

≪ (log D)^{2n+1} ∑_{χ∈C(D)} ∫_{|ξ−s_{0}|=r} |L^{(n)}(M_{χ}, ξ)|^{2}dξ

≪ (log D)^{2n} sup_{|ξ−s_{0}|≤(log D)^{−1}} ∑_{χ∈C(D)} |L(M_{χ}, ξ)|^{2}

≪ D^{1+ϵ}.

Now from Theorem 1 and the above we obtain, by application of the Cauchy-Schwarz inequality, that

D(log D)^{n} ≪ ∑_{χ∈C(D)} |L^{(n)}(M_{χ}, \frac{w + 1}{2})| ≤ ( ∑_{χ∈C(D)} |L^{(n)}(M_{χ}, \frac{w + 1}{2})|)^{\frac{1}{2}} N_{n}(D, M)^{\frac{1}{2}} .

Hence

N_{n}(D, M) ≫ D^{1−2ϵ}

and Theorem 3 follows.

5. Applications to a bound for the order of the (twisted) Shafarevich-Tate groups

Let A be an abelian variety over ℚ of rank d and conductor c(A); let p be a prime dividing c(A). Let A_{p} be a Néron model of A over ℚ_{p} (see [BLR] for definition). We are interested in a bound for the order of the group Φ_{p}(A) of connected components of A_{p}.

In a case of an elliptic curve E with integral j-invariant we have ♯Φ_{p}(E) ≤ 4 (see [T1]).

In a general case W.G. McCallum [Mc] proved the following result. Assume that a Néron model A_{p} of A has no multiplicative component in its reduction; then such a bound exists and depends only on the dimension d of A: ♯Φ_{p}(A) ≤ C_{d}.

Let A_{χ} denote the twist of A by quadratic character χ; it is again an abelian variety over ℚ. Assume that for any p dividing c(A) a Néron model A_{p} has no multiplicative component in its reduction; if (c(χ), c(A)) = 1 the same holds for any A_{χ}. Hence, in this case

∏_{p|c(A_{t})} ♯Φ_{p}(A_{t}) ≪ C_{d}^{ψ(t)},

where A_{t} = A_{χ_{t}}, and ψ(t) denotes the number of prime divisors of t. As a consequence, we obtain

(5.1)  ∏_{p|c(A_{χ_{t}})} ♯Φ_{p}(A_{χ_{t}}) = O(t^{ϵ}).

Proof of Theorem 4. We start with the following estimate for the real period of A:

(5.2)  π_{∞}(A_{t}) ≪ t^{− \frac{d}{2}} .

Theorem 3 implies that there exist infinitely many t such that L(A_{t}, 1) ≫ 1. Assuming the Birch and Swinnerton-Dyer conjecture for all such A_{t} we obtain

♯III(A_{t}) ≫ \frac{1}{π_{∞}(A_{t})π_{bad}(A_{t})}.

Now, the assertion follows from (5.1), (5.2) and Lemma 1.

Remark. If M is a pure motive of weight −1, then the same arguments (at least conjecturally) may be used to state the analogue of Theorem 4 in this case.

6. Concluding remarks

6.1. Let E be an elliptic curve over ℚ of conductor N → ∞. It is conjectured [GS] that

(6.1)  ♯III(E) = O(N^{\frac{1}{2} +ϵ}) .

Let’s also remark the following consequence of [GS]: if ♯III_{l}(E) ≠ 1 for some prime l ≫ N^{\frac{1}{2} +ϵ}, then either (6.1) is false or the Birch-Swinnerton-Dyer conjecture is false or the Shimura-Taniyama-Weil conjecture is false. Note however that the last conjecture has been proved by A. Wiles for all semistable elliptic curves.

It is an important question whether there exist elliptic curves with arbitrary high rank and what is the frequency of these high ranks. The work of F. Gouvea and B. Mazur [GM] shows that

♯{D ∈ ℤ : D ≡ 1(mod 8), 1 ≤ D ≤ X, D square free, ord_{s=1}L(E_{D}, s) ≥ 2}

≫ X^{\frac{1}{2} −ϵ}.

On the other hand we have the following conjecture due to R. Greenberg [G].

Conjecture. Let Σ be a finite set of primes. Let f vary over all normalized newforms of weight 2 for Γ_{0}(q), where q is divisible only by primes in Σ. Then ord_{s=1}L(f, s) is 0 or 1, except for at most finitely many such f’s.

6.2. The following possible application was suggested by ([BFH2], p.169). Suppose that f and g are two newforms of the same weight k on some Γ_{0}(q). One asks whether there are infinitely many quadratic characters χ such that

L(M(f)_{χ}, \frac{k}{2} ) L(M(g)_{χ}, \frac{k}{2} ) ≠ 0.

