Low-lying Zeros of Number Field L -functions
aa r X i v : . [ m a t h . N T ] M a r LOW-LYING ZEROS OF NUMBER FIELD L -FUNCTIONS STEVEN J. MILLER AND RYAN PECKNERA
BSTRACT . One of the most important statistics in studying the zeros of L -functionsis the 1-level density, which measures the concentration of zeros near the central point.Fouvry and Iwaniec [FI] proved that the 1-level density for L -functions attached toimaginary quadratic fields agrees with results predicted by random matrix theory. Inthis paper, we show a similar agreement with random matrix theory occurring in moregeneral sequences of number fields. We first show that the main term agrees withrandom matrix theory, and similar to all other families studied to date, is independentof the arithmetic of the fields. We then derive the first lower order term of the 1-leveldensity, and see the arithmetic enter.
1. I
NTRODUCTION
Background.
While studying class numbers in the early 1970s, Montgomery madethe remarkable observation that the zeros of the Riemann zeta function appear to becorrelated in precisely the same way as the eigenvalues of Gaussian random matrices[Mon]. This was based on a chance encounter with Freeman Dyson, who had calculatedthe eigenvalue pair correlation function for the Gaussian Unitary Ensemble and foundit to be − (cid:18) sin πxπx (cid:19) , exactly the distribution conjectured by Montgomery for the zeros of the zeta function.Extensive numerical computations by Odlyzko [Od1, Od2] support this unexpected cor-respondence to impressive heights on the critical line.Attempts to explain this connection rigorously in the number field case have thusfar been unsuccessful. However, groundbreaking theoretical work by Katz and Sarnakhas put this goal within reach in the function field setting. They proved that, as oneaverages over the zeros of suitable families of L -functions obtained from geometry,the scaling limit of the spacing measures of the normalized zeros tends to a ‘universalmeasure’ which is the limit of the spacing measures of the eigenvalues of Gaussianrandom matrices (see [KaSa1, KaSa2] for details, as well as the survey article [FM]for a description of the development of random matrix theory from nuclear physics tonumber theory). Moreover, their work predicts that associated to an appropriate family Date : October 28, 2018.2000
Mathematics Subject Classification.
Key words and phrases. -level density, Hecke characters, low-lying zeros, symmetry, CM-fields,class number, lower order terms.This work was done at the 2009 SMALL Undergraduate Research Project at Williams College, fundedby NSF GRANT DMS0850577 and Williams College; it is a pleasure to thank them for their support.We also thank Michael Rosen and the participants of the 2009 YMC at Ohio State for many enlighteningconversations. The first named author was also partly supported by NSF grant DMS0600848. E of elliptic curves over Q is a classical compact matrix group G ( E ) (which may beviewed as a group of random matrices under normalized Haar measure) in such a waythat for any compactly supported even Schwartz function h on R , we have Z R h ( x ) W G ( E ) ( x ) dx = lim X →∞ / X n ≤ X |E n | ! X n ≤ X,E ∈E n h (cid:18) γ E,j log( N E )2 π (cid:19) (1.1)where N E denotes the conductor of the curve E , E n = { Q − isogeny classes of E ∈ E : N E = n } and / ± iγ E,j are the zeros of L ( S, E/ Q ) (normalized to have functional equation s → − s ). The distribution W G ( E ) is canonically associated to the scaling limit of aclassical compact group, and gives the density of the normalized spacings between theeigenangles. Katz and Sarnak [KaSa1, KaSa2] showed that for test functions φ withFourier transforms supported in ( − , , the one-level densities of the scaling limits ofthe classical compact groups are given by Z φ ( x ) W SO(even) ( x ) dx = b φ (0) + 12 φ (0) Z φ ( x ) W SO(odd) ( x ) dx = b φ (0) + 12 φ (0) Z φ ( x ) W O ( x ) dx = b φ (0) + 12 φ (0) Z φ ( x ) W USp ( x ) dx = b φ (0) − φ (0) Z φ ( x ) W U ( x ) dx = b φ (0) . (1.2)The quantity on the right side of (1.1), which due to the normalization by log( N E )2 π measures the low-lying zeros of the L -functions, is known as the 1-level density for thefamily. Thus, this conjecture is often referred to as the ‘density conjecture’.One expects that an analogue of this conjecture should hold for all suitable familiesof automorphic L -functions, not just those associated to elliptic curves. Indeed, thedensity conjecture has been verified (up to small support) for a wide variety of families,including all Dirichlet characters, quadratic Dirichlet characters, elliptic curves, weight k level N cuspidal newforms, symmetric powers of GL(2) L -functions, and certainfamilies of GL(4) and GL(6) L -functions; see [DM1, DM2, HR, HM, ILS, KaSa2,Mil3, OS, RR, Ro, Rub, Yo2]. We have two goals in this paper. The first is to verifythe density conjecture for as large a class of test functions as possible for L -functionscoming from a patently different situation than that of elliptic curves, namely the L -functions of ideal class characters of number fields. As in all other families studied todate, the main term is independent of the arithmetic of the family. Our second goal For the purposes of this paper, the following formulas suffice as we only need to know the one-leveldensities when supp( b φ ) ⊂ ( − , . See [KaSa1, KaSa2] for determinantal formulas for the n -leveldensities for arbitrary support. OW-LYING ZEROS OF NUMBER FIELD L -FUNCTIONS 3 is to see the effects of the arithmetic in the lower order terms, thereby distinguishingdifferent families.To make things precise, let F be a family of number fields, and define for each field K ∈ F the -level density D \ CL ( K ) ( φ ) = 1 h K X χ ∈ \ CL ( K ) X γ χ L (1 / iγ χ ,χ )=0 φ (cid:18) γ χ log ∆ K π (cid:19) (1.3)where φ is an even Schwartz function whose Fourier transform has compact support, h K is the class number of K , ∆ K is the absolute value of its discriminant, and the outersum runs over the characters of the ideal class group CL ( K ) of K . Again, due to therapid decay of φ and the scaling factor log ∆ K π , only the low-lying zeros contribute tothis sum in the limit as ∆ K → ∞ . Since for a given number X there are only finitelymany number fields of (absolute value of) discriminant less than X , the discriminantmust tend to infinity in any infinite family of number fields. Moreover, ordering thefamily F according to the increasing parameter ∆ K , we may consider the limit D F ( φ ) = lim ∆ K →∞ D \ CL ( K ) ( φ ) , and this is independent of rearranging fields which have the same value of ∆ K . How-ever, there is no good reason to expect this limit to exist if F is just an arbitrary