Rudnick and Soundararajan’s Theorem for Function Fields

# Rudnick and Soundararajan’s Theorem for Function Fields

## Abstract.

In this paper we prove a function field version of a theorem by Rudnick and Soundararajan about lower bounds for moments of quadratic Dirichlet –functions. We establish lower bounds for the moments of quadratic Dirichlet –functions associated to hyperelliptic curves of genus over a fixed finite field in the large genus limit.

###### Key words and phrases:
function fields and finite fields and hyperelliptic curves and lower bounds for moments and moments of –functions and quadratic Dirichlet –functions and random matrix theory
###### 2010 Mathematics Subject Classification:
Primary 11M38; Secondary 11G20, 11M50, 14G10

## 1. Introduction

It is a fundamental problem in analytic number theory to estimate moments of central values of –functions in families. For example, in the case of the Riemann zeta function the question is to establish asymptotic formulae for

 (1.1) Mk(T):=∫T1|ζ(12+it)|2kdt,

where is a positive integer and .

A believed folklore conjecture asserts that, as , there is a positive constant such that

 (1.2) Mk(T)∼CkT(logT)k2.

Due to the work of Conrey and Ghosh [3] the conjecture above assumes a more explicit form, namely

 (1.3) Ck=akgkΓ(k2+1),

where

 (1.4) ak=∏p prime[(1−1p)k2∑m≥0dk(m)2pm],

is an integer when is an integer and is the number of ways to represent as a product of factors.

Asymptotics for are only known for , due to Hardy and Littlewood [7]

 (1.5) M1(T)∼TlogT,

and for , due to Ingham [10]

 (1.6) M2(T)∼12π2Tlog4T.

Unfortunately the recent technology does not allow us to obtain asymptotics for higher moments of the Riemann zeta function. The same statement applies for the higher moments of other –functions. However, due to the precursor work of Keating and Snaith [14, 15] and, subsequently, due to the work of Conrey, Farmer, Keating, Rubinstein and Snaith [4], and Diaconu, Goldfeld and Hoffstein [5], there are now very elegant conjectures for moments of –functions.

The work of Katz and Sarnak [12, 13] associates a symmetry group for each family of –function and the moments are sensitive and take different forms for each one of these groups. In other words the conjectured asymptotic formulas for the moments of families of –function depends whether the symmetry group attached to the family is unitary, orthogonal or symplectic. For a recent and detailed discussion about a working definition of a family of –functions see [21].

We will typify the conjectures above by considering different families of –functions. For example, the family of all Dirichlet –functions , as varies over primitive characters , is an example of a unitary family, and it is conjectured that

 (1.7) ∑∑∗χ(modq)|L(12,χ)|2k∼CU(N)(k)q(logq)k2,

where and is a positive constant. For a symplectic family of –functions we consider the quadratic Dirichlet –functions associated to the quadratic character , as varies over fundamental discriminants. In this case it is conjectured that

 (1.8) ∑∑♭|d|≤XL(12,χd)k∼CUSp(2N)(k)X(logX)k(k+1)/2,

where and is a positive constant. And finally we consider the family of –functions associated to Hecke eigencuspforms of weight for the full modular group as varies in the set of Hecke eigencuspforms. This is an example of an orthogonal family and it is conjectured that

 (1.9) ∑∑hf∈HkL(12,f)r∼CO(N)(r)(logk)r(r−1)/2,

where is a positive constant, and

 (1.10) ∑∑hf∈HkL(12,f)r:=∑f∈Hk1ωfL(12,f)r,

with

 (1.11) ωf:=(4π)k−1Γ(k−1)⟨f,f⟩=k−12π2L(1,Sym2f),

where denotes the Petersson inner product. For more details on Hecke eigencuspforms –functions see Iwaniec [11].