In a case f and g correspond to elliptic curves E_{f} and E_{g} respectively, it would then imply, combined with the Kolyvagin’s work, that there are infinitely fundamental discriminants t such that the ranks of the Mordell-Weil groups of the twisted elliptic curves E_{f}^{(t)} and E_{g}^{(t)} are both zero. By this method it reduces to a problem concerning meromorphic continuality of D(M(f) ⊕ M(g), s) to Re(s) > \frac{k−1}{2} − ϵ, and certainly depends on a variant of Rankin-Selberg method for GL(4).

Appendix

It is quite easy to continue D(ℚ(0)^{⊕r}, s) meromorphically to Re(s) > \frac{1}{2} −ϵ. Here the procedure is explicit and gives information about the behaviour of our series at s = \frac{1}{2}: it always has a pole at this point. Let us perform the calculations for r = 1, 2.

(i) Consider D(ℚ(0), s). We have

D(ℚ(0), s) = (1 + 2^{−s}) ∑_{l=1}^{∞} ∏_{p|l, p≠2} (1 + \frac{1}{p})^{−1} l^{−2s}, Re(s) > \frac{1}{2}.

Let f be a multiplicative function, defined on powers of primes by

f(1) = 1,  f(2^{k}) = 1, k ≥ 1,  f(p^{k}) = (1 + \frac{1}{p})^{−1} , p ≠ 2, k ≥ 1.

One checks that

f = ϕ_{1} ⋆ ϵ,

where ϕ is the Euler function, ϕ_{1}(n) := n^{−1}ϕ(n) and ϵ is a multiplicative function defined by

ϵ(1) = 1, ϵ(2^{k}) = 2^{−k}, k ≥ 1, ϵ(p^{k}) = \frac{1}{p^{k}(p + 1)}, p ≠ 2, k ≥ 1.

Therefore

D(ℚ(0), s) = (1 + 2^{−s}) (∑_{n=1}^{∞} ϕ(n)n^{−2s−1}) (∑_{n=1}^{∞} ϵ(n)n^{−2s}) .

Now

∑_{n=1}^{∞} ϵ(n)n^{−2s} = (1 − 2^{−1−2s})^{−1} ∏_{p≠2} (1 + \frac{1}{p + 1} \frac{p^{−1−2s}}{1 − p^{−1−2s}} )

is convergent and non-zero at s = \frac{1}{2}, and

∑_{n=1}^{∞} ϕ(n)n^{−2s−1} = \frac{ζ(2s)}{ζ(2s + 1)}.

Hence D(ℚ(0), s) has a simple pole at s = \frac{1}{2}.

(ii) Let d(n) denote the number of divisors of n. We have

D(ℚ(0) ⊕ ℚ(0), s) = ∑_{r=0}^{∞} ∑_{l odd} d(2^{r}l^{2}) ∏_{p|l} (1 + \frac{1}{p})^{−1} 2^{−rs}l^{−2s}

= ∑_{r=0}^{∞} d(2^{r})2^{−rs} ∑_{n odd} d(n^{2}) ∏_{p|n} (1 + \frac{1}{p})^{−1} n^{−2s}, Re(s) > \frac{1}{2}.

The first sum is convergent and non-zero for s = \frac{1}{2}. Let ϵ be a multiplicative function defined by ϵ(1) = 1, ϵ(2) = −1, ϵ(2^{r}) = 0, (r ≥ 2) and

ϵ(p) = \frac{2p − 1}{p + 1}, p ≠ 2, ϵ(p^{r}) = \frac{2p}{p + 1}, p ≠ 2, r ≥ 2.

Let f be defined by f(1) = 1 and

f(2^{r}) = 0, r ≥ 1, f(p^{r}) = \frac{p}{p + 1}d(p^{2r}), p ≠ 2, r ≥ 1.

Then f = ϵ ⋆ 1 and we obtain

∑_{n odd} d(n^{2}) ∏_{p|n} (1 + \frac{1}{p})^{−1} n^{−2s} = (1 − 2^{−2s})^{2}ζ(2s)^{2} ∏_{p≠2} (1 − (\frac{2}{p + 1} + p^{−2s})p^{−2s}) .

Hence D(ℚ(0) ⊕ ℚ(0), s) has a double pole at s = \frac{1}{2}.

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Andrzej Da̧browski: Max-Planck-Institut für Mathematik, Gottfried-Claren-Str. 26, 53225 Bonn, Germany and University of Szczecin, Department of Mathematics, ul. Wielkopolska 15, 70-451 Szczecin, Poland

E-mail: dabrowsk@mpim-bonn.mpg.de and dabrowsk@sus.univ.szczecin.pl

Jacek Pomykała: University of Warsaw, Institute of Mathematics, ul. Banacha 2, 02-097 Warsaw, Poland

E-mail: pomykala@impan.impan.gov.pl