collec-tion of number fields; thus we reserve the term ‘family’ for a collection F of numberfields whose members have similar arithmetic properties and for which the -level den-sity actually exists. This is by no means an attempt at an actual definition of the term‘family’, which is an ongoing subject, but it suffices for our purposes, wherein the com-mon arithmetic origin of our fields will be obvious.Among the wide variety of families for which the density conjecture has been inves-tigated, few have arisen from the number field context. In fact, to our knowledge, theonly work to date analyzing the 1-level density for Hecke characters is that of Fouvry-Iwaniec [FI], who showed that, in the notation above, the -level density D F ( φ ) for F the family Q ( − D ) with − D a fundamental discriminant is given by the symplecticdistribution. In addition, recent unpublished work of Andrew Yang [Ya] indicates thatthe 1-level density for the Dedekind zeta functions of cubic fields is governed by thesymplectic distribution. In this paper, we extend the results of [FI] to the family ofall CM-fields over a fixed totally real field (see below for definitions). Since infinitelymany such families exist, we also derive the first lower order term of the 1-level den-sity (under certain conditions), which allows us to distinguish different families by theirarithmetic.1.2. In this paper, K will denote a number field of fixed degree N over Q , h K its class number, ∆ K the absolute value of its discriminant, r and r the numberof real resp. half the number of complex embeddings , and R K the regulator. Although K will vary, we will generally omit the subscriptsfrom our notation; thus h = h K , et cetera. Thus r + 2 r = N . STEVEN J. MILLER AND RYAN PECKNER
Let χ be a character of the ideal class group of K , and let φ be an even function inthe Schwartz space S ( R ) such that the function b φ has compact support; here b φ repre-sents the Fourier Transform b φ ( y ) = Z ∞−∞ φ ( x ) e − πixy dx. (1.4)Assume the generalized Riemann hypothesis, so we may write the zeros of L ( s, χ ) as / iγ χ with γ χ ∈ R . Then Weil’s explicit formula, as simplified by Poitou, reads[Po, BDF, La1] X γ χ φ (cid:18) γ χ log ∆2 π (cid:19) = 1log ∆ (cid:20) δ χ Z ∞ b φ (cid:18) x log ∆ (cid:19) cosh ( x/ dx + b φ (0)(log ∆ − N γ EM − N log 8 π − r π − X p log N p ∞ X m =1 b φ (cid:18) m log N p log ∆ (cid:19) N p m/ ( χ ( p ) m + χ ( p ) − m )+ r Z ∞ b φ (0) − b φ ( x )2 cosh ( x/ dx + N Z ∞ b φ (0) − b φ ( x )2 sinh ( x/ dx , (1.5)where the sum on the left is over the imaginary parts γ χ of the zeros of L ( s, χ ) , thesum on the right is over the prime ideals of the ring of integers of K , γ EM is the Euler-Mascheroni constant and δ χ is the indicator of the trivial character (i.e., it is 1 if χ isthe trivial character and 0 otherwise). As is standard, we rescaled the zeros by log ∆ tofacilitate applications to studying the zeros near the central point.We now wish to average this formula over all characters χ of the ideal class group CL ( K ) of K . We denote its dual by \ CL ( K ) , and note that its cardinality is the classnumber h . By χ ( p ) we of course mean the value of χ on the ideal class of p . For anynon-zero integer m and any prime p of K we have X χ ∈ \ CL ( K ) χ ( p ) m = ( h if p is principal h if p is not principal and m | ord CL ( K ) ( p )0 otherwise. (1.6) Note other works may use a different normalization, using e − ixy instead of e − πixy . OW-LYING ZEROS OF NUMBER FIELD L -FUNCTIONS 5 Averaging the explicit formula over the family yields the one-level density D \ CL ( K ) ( φ ) := 1 h X χ ∈ \ CL ( K ) X γ χ L (1 / iγ χ , χ )=0 φ (cid:18) γ χ log ∆2 π (cid:19) = 1log ∆ " h Z ∞ b φ (cid:18) x log ∆ (cid:19) cosh (cid:16) x (cid:17) dx + b φ (0) · (cid:16) log ∆ − N γ EM − N log 8 π − r π (cid:17) − X p non − principal log N p X m ≥ p m principal b φ (cid:16) m log N p log ∆ (cid:17) N p m/ + X p principal log N p ∞ X m =1 b φ (cid:16) m log N p log ∆ (cid:17) N p m/ + r Z ∞ b φ (0) − b φ ( x )2 cosh( x/ dx + N Z ∞ b φ (0) − b φ ( x )2 sinh( x/ dx . (1.7)We wish to ascertain the behavior of this average as ∆ → ∞ .We recall some relevant facts from algebraic number theory (see Chapter 4, Part 1of [La1] or [Wa] for more details). A number field K is called totally real if everyembedding of K into C has image contained in R , i.e. K is generated over Q byan algebraic number all of whose conjugates are real. On the other hand, a numberfield K is called totally imaginary if no embedding of K into C has image contained in R . A CM-field is a totally imaginary number field which forms a quadratic extensionof a totally real number field. This totally real field is unique and is denoted K + . K then takes the form K = K + ( √ β ) , where β is a square-free element of O K + whichis totally negative, e.g. σ ( β ) < for every embedding σ : K + ֒ → R . Any totallyreal field obviously has infinitely many CM-fields over it, and CM-fields form a richand abundant class of number fields. Indeed, any finite abelian extension of Q is eithertotally real or is a CM-field (by the Kronecker-Weber theorem), and the abbreviationCM reflects the strong connection between CM-fields and the theory of abelian varietieswith complex multiplication (see IV.18 of [Sh] for details).We now describe our family of number fields. Fix a totally real number field K / Q of class number one and degree N over Q , and let { K ∆ } be the family of all CM-fieldsfor which K +∆ = K , ordered by (absolute value of) discriminant ∆ . Although it maybe the case that several K share the same value of ∆ , there are by standard results onlyfinitely many which do ([La1], pg. 121), so their ordering is irrelevant. Each of thesefields has degree N over Q . We denote the class number of K ∆ by h ∆ .Define distributions S (∆ , · ) , S (∆ , · ) by S (∆ , φ ) := − X p non − principal log N p X m ≥ p m principal b φ (cid:16) m log N p log ∆ (cid:17) N p m/ S (∆ , φ ) := − X p principal log N p ∞ X m =1 b φ (cid:16) m log N p log ∆ (cid:17) N p m/ ; (1.8) STEVEN J. MILLER AND RYAN PECKNER note the m -sum for S (∆ , φ ) starts at and not because p is not principal but p m is.In terms of this notation, (1.7) yields Theorem 1.1 (Expansion for the 1-level density) . Notation as above, if φ is an evenSchwartz function with supp( b φ ) ⊂ ( − σ, σ ) , then D \ CL ( K ) ( φ ) := 1 h ∆ X χ ∈ \ CL ( K ∆ ) X γ χ L (1 / iγ χ , χ )=0 φ (cid:18) γ χ log ∆2 π (cid:19) = 1log ∆ " h ∆ Z ∞ b φ (cid:18) x log ∆ (cid:19) cosh (cid:16) x (cid:17) dx + b φ (0) · (log ∆ − N γ EM − N log 8 π )+ S (∆ , φ ) + S (∆ , φ ) + 2 N Z ∞ b φ (0) − b φ ( x )2 sinh( x/ dx . (1.9)Note that we’ve used r = 0 , since K is totally imaginary.1.3. Main results.