The conjectures (1.2), (1.7) and (1.8) can be verified for small values of and the same holds for (1.9), where it can be verified only for small values of . Ramachandra [17] showed that

 (1.12) ∫T1|ζ(12+it)|2kdt≫T(logT)k2,

for positive integers . Titchmarsh [24, Theorem 7.19] had proved a smooth version of these lower bound for positive integer . The work of Heath–Brown [8] extends (1.12) for all positive rational numbers . Recently Radziwiłł and Soundararajan [16] proved that

 (1.13) Mk(T)≥e−30k4T(logT)k2,

for any real number and all large . For other families of –functions, as those given above, the lower bounds for moments were proved by Rudnick and Soundarajan in [19, 20] where they have established that

 (1.14) ∑∑∗χ(modq)χ≠χ0|L(12,χ)|2k≫kq(logq)k2,

for a fixed natural number and all large primes . They also proved in [20] that

 (1.15) ∑∑hf∈HkL(12,f)r≫r(logk)r(r−1)/2,

for any given natural number , and weight with . And for the symplectic family they showed that for every even natural number

 (1.16) ∑∑♭|d|≤XL(12,χd)k≫kX(logX)k(k+1)/2,

where the sum is taken over fundamental discriminants . Radziwiłł and Soundararajan [16] pointed out that their method may easily be modified to provide lower bounds for moments to the case of –functions in families, for any real number .

Recently, in a beautiful paper, Tamam [23] proved the function field analogue of (1.14). In this paper we consider the function field analogue of equation (1.16) for quadratic Dirichlet –functions associated to a family of hyperelliptic curves over . See next section.

## 2. Main Theorem

Before we enunciate the main theorem of this paper we need a few basic facts about rational function fields. We start by fixing a finite field of odd cardinality with a prime. And we denote by the polynomial ring over and by the rational function field over .

The zeta function associated to is defined by the following Dirichlet series

 (2.1) ζA(s):=∑f∈Amonic1|f|s        for Re(s)>1,

where for and for . Surprisingly the zeta function associated to is a much simpler object than the usual Riemann zeta function and can be showed that

 (2.2) ζA(s)=11−q1−s.

Let be a square–free monic polynomial in of degree odd. Then we define the quadratic character attached to by making use of the quadratic residue symbol for by

 (2.3) χD(f)=(Df).

In other words, if is monic irreducible we have

 (2.4) χD(P)={0,if P∣D,1,if P⧸|D and D is a square modulo P,−1,if P⧸|D and D is a non square modulo P.

For more details about Dirichlet characters for function fields see [18, Chapter 3] and [6].

We attach to the character the quadratic Dirichlet –function defined by

 (2.5) L(s,χD)=∑f∈Af monicχD(f)|f|s        %forRe$(s)>1$.

If , where

 (2.6) H2g+1,q={D∈A, square--free, monic and deg(D)=2g+1},

then the –function associated to is indeed the numerator of the zeta function associated to the hyperelliptic curve and therefore is a polynomial in of degree given by

 (2.7) L(s,χD)=2g∑n=0∑f monicdeg(f)=nχD(f)q−ns.

(see [18, Propositions 14.6 and 17.7] and [1, Section 3]).

This –function satisfies a functional equation, namely

 (2.8) L(s,χD)=(q1−2s)gL(1−s,χD),

and the Riemann hypothesis for curves proved by Weil [25] tell us that all the zeros of have real part equals .

The main result of this paper is now presented:

###### Theorem 2.1.

For every even natural number we have,

 (2.9) 1#H2g+1,q∑D∈H2g+1,qL(12,χD)k≫k(logq|D|)k(k+1)/2.
###### Remark 2.1.

To avoid any misunderstanding concerning the notation and conventions presented in this paper it is necessary a note about the notation used in the theorem above and in the rest of this note. On the formula above the right-hand side of the main lower bound appears while is the summation variable on the left-hand side of that same formula. This is done because the function is constant within and so we can always write

 ∑D∈H2g+1,q|D|=|D|∑D∈H2g+1,q1.