Our first result is the following.
Theorem 1.2.
Assume the Generalized Riemann Hypothesis for all Hecke L -functions.Let φ be an even Schwartz function whose Fourier transform is supported in ( − , .Fix a normal, totally real number field K / Q of class number one and degree N over Q , and let { K ∆ } be the family of all CM-fields for which K +∆ = K , ordered by theabsolute value of the discriminant ∆ . Then D \ CL ( K ) ( φ ) = b φ (0) − φ (0) + O (cid:18) log log ∆log ∆ (cid:19) , (1.10) which implies that the one-level density agrees with the scaling limit of symplectic butnot unitary or orthogonal matrices (see (1.2) ). Frequently in computing 1-level densities of families, we are able to improve oursupport or isolate lower order terms if we restrict to a sub-family of the original familywhich is more amenable to averaging. See for instance the results of Gao [Gao] andMiller [Mil4] for sub-families of the family of quadratic Dirichlet characters with evenfundamental discriminants at most X , or [Mil3] for families of elliptic curves. Thesituation is similar here; to derive the lower order terms of the 1-level density, we makethe additional assumption that the class number of K in the narrow sense is 1. Recallthat the narrow class group of K is defined similarly to the ordinary ideal class group,except that ideals are considered equivalent if and only if they differ by a totally positiveelement of K rather than an arbitrary one.By restricting the family of number fields we study a little bit, we are able to isolatethe first lower order term, which depends on the arithmetic of the field. Theorem 1.3 (First Lower Order Term) . Assume the Generalized Riemann Hypothesisfor all Hecke L -functions. Let φ be an even Schwartz function whose Fourier transform The sub-family studied is { d : 0 < d ≤ X ; d an odd, positive square-free fundamentaldiscriminant } ; this extra restriction facilitates the application of Poisson summation. OW-LYING ZEROS OF NUMBER FIELD L -FUNCTIONS 7 is supported in ( − , . Fix a normal, totally real number field K / Q whose classnumber in the narrow sense is 1, and let { K ∆ } be the family of all CM-fields of oddclass number (in the usual sense) for which K +∆ = K , ordered by the absolute value ofthe discriminant ∆ . For a number field E/ Q , let ρ E be the residue of its Dedekind zetafunction at the simple pole s = 1 ρ E = res s =1 ζ E ( s ) = 2 r (2 π ) r h E R E w E p | D E/ Q | , (1.11) and let γ E denote its Euler constant γ E = dds [( s − ζ E ( s )] s =1 = lim s → (cid:18) ζ E ( s ) − ρ E s − (cid:19) . (1.12) Let γ EM be the Euler-Mascheroni constant. Then the 1-level density is given by D \ CL ( K ) ( φ ) = b φ (0) − φ (0) + 1log ∆ (cid:16) b φ (0) τ (∆) + L (∆) (cid:17) + O (cid:18) ∆ (cid:19) (1.13) where L (∆) = 4 h ∆ Z ∞ b φ (cid:18) x log ∆ (cid:19) cosh (cid:16) x (cid:17) dx + b φ (0) · ( − N γ EM − N log 8 π )+2 N Z ∞ b φ (0) − b φ ( x )2 sinh( x/ dx (1.14) and τ (∆) = 4 γ K ρ K − γ K ρ K − X q ⊂O K inert in K log N q N q − . (1.15) Moreover, τ (∆) = O (1) , with the implied constant depending on K . Remark 1.4.
As is common in many families of L -functions (see for example [FI, HKS,Mil2, Mil3, Mil4, Mil5, MilMo, Ya, Yo1] ), the main term in the 1-level density is inde-pendent of the arithmetic of the family, which only surfaces in the lower order terms. This paper is organized as follows. After analyzing part of the first lower order term,we prove a lemma on CM-fields that allows us to bound sums over principal primes ofdegree 1. We proceed to reduce sums over K to sums over K , which are then han-dled using standard algebraic number theory. To deal with sums over degree 2 primes,we introduce a variant of the Dedekind zeta function of K and show that integrationagainst its logarithmic derivative yields the desired quantities (up to reasonably smallerror), from which we obtain the result. In Section 3, we restrict our class of numberfields in order to obtain complete control of the ramification behavior, which allowsus to reduce the error terms significantly. We then extract the first lower order termby closely studying the arithmetic of the families in question, in the process proving adiscriminant-independent bound on number field Euler constants that we haven’t seenelsewhere in the literature (see Proposition 3.3 and Appendix A). STEVEN J. MILLER AND RYAN PECKNER
2. P
ROOF OF T HEOREM S i (∆ , φ ) are readily analyzed. To see this, we first need a lemmarelating the size of h ∆ to ∆ . Lemma 2.1.
We have log h ∆ ∼ log ∆ as ∆ → ∞ .Proof. Since the fields K ∆ all have the same degree over Q , we have by the Brauer-Siegel Theorem ([La1], Chapter XVI) that log( h ∆ R ∆ ) ∼
12 log ∆ as ∆ → ∞ . (2.1)The regulator R ∆ satisfies ([Wa], pg. 41) R ∆ R K + = 1 Q N − (2.2)where Q = 1 or , and therefore R ∆ is bounded by a constant independent of ∆ . Thisproves the claim. (cid:3) Lemma 2.2.
Assume supp( b φ ) ⊂ ( − σ, σ ) with σ < . Then the terms involving cosh and sinh in Theorem 1.1 are O (1 / log ∆) .Proof. The last two terms, where the hyperbolic trig functions are in the denominator,are readily analyzed. As cosh( x/ ≫ and decays exponentially, the integrand with cosh in the denominator is O (1) . The sinh integral is handled similarly (note everythingis well-behaved near x = 0 because φ is differentiable, and by L’Hopital’s rule thequotient is bounded near x = 0 ).We are left with handling the integral of b φ against cosh . Changing variables ( u = x/ log ∆ ) gives h ∆ log ∆ Z ∞ b φ (cid:18) x log ∆ (cid:19) cosh (cid:16) x (cid:17) dx = 4 h ∆ Z ∞ b φ ( u ) cosh (cid:18) u log ∆2 (cid:19) du. (2.3)Using t ) = e t + e − t , we see this integral is dominated by h ∆ Z ∞ (cid:12)(cid:12)(cid:12) b φ ( u ) (cid:12)(cid:12)(cid:12) ∆ u/ du ≪ σ ∆ σ/ h ∆ , (2.4)which tends to zero by Lemma 2.1 as σ < . (cid:3) Thus, by the above lemma, the asymptotic behavior of F (∆ , φ ) for fixed φ is deter-mined by that of S and S . While the hyperbolic integrals will contribute lower orderterms of size / log ∆ , the values of these integrals are independent of the family. In what follows, we drop ∆ from our number field notation;thus K = K ∆ , h = h ∆ , et cetera. Before analyzing S and S , we first prove some lemmas on CM-fields which will beessential in our investigations. OW-LYING ZEROS OF NUMBER FIELD L -FUNCTIONS 9 Lemmas on CM-fields.
Just as in the case of quadratic fields over Q , one easilyproves the following. Lemma 2.3.
Let K be a CM-field and β ∈ O K + a totally negative, square-free elementsuch that K = K + ( √ β ) . Then either O K = O K + [ p β ] or O K = O K + (cid:20) √ β (cid:21) . Indeed, the minimal polynomial of an element α = x + y √ β ∈ K, x, y ∈ K + over K + is t − xt + x − βy so by transitivity of integral closure, α ∈ O K if and only if x, x − βy ∈ O K + . Thetwo possibilities of the lemma then correspond to whether x ∈ O K + or x ∈ O K + .The following lemma is crucial, as it allows us to bound sums over degree 1 principalprimes (by showing the sums are vacuous if the support is restricted as in Theorem 1.2). Lemma 2.4.
Let K be a CM-field with maximal real subfield K + . Choose β ∈ K + which is totally negative and such that K = K + ( √ β ) . Let p ⊂ O K be a principalprime ideal of degree 1 with norm N p = p . Then p ≥ C ∆ , where C is a constantdepending only on K + .Proof. We assume that O K = O K + [ √ β ] ; the other case is similar. We first claim that p ≥ | N K + Q ( β ) | . Since p is principal, there exist x, y ∈ O K + such that p = ( x + y √ β ) .Suppose y = 0 ; then N p := N K Q ( p ) = N K + Q ( N KK + ( p ))= N K + Q ( x )= N K + Q ( x ) which is a contradiction since p = N p is a prime number ( | N K + Q ( x ) | > because x can’t be a unit). Thus y = 0 .Assume now y = 0 . Recall the minimal polynomial of x + y √ β over K + is t − xt + x − βy , (2.5)so N KK + ( p ) = N KK + ( x + y √ β ) = x − βy . Hence, since the degree is multiplicativeover towers, p = | N K + Q ( x − βy ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Y σ : K + → C σ ( x − βy ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Y σ : K + → C ( σ ( x ) − σ ( β ) σ ( y ) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (2.6) We now use our assumption that β is totally negative, which implies that σ ( β ) < foreach σ . We have − σ ( β ) = | σ ( β ) | and so (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Y σ ( σ ( x ) − σ ( β ) σ ( y ) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Y σ ( σ ( x ) + | σ ( β ) | σ ( y ) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (2.7)Since x, y ∈ K + and K + is totally real, we have σ ( x ) , σ ( y ) ∈ R for each σ . Therefore σ ( x ) ≥ , σ ( y ) > and so (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Y σ ( σ ( x ) + | σ ( β ) | σ ( y ) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = Y σ ( σ ( x ) + | σ ( β ) | σ ( y ) ) ≥ Y σ σ ( x ) + Y σ | σ ( β ) | σ ( y ) = N K + Q ( x ) + | N K + Q ( β ) | · N K + Q ( y ) . (2.8)Since y = 0 and y ∈ O K + , N K + Q ( y ) is a positive integer. Thus the last expression is atleast | N K + Q ( β ) | , which proves the claim.By the relative discriminant formula, and since [ K : K + ] = 2 , we have D K/ Q = N K + Q ( D K/K + ) · D K + / Q (2.9)where for an extension of number fields K/E , D K/E denotes the relative discriminant(which we take to be an integer if E = Q , although it is an ideal of O E in general).Since D K/K + = (4 β ) , we have N K + Q ( D K/K + ) = 4 N | N K + Q ( β ) | . Therefore, by the aboveclaim, we have p ≥ | N K + Q ( β ) | = | D K/ Q | N D K + / Q = ∆4 N D K + / Q (2.10)Finally, note that / (4 N D K + / Q ) depends only on K + . (cid:3) In particular, since in our setting K + = K is fixed, we see that C is independentof ∆ . This observation will be crucial in what follows, in that it allows us to assertthe vacuity of certain sums since they only involve primes whose norms lie outside thesupport of b φ . Remark 2.5.
The CM structure is crucial to obtain such a strong lower bound on thenorm of degree 1 principal primes. In general, the results of Lagarias, Montgomeryand Odlyzko [LMO] and Oesterlé [Oe] guarantee that for
L/K a Galois extension ofnumber fields, there exists a prime p of K of norm at most | D L/ Q | ) . One musttherefore avoid number fields with extensions of small discriminant in order to obtainsuch a bound. Evaluation of S .Lemma 2.6. Assume supp( b φ ) ⊂ ( − σ, σ ) . If σ < , we have S (∆ , φ ) = O (log log ∆) as ∆ → ∞ . (2.11) OW-LYING ZEROS OF NUMBER FIELD L -FUNCTIONS 11 Proof.