In the case the reader feel uncomfortable with the above notation he/she can always remember that .

###### Remark 2.2.

For simplicity, we will restrict ourselves to the fundamental discriminants , monic and . But the calculations are analogous for the even case, i.e., .

Using the same techniques developed by Rudnick and Soundararajan in [19, 20] and extended for function fields in this paper we can also prove the following theorem.

###### Theorem 2.2.

For every even natural number and or we have,

 (2.10) 1πA(n)∑P monicirreducibledeg(P)=nL(12,χP)k≫k(logq|P|)k(k+1)2,

where and the prime number theorem for polynomials [18, Theorem 2.2] says that .

## 3. Necessary Tools

In this section we present some auxiliary lemmas that will be used in the proof of the main theorem. We start with:

###### Lemma 3.1 (“Approximate” Functional Equation).

Let . Then can be represented as

 (3.1) L(s,χD)=∑f1 monicdeg(f1)≤gχD(f1)|f1|s+(q1−2s)g∑f2 monicdeg(f2)≤g−1χD(f2)|f2|1−s.
###### Proof.

The proof of this Lemma can be found in [1, Lemma 3.3]. ∎

The following lemma is the function field analogue of Pólya–Vinogradov inequality for character sums.

###### Lemma 3.2 (Pólya–Vinogradov inequality for Fq(T)).

Let be a non–principal Dirichlet character modulo such that is odd. Then we have,

 (3.2) ∑deg(f)=xχ(f)≪|Q|1/2.
###### Proof.

The proof of this Lemma can be found in [9, Proposition 2.1]. ∎

The next lemma is taken from Andrade-Keating [1, Proposition 5.2] and it is about counting the number of square–free polynomials coprime to a fixed monic polynomial.

###### Lemma 3.3.

Let be a fixed monic polynomial. Then for all we have that

 (3.3) ∑D∈H2g+1,q(D,f)=11=|D|ζA(2)∏P monicirreducibleP∣f(|P||P|+1)+O(|D|12|f|ε).

## 4. Proof of Theorem 2.1

In this section we prove Theorem 2.1.

Let be a given even number, and set . We define

 (4.1) A(D)=∑deg(n)≤xχD(n)√|n|,

and let

 (4.2) S1:=∑D∈H2g+1,qL(12,χD)A(D)k−1,

and

 (4.3) S2:=∑D∈H2g+1,qA(D)k.

An application of Triangle inequality followed by Hölder’s inequality gives us that,

 ∣∣∣∑D∈H2g+1,qL(12,χD)A(D)k−1∣∣∣≤∑D∈H2g+1,q|L(12,χD)||A(D)|k−1 (4.4) ≤⎛⎝∑D∈H2g+1,qL(12,χD)k⎞⎠1/k⎛⎝∑D∈H2g+1,qA(D)k⎞⎠k−1k.

From (4) we have

 ∑D∈H2g+1,qL(12,χD)k ≥(∑D∈H2g+1,qL(12,χD)A(D)k−1)k(∑D∈H2g+1,qA(D)k)k−1 (4.5) =Sk1Sk−12.

Hence from (4) we can see that to prove Theorem 2.1 we only need to give satisfactory estimates for and . We start with .

### 4.1. Estimating S2

We have that

 (4.6) A(D)k=∑n1,…,nkdeg(nj)≤xj=1,…,kχD(n1…nk)√|n1|…|nk|.

So,

 S2 =∑D∈H2g+1,qA(D)k (4.7) =∑n1,…,nkdeg(nj)≤xj=1,…,k1√|n1|…|nk|∑D∈H2g+1,q(Dn1…nk).

At this stage we need an auxiliary Lemma. It is called orthogonal relations for quadratic characters and it has appeared in a different form in [1, 2, 6].

###### Lemma 4.1.

If is not a perfect square then

 (4.8) ∑D∈H2g+1,qn≠□(Dn)≪|D|1/2|n|1/4.