First, we claim that S (∆ , φ ) = − X p non − principal p principal log N p N p b φ (cid:18) N p log ∆ (cid:19) + O (1) . (2.12)Indeed, since ˆ φ is bounded, and since each rational prime p has at most N prime idealslying over it in K , the sum X p non − principal log N p ∞ X m =3 b φ (cid:16) m log N p log ∆ (cid:17) N p m/ (2.13)is bounded by a constant times a convergent series, namely X p X m log pp m ≪ X p log pp ≪ . (2.14)This proves (2.12).For K/E an extension of number fields and p a prime ideal of O K , we denote by f K/E ( p ) the residue degree of p over E , so that N KE ( p ) = q f K/E ( p ) , where q = p ∩ O E .Notice that X p non − principal p principal log N p N p b φ (cid:18) N p log ∆ (cid:19) = X p non − principal p principal f K/ Q ( p )=1 log N p N p b φ (cid:18) N p log ∆ (cid:19) + O (1) since the complementary sum is again bounded up to a constant by the convergent series P p log pp . By the compact support of b φ , we have X p non − principal p principal f K/ Q ( p )=1 log N p N p b φ (cid:18) N p log ∆ (cid:19) = X p non − principal p principal f K/ Q ( p )=1log N p < σ log ∆2 log N p N p b φ (cid:18) N p log ∆ (cid:19) . (2.15)Let p be a prime of degree 1 over Q such that p is principal, say p = ( α ) . Either α ∈ O K + or α ∈ O K \ O K + . Denote these contributions by S , (∆ , φ ) and S , (∆ , φ ) .Suppose first that α ∈ O K + . Then α O K + is a prime ideal of O K + since N K/ Q ( p ) = N K + / Q ( α ) , and it ramifies in K . Therefore, since f K/ Q ( p ) = 1 implies that p = N p is a rational prime, p ramifies in K . As the ramified rational primes in K are preciselythose dividing ∆ , we find S , (∆ , φ ) := X p non − principal p =( α ) ,α ∈O K + f K/ Q ( p )=1log N p < σ log ∆2 log N p N p b φ (cid:18) N p log ∆ (cid:19) (2.16) ≪ X pp | ∆ log pp = O (log log ∆) , (2.17) where we used the standard fact that P p | ∆ log pp ≪ log log ∆ .Now consider the case when α ∈ O K \O K + . Let S , (∆ , φ ) := X p non − principal p =( α ) ,α ∈O K \O K + f K/ Q ( p )=1log N p < σ log ∆2 log N p N p b φ (cid:18) N p log ∆ (cid:19) . (2.18)In this situation, we have N K/ Q ( p ) = N K/ Q ( α ) , so the proof of Lemma 2.4 shows that N p ≥ C √ ∆ , where C is a positive constant independent of ∆ . Hence, since σ < ,the condition log N p < σ log ∆2 on the sum implies that S , (∆ , φ ) is zero for sufficientlylarge ∆ . Putting things together, we have for σ < that S (∆ , φ ) = S , (∆ , φ ) + S , (∆ , φ ) + O (1) = O (log log ∆) , (2.19)which proves the claim. (cid:3) Reduction of S . In this subsection we replace S with sums which are easier toevaluate. We determine those sums in the next subsection, which will complete theanalysis of S .We write S as a sum S (∆ , φ ) = S , (∆ , φ ) + S , (∆ , φ ) (2.20)where S , (∆ , φ ) := − X p principal log N p X m ≥ m,h ∆ )=1 b φ (cid:16) m log N p log ∆ (cid:17) N p m/ S , (∆ , φ ) := − X p principal log N p X m ≥ m,h ∆ ) > b φ (cid:16) m log N p log ∆ (cid:17) N p m/ . (2.21)Note that the proof of Lemma 2.6 did not actually use the non-principality of the primeideals involved in the sum, but only the fact that the primes have principal square, aswell as Lemma 2.4 and the fact that the sum began at m = 2 . Since the principalityof p of course implies the principality of p , and since the condition ( m, h ) > in thedefinition of S , (∆ , φ ) implies that the sum again begins at least at m = 2 , the sameargument given in Lemma 2.6 shows that S , (∆ , φ ) ≪ X pp | ∆ log pp = O (log log ∆) . (2.22) Note log uu is decreasing for u ≥ , so the sum is maximized when ∆ is a primorial. If · · · · p r = ∆ then p r ∼ log ∆ , and the claim follows from partial summation. OW-LYING ZEROS OF NUMBER FIELD L -FUNCTIONS 13 We now analyze S , (∆ , φ ) . Note that S , (∆ , φ ) = − X p principal f K/ Q ( p ) ≤ log N p X m ≥ m,h ∆ )=1 b φ (cid:16) m log N p log ∆ (cid:17) N p m/ + O (1) (2.23)since, as before (see Lemma 2.6), the sum X p principal f K/ Q ( p ) > log N p X m ≥ m,h ∆ )=1 b φ (cid:16) m log N p log ∆ (cid:17) N p m/ isbounded by a convergent series. Moreover, observe that X p principal f K/ Q ( p )=1 log N p X m ≥ m,h ∆ )=1 b φ (cid:16) m log N p log ∆ (cid:17) N p m/ = X p principal f K/ Q ( p )=1 N p < ∆ σ log N p X m ≥ m,h ∆ )=1 b φ (cid:16) m log N p log ∆ (cid:17) N p m/ , (2.24)and if σ < then this sum is zero for sufficiently large ∆ by Lemma 2.4. Thus, letting S , (∆ , φ ) = − X p principal f K/ Q ( p )=2 log N p X m ≥ m,h ∆ )=1 b φ (cid:16) m log N p log ∆ (cid:17) N p m/ , (2.25)we find that S , (∆ , φ ) = S , (∆ , φ ) + O (1) (2.26)and so, by (2.20) and (2.22), we find that S (∆ , φ ) = S , (∆ , φ ) + O (log log ∆) . (2.27) Proposition 2.7.
We have S , (∆ , φ ) = − X p principal f K/ Q ( p )=2 log N p N p / b φ (cid:18) log N p log ∆ (cid:19) + O (1) . (2.28) Proof.
Let A (∆ , φ ) be the difference between S , (∆ , φ ) and the main term on theright hand side of (2.28). Thus A (∆ , φ ) = − X p principal f K/ Q ( p )=2 log N p X m ≥ m,h ∆ )=1 b φ (cid:16) m log N p log ∆ (cid:17) N p m/ . (2.29) Since b φ is bounded and N p ≥ , we have A (∆ , φ ) ≪ X p principal f K/ Q ( p )=2 log N p ∞ X m =2 N p m/ ≪ X p principal f K/ Q ( p )=2 log N p N p , (2.30)where the last statement is derived by summing the geometric series. Since each rationalprime p has at most N prime ideals of degree 2 lying above it in K , we find A (∆ , φ ) ≪ X p N log pp . (2.31)This sum is convergent, since it is dominated by a convergent series. Hence A (∆ , φ ) = O (1) as claimed. (cid:3) We now express S , (∆ , φ ) in terms of primes of K + . Proposition 2.8.
We have S , (∆ , φ ) = − " X q ⊂O K + q inert in Kf K + / Q ( q )=1 log N q N q b φ (cid:18) N q log ∆ (cid:19) + O (log log ∆) . (2.32) Proof.
Let M (∆ , φ ) be the main term in the expression for S , (∆ , φ ) given by Propo-sition 2.7: M (∆ , φ ) = − X p principal f K/ Q ( p )=2 log N p N p / b φ (cid:18) log N p log ∆ (cid:19) . (2.33)Divide this sum by degree over K + : M (∆ , φ ) = − " X p principal f K/K + ( p )= f K/ Q ( p )=2 log N p N p / b φ (cid:18) log N p log ∆ (cid:19) + X p principal f K/K + ( p )=1 ,f K/ Q ( p )=2 log N p N p / b φ (cid:18) log N p log ∆ (cid:19) := M (∆ , φ ) + M (∆ , φ ) . (2.34)For M (∆ , φ ) , f K/K + ( p ) = 1 implies that q = p ∩ O K + either splits or is ramified in K . It follows as before from Lemma 2.4 that the contribution from split primes is zerofor large enough ∆ as supp( b φ ) ⊂ ( − , . The contribution from those p which lie over OW-LYING ZEROS OF NUMBER FIELD L -FUNCTIONS 15 ramified primes in K + and for which f K/ Q ( p ) = 2 is bounded (up to a constant) by X p | ∆ log pp ≪ log log ∆ . (2.35)Therefore M (∆ , φ ) = O (log log ∆) .Denote the main term in (2.32) by M ′ (∆ , φ ) , so M ′ (∆ , φ ) := − X q ⊂O K + q inert in K f K + / Q ( q )=1 log N q N q b φ (cid:18) N q log ∆ (cid:19) . (2.36)As M (∆ , φ ) = O (log log ∆) it suffices to show M ′ (∆ , φ ) = M (∆ , φ ) to completethe proof.Let q be a prime of K + of degree 1 over Q that is inert in K . Then, since h K + =1 , p = q O K is principal. Moreover, f K/K + ( p ) = f K/ Q ( p ) = 2 and N p = N q .Conversely, if p is a prime of K such that f K/K + ( p ) = f K/ Q ( p ) = 2 , then q = p ∩ O K + has degree 1 over Q and is inert in K . Therefore M ′ (∆ , φ ) = − " X p ⊂O K principal f K/K + ( p )= f K/ Q ( p )=2 log( N p / ) N p / b φ (cid:18) N p / )log ∆ (cid:19) = − " X p principal f K/K + ( p )= f K/ Q ( p )=2 log N p N p / b φ (cid:18) log N p log ∆ (cid:19) = M (∆ , φ ) . (2.37)Hence, S , (∆ , φ ) = M (∆ , φ ) + O (1) = M (∆ , φ ) + M (∆ , φ ) + O (1) = M ′ (∆ , φ ) + O (log log ∆) , as claimed. (cid:3) Evaluation of S . We now complete the analysis of S . Let χ be the unique non-trivial character of G := Gal ( K/K + ) . For q a prime of K + unramified in K , define χ ( q ) := χ (cid:18)(cid:18) q K/K + (cid:19)(cid:19) where (cid:18) q K/K + (cid:19) is the Artin symbol. Thus χ ( q ) = (cid:26) − if q is inert in K if q splits in K. (2.38)The Artin L-function associated to χ is L ( s, χ ) = Y q unramified in K (cid:18) − χ ( q ) N q s (cid:19) − . (2.39)Since χ is the character of a non-trivial one-dimensional representation of G , L ( s, χ ) isentire and has no zeros on the line ℜ s = 1 . Define a function U ( s ) by U ( s ) = ( s − ζ K + ( s ) L ( s, χ ) ζ ram ( s ) . (2.40) Here ζ ram ( s ) is given by the partial Euler product for ζ K + ( s ) restricted to those primeswhich ramify in K . One has ([La1], pg. 161) that ζ K + ( s ) is analytic for ℜ s > − /N except for a simple pole at s = 1 . Since the factor of ( s − cancels this pole, U ( s ) isanalytic for ℜ s > − /N . In this region, we have U ( s ) = ( s − Y q inert in K (cid:18) N q s − N q s + 1 (cid:19) − . (2.41)Therefore, for ℜ s > − /N one has U ′ U ( s ) = 1 s − − X q inert in K ∞ X m =0 log N q ( N q s ) m +1 . (2.42)Consider the integral Z ∞−∞ φ ( x ) U ′ U (cid:18) πix log ∆ (cid:19) dx. (2.43)We substitute the expansion from (2.42) above. The first piece is the integral Z ∞−∞ φ ( x ) log ∆ dx πix = log ∆2 12 πi Z ∞−∞ φ ( x ) dxx , (2.44)which is just φ (0) log ∆ from complex analysis. The second piece becomes the in-tegral of φ ( x ) against factors such as ( N q ) s (2 m +1) with s = 1 + πix log ∆ . The integrationagainst x gives the Fourier transform of φ . Specifically, these terms contribute φ (0) log ∆ − X q inert in K ∞ X m =0 log N q N q m +1 b φ (cid:18) m + 1) log N q log ∆ (cid:19) , (2.45)where φ (0) log ∆ appears as half the residue of φ ( s ) s − log ∆ at s = 0 . Similarly tothe above, one has X q inert in K ∞ X m =0 log N q N q m +1 b φ (cid:18) m + 1) log N q log ∆ (cid:19) = X q inert in K log N q N q b φ (cid:18) N q log ∆ (cid:19) + O (1)= X q inert in Kf K + / Q ( q )=1 log N q N q b φ (cid:18) N q log ∆ (cid:19) + O (1) . (2.46)Therefore, by Proposition 2.8, we have shown Lemma 2.9. S , (∆ , φ ) = − φ (0) log ∆ + 2 Z ∞−∞ φ ( x ) U ′ U (cid:18) πix log ∆ (cid:19) dx + O (log log ∆) . (2.47) Remember that φ is an even function. The extra factor of / is due to the pole lying on the line ofintegration. OW-LYING ZEROS OF NUMBER FIELD L -FUNCTIONS 17 Write U ′ U ( s ) = 1 s − ζ ′ K + ζ K + ( s ) − L ′ L ( s, χ ) − ζ ′ ram ζ ram ( s ) . (2.48)We have the following important fact (Theorem 5.17 of [IK]). Theorem 2.10.