And if is a perfect square then

 (4.9) ∑D∈H2g+1,qn=□(Dn)=|D|ζA(2)∏P monicirreducibleP∣n(|P||P|+1)+O(|D|12|n|ε),

for any .

###### Remark 4.2.

Equation (4.8) can be seen as an improvement on the estimate given in [6, Lemma 3.1]. And the same equation (4.8) can be used to improve the error term in the first moment of quadratic Dirichlet –functions over function fields as given in [1, Theorem 2.1].

###### Proof.

If , then

 (4.10) ∑D∈H2g+1,qn=□(Dn)=∑D∈H2g+1,q(D,n)=11.

By invoking Lemma 3.3 we establish equation (4.9).

For (4.8) we write

 ∑D∈H2g+1,q(Dn)=∑2α+β=2g+1∑deg(B)=β∑deg(A)=αμ(A)(A2Bn) =∑0≤α≤g∑deg(A)=αμ(A)(A2n)∑deg(B)=2g+1−2α(Bn) (4.11) ≤∑0≤α≤g∑deg(A)=α∑deg(B)=2g+1−2α(Bn).

If then is a character sum to a non–principal character modulo . So using Lemma 3.2 we have that

 (4.12) ∑deg(B)=2g+1−2α(Bn)≪|n|1/2.

Further we can estimate trivially the non–principal character sum by

 (4.13) ∑deg(B)=2g+1−2α(Bn)≪|D||A|2=q2g+1−2α.

Thus, if , we obtain that

 ∑D∈H2g+1,q(Dn) ≪∑0≤α≤g∑deg(A)=αmin(|n|1/2,|D||A|2) (4.14) ≪|D|12|n|14,

upon using the first bound (4.12) for and the second bound (4.13) for larger . And this concludes the proof of the lemma. ∎

Using Lemma 4.1 in (4.1) we obtain that

 S2 =∑n1,…,nkdeg(nj)≤xj=1,…,kn1…nk=□1√|n1|…|nk|(|D|ζA(2)∏P monicirreducibleP∣n1…nk(|P||P|+1)) +∑n1,…,nkdeg(nj)≤xj=1,…,kn1…nk=□1√|n1|…|nk|O(|D|12|n1…nk|ε) (4.15) +∑n1,…,nkdeg(nj)≤xj=1,…,kn1…nk≠□1√|n1|…|nk|O(|D|12|n1…nk|14).

After some arithmetic manipulations with the –terms we get that

 S2 =|D|ζA(2)∑n1,…,nkdeg(nj)≤xj=1,…,kn1…nk=□1√|n1|…|nk|∏P monicirreducibleP∣n1…nk(|P||P|+1) (4.16)

Since , the error term above is . So,

 (4.17) S2=|D|ζA(2)∑n1,…,nkdeg(nj)≤xj=1,…,kn1…nk=□1√|n1|…|nk|∏P monicirreducibleP∣n1…nk(|P||P|+1)+O(|D|35).

Writing we see that

 ∑m2 monicdeg(m2)≤xdk(m2)|m|∏P monicirreducibleP∣m(|P||P|+1) ≤∑n1,…,nkdeg(nj)≤xj=1,…,kn1…nk=□=m21√|n1|…|nk|∏P monicirreducibleP∣n1…nk(|P||P|+1) (4.18) ≤∑m2 monicdeg(m2)≤kxdk(m2)|m|∏P monicirreducibleP∣m(|P||P|+1),

where represents the number of ways to write the monic polynomial as a product of factors.

We need to obtain an estimate for

 (4.19) ∑m monicdeg(m)=xdk(m2)am,

where .