Assume the Generalized Riemann Hypothesis. Let L ( s, ρ ) be the Artin L -function associated to a (possibly trivial) one-dimensional representation ρ of G . Let r be the order of the pole of this L -function at s = 1 , and let q ( χ, s ) be the analyticconductor of the associated Hecke character. Then − L ′ L (1 + it, ρ ) = rs − O (log log q ( χ, s )) , (2.49) the implied constant being absolute. In our situation, we have a factorization of the Dedekind zeta-function of K just asin the case of imaginary quadratic fields: ζ K ( s ) = ζ K + ( s ) L ( s, χ ) , (2.50)which may be proven by checking the local factors at each prime ideal of K . Thusevery rational prime dividing q ( χ ) (the ordinary conductor) must also divide ∆ . Butwe also have q ( χ ) = | D K + / Q | N K + Q f ( χ ) for an integral ideal f ( χ ) of K + ([IK], pg.142), and since each prime in the factorization of this ideal has degree at most N over Q , we find q ( χ ) ≤ | D K + / Q | ∆ N . Thus, since | D K + / Q | is independent of ∆ , we find q ( χ, s ) ≪ ∆ N | s | N . Since L ( s, χ ) is entire, we therefore obtain by Theorem 2.10 theestimates − ζ ′ K + ζ K + (1 + it ) = 1 s − O (log log(∆ | t | N )) − L ′ L (1 + it, χ ) ≪ log log(∆ N | t | N ) . (2.51)Combining these estimates with the fact that ζ ′ ram ζ ram (1 + it ) ≪ log log ∆ (2.52)(use X p | ∆ log pp ≪ log log ∆ ), one finds since φ is Schwartz that Z ∞−∞ φ ( x ) U ′ U (cid:18) πix log ∆ (cid:19) dx ≪ log log ∆ , (2.53)where the implied constant depends only on φ and N . Combined with the previouslemma, this proves Lemma 2.11.
We have S , (∆ , φ ) = − φ (0) log ∆ + O (log log ∆) . (2.54) Thus, by (2.27) , we have S (∆ , φ ) = − φ (0) log ∆ + O (log log ∆) (2.55) as well. We are now ready to prove the main theorem.2.5.
Proof of Theorem 1.2.
Our main result trivially follows from our analysis of S and S . Proof of Theorem 1.2.
By (1.9), we have D \ CL ( K ) ( φ ) = 1log ∆ " h ∆ Z ∞ b φ (cid:18) x log ∆ (cid:19) cosh (cid:16) x (cid:17) dx + b φ (0) · (cid:16) log ∆ − N γ EM − N log 8 π (cid:17) + S (∆ , φ ) + S (∆ , φ )+2 N Z ∞ b φ (0) − b φ ( x )2 sinh( x/ dx . (2.56)By Lemmas 2.2, 2.6 and 2.11, and since N is fixed and r ≤ N , this entire expressionequals (cid:20) b φ (0) log ∆ − φ (0) log ∆ + O (log log ∆) (cid:21) , (2.57)which completes the proof. (cid:3)
3. L
OWER O RDER T ERMS
In this section, we prove Theorem 1.3, which gives the lower order terms for a sub-family of our original family. Similar to investigations of the 1-level density in otherfamilies (such as [Gao, Mil4]), we are able to isolate lower order terms if we restrictto a sub-family which simplifies some of the terms. To derive the lower order terms ofthe 1-level density, we make the additional assumption that the class number of K inthe narrow sense is 1 (recall that the narrow class group of K is defined similarly tothe ordinary ideal class group, except that ideals are considered equivalent if and onlyif they differ by a totally positive element of K rather than an arbitrary one). We willmake use of the following facts, which rephrase Theorems 1 and 2 of [Ho]. Proposition 3.1.
The family { K ∆ } of CM-fields for which K + = K contains infinitelymany fields of odd class number (in the usual sense). Thus we may consider { K ∆ : 2 ∤ h ∆ } as a sub-family of { K ∆ } . Unless otherwise stated, K = K ∆ denotes a CM-field of oddclass number such that K + = K . Proposition 3.2.
Let K be a CM-field such that K + has class number 1, and supposethat the class number of K is odd. Then at most one finite prime of K + ramifies in K . OW-LYING ZEROS OF NUMBER FIELD L -FUNCTIONS 19 Writing K = K + ( √ β ) , this implies that the relative discriminant D ( K/K + ) is divis-ible by at most one prime of O K + , which we denote q K/K + = q . Since the CM-fields K for which O K = O K + [ √ β ] have discriminant (4 β ) , which is divisible by more than oneprime, any K as in the proposition must have ring of integers O K = O K + (cid:20) √ β (cid:21) and relative discriminant D K/K + = ( β ) . Since β is square-free and h K + = 1 , theproposition then implies that D K/K + is prime. Arguing as in the end of the proof ofLemma 2.4, we moreover have N K + Q ( D K/K + ) = | N K + Q ( β ) | = ∆ D K + / Q . (3.1)Thus the contribution from the ramified prime of O K + to terms like log N q N q / is O (cid:18) log ∆∆ / (cid:19) ,where the implied constant depends only on K + = K . Since we’re only interested interms of size , we may therefore ignore the ramified prime in what follows.3.1. Evaluation of S (Redux). With all notation as before, we again consider S (∆ , φ ) .Our goal is to improve the calculation to terms of size / log ∆ . Recall (cf. (2.12)) that S (∆ , φ ) = − X p non − principal p principal log N p N p b φ (cid:18) N p log ∆ (cid:19) + O (1) . (3.2) Since now the class number of K is odd , no non-principal prime has principal square,so in fact S (∆ , φ ) = − X p non − principal log N p X m ≥ p m principal b φ (cid:16) m log N p log ∆ (cid:17) N p m/ . (3.3)Observe that if p is non-principal, then f K/K + ( p ) = 1 , since otherwise p lies overan inert prime of K + and so must be principal since h K + = 1 . Let m > be aninteger such that p m is principal. Let p m = α O K , and suppose α ∈ O K + . Then N KK + ( p m ) = ( α ) . Since f K/K + ( p ) = 1 , the ideal q = N KK + ( p ) of O K + is prime, sounique factorization into primes implies that m must be even. Consequently, since thefact that h K is odd implies that the order d of p in CL ( K ) must be odd as well, wemust have α ∈ O K \O K + if p d = ( α ) . Hence, we may write α = x + y √ β , where x, y ∈ O K + and y = 0 . Thus N K Q ( p d ) = | N K Q ( α ) | , (3.4)so the proof of Lemma 2.4 implies that N K Q ( p ) ≥ ( C ∆) /d , (3.5)where C depends only on K + = K .Since p m is principal if and only if d | m , we have (writing d = d p to specify the prime), S (∆ , φ ) = − X p non − principal log N p ∞ X k =1 b φ (cid:16) d p k log N p log ∆ (cid:17) N p d p k/ = − X p non − principallog N p < σ log ∆ d p log N p ∞ X k =1 b φ (cid:16) d p k log N p log ∆ (cid:17) N p d p k/ (3.6)so (3.5) and the fact that σ < imply that S (∆ , φ ) = 0 for sufficiently large ∆ becausethe sum is vacuous.3.2. Evaluation of S (Redux). We have S (∆ , φ ) = − X p principal log N p ∞ X m =1 b φ (cid:16) m log N p log ∆ (cid:17) N p m/ . (3.7)As argued above, the contribution from the ramified prime is negligible, while the con-tribution from the primes of degree 1 over K is ultimately zero. Consequently, for ∆ large enough, we have (up to the O (cid:0) log ∆∆ / (cid:1) error from the ramified prime) S (∆ , φ ) = − X p principal f K/K ( p )=2 log N p ∞ X m =1 b φ (cid:16) m log N p log ∆ (cid:17) N p m/ = − X q ⊂O K inert in K log N q ∞ X m =1 b φ (cid:16) m log N q log ∆ (cid:17) N q m . (3.8)Recall from Section 2.4 that Z ∞−∞ φ ( x ) U ′ U (cid:18) πix log ∆ (cid:19) dx = 14 φ (0) log ∆ − X q ⊂O K inert in K X m ≥ log N q N q m b φ (cid:18) m log N q log ∆ (cid:19) . (3.9) OW-LYING ZEROS OF NUMBER FIELD L -FUNCTIONS 21 Thus, using Lemma 2.4 and the fact that contribution from the ramified prime is negli-gible, we have by the compact support of b φS (∆ , φ ) = − X q ⊂O K inert in K log N q ∞ X m =1 b φ (cid:16) m log N q log ∆ (cid:17) N q m = − φ (0) log ∆ + 2 Z ∞−∞ φ ( x ) U ′ U (cid:18) πix log ∆ (cid:19) dx − X q ⊂O K inert in K log N q X m ≥ b φ (cid:16) m log N q log ∆ (cid:17) N q m . (3.10)Therefore, to complete the analysis of the lower-order terms, we must show that Z ∞−∞ φ ( x ) U ′ U (cid:18) πix log ∆ (cid:19) dx − X q ⊂O K inert in K log N q X m ≥ b φ (cid:16) m log N q log ∆ (cid:17) N q m (3.11)equals c K + o (1) , with c K bounded independently of K . Note that in the explicit formulathe terms S (∆ , φ ) and S (∆ , φ ) are multiplied by / log ∆ ; thus if we show the termabove is c K + o (1) , we will have isolated its contribution to the first lower order term.First, note that since the compact support of b φ restricts the sums to be finite, we haveusing Taylor series X q ⊂O K inert in K log N q X m ≥ b φ (cid:16) m log N q log ∆ (cid:17) N q m = b φ (0) X q ⊂O K inert in K log N q X m ≥ N q m + O (cid:18) (cid:19) = b φ (0) X q ⊂O K inert in K log N q N q − O (cid:18) (cid:19) (3.12)and since each prime of O K lies over at most N rational primes, this is dominated bya convergent p -series independent of K , and thus is O (1) .To analyze the integral of φ against the logarithmic derivative of U ( s ) , let β k (∆) denote the k -th coefficient in the power series expansion of the logarithmic derivativeof U ( s ) about s = 1 ; thus U ′ U (cid:18) πix log ∆ (cid:19) = log ∆4 πix + ∞ X k =0 β k (∆) (cid:18) πix log ∆ (cid:19) k . (3.13)To get rid of the term log ∆ / πix , observe that ℑ U ′ U (1 + 4 πix/ log ∆) is an odd func-tion of x , so that Z ∞−∞ φ ( x ) U ′ U (cid:18) πix log ∆ (cid:19) dx = Z ∞−∞ φ ( x ) ℜ U ′ U (cid:18) πix log ∆ (cid:19) dx (3.14) and ℜ U ′ U (cid:18) πix log ∆ (cid:19) = ∞ X k =0 β k (∆) (cid:18) πix log ∆ (cid:19) k . (3.15)Recall from 2.4 that U ( s ) = ( s − ζ K ( s ) L ( s, χ ) ζ ram ( s ) (3.16)and that U ( s ) is analytic and non-zero at s = 1 . A straightforward computation, usingthe fact that L ( s, χ ) = ζ K ( s ) /ζ K ( s ) , then yields β (∆) = U ′ U (1) = 2 γ K ρ K − γ K ρ K + O (cid:18) (cid:19) (3.17)where for a number field E/ Q , ρ E is the residue of its Dedekind zeta function at thesimple pole s = 1 ρ E = res s =1 ζ E ( s ) = 2 r (2 π ) r h E R E w E p | D E/ Q | (3.18)and γ E denotes its Euler constant γ E = dds [( s − ζ E ( s )] s =1 = lim s → (cid:18) ζ E ( s ) − ρ E s − (cid:19) . (3.19)The O (1 / log ∆) term in (3.17) comes from ζ ram ( s ) . We claim that β (∆) = O (1) as ∆ → ∞ , with the implied constant depending only on K .We use the following bound for the number field Euler constant, which is Theorem7 of [MO]. Let E be a number field of degree n over Q , with r real and r complexembeddings. Denote the embeddings K ֒ → K ( i ) , and arrange them in such a waythat K ֒ → K ( i ) is real for ≤ i ≤ r , imaginary for r + 1 ≤ i ≤ r + r , and K ( i + r ) = K ( i ) . Let ǫ , ..., ǫ r be an independent set of generators for the unit group of O E modulo roots of unity, where r = r + r − . Let M be the largest of the values | log | ǫ ( i ) j || for ≤ i, j ≤ r . Also, choose an integral basis β , ..., β n for O E over Q , andlet ( γ ij ) be the inverse of the non-singular matrix ( β ( i ) j ) . Finally, set γ = max i,j | γ ij | .Then we have Proposition 3.3. | γ E | ≤ ρ E (1 + n n max(1 , Φ n )) (3.20) where Φ = 2 n − n n γ n − e rM ( n − . In our setting (e.g. CM-fields of odd class number over a fixed totally real field ofstrict class number 1), the values γ and M , which a priori depend on K = K ∆ , can infact be made independent of ∆ (see Appendix A for justification). Combining this factwith the above proposition and (3.17), as well as the fact that n = [ K : Q ] = 2 N is OW-LYING ZEROS OF NUMBER FIELD L -FUNCTIONS 23 fixed, we find β (∆) = 2 γ K ρ K − γ K ρ K + O (cid:18) (cid:19) ≪ γ K ρ K + ρ K (1 + 2 N N max(1 , Φ N )) ρ K + O (cid:18) (cid:19) = 2 γ K ρ K + 1 + 2 N N max(1 , Φ N ) + O