To obtain the desired estimate we consider the corresponding Dirichlet series

 (4.20) ζf(s)=∑m monicdk(m2)am|m|s=∞∑n=0∑deg(m)=xdk(m2)amux=Zf(u),

with . Writing the above as an Euler product

 (4.21) ζf(s)=∏P monicirreducible(1+dk(P2)aP|P|s+dk(P4)aP2|P|2s+⋯),

we can identify the poles of . Similar calculations carried out in the classical case by Soundararajan and Rudnick [20, page 9] and Selberg [22, Theorem 2], and for function fields by Andrade and Keating [2, Section 4.3] shows us that has a pole at of order . Therefore we can write

 ζf(s) =∏P monicirreducible(1−1|P|s)−k(k+1)2 (4.22) ×∏P monicirreducible(1+(|P||P|+1∞∑j=1dk(P2j)|P|js))(1−1|P|s)k(k+1)2,

where the first product has a pole at of order and the second product above (4.1) is convergent for and holomorphic in with

 (4.23) B={s∈C | −πilog(q)≤I(s)<πilog(q)}.

Thus we can use Theorem 17.4 from [18] to obtain the desired estimate. But we sketch below how this can be done. A standard contour integration (Cauchy’s theorem)

 (4.24) 12πi∮Cε+CZf(u)ux+1du=∑Res(Zf(u)u−x−1),

where is the boundary of the disc for some and a small circle about oriented clockwise. There is only one pole in the integration region and it is located at as can be seen from (4.1). To find the residue there, we expand both and in Laurent series about , multiply the results together, and pick out the coefficient of . After this residue calculation we obtain that

 (4.25) ∑m monicdeg(m)=xdk(m2)am∼C(k)qxxk(k+1)2−1,

for a positive constant explicitly given by

 (4.26) C(k)=log(q)k(k+1)2(k(k+1)2−1)!α,

with

 (4.27) α=lims→1[(s−1)k(k+1)2ζf(s)].

In the end we obtain that

 (4.28) ∑m monicdeg(m)≤zdk(m2)|m|∏P monicirreducibleP∣m(|P||P|+1)∼C(k)(z)k(k+1)/2.

Therefore we can conclude that

 (4.29) S2≪|D|(logq|D|)k(k+1)/2.

### 4.2. Estimating S1.

It remains to evaluate and for that we need an “approximate” functional equation for . Using Lemma 3.1 with we have that

 S1 =∑D∈H2g+1,q⎛⎝∑deg(f1)≤gχD(f1)|f1|1/2+∑deg(f2)≤g−1χD(f2)|f2|1/2⎞⎠ ×(∑n1,…,nk−1deg(nj)≤xj=1,…,k−1χD(n1…nk−1)√|n1|…|nk−1|) =∑deg(f1)≤g1√|f1|∑n1,…,nk−1deg(nj)≤xj=1,…,k−11√|n1|…|nk−1|∑D∈H2g+1,q(Df1n1…nk−1) (4.30) +∑deg(f2)≤g−11√|f2|∑n1,…,nk−1deg(nj)≤xj=1,…,k−11√|n1|…|nk−1|∑D∈H2g+1,q(Df2n1…nk−1).

In the last equality in equation (4.2) the sums over and are exactly the same, with the only difference being the size of the sums, i.e., and . We estimate only the sum in the last equality and the result being the same for the sum just replacing by .

If is not a square then an application of Lemma 4.1 gives us that

 ∑deg(f1)≤g1√|f1|∑n1,…,nk−1deg(nj)≤xj=1,…,k−11√|n1|…|nk−1|∑D∈H2g+1,q(Df1n1…nk−1) ≪∑deg(f1)≤g1√|f1|∑n1,…,nk−1deg(nj)≤xj=1,…,k−11√|n1|…|nk−1||D|12|f1n1…nk−1|14 =|D|12∑deg(f1)≤g|f1|−14∑deg(n1)≤x|n1|−14⋯∑% deg(nk−1)≤x|nk−1|−14 (4.31) ≪|D|12q34g(qx)(k−1)34.

With our choice of , we have that for not a square

 ∑deg(f1)≤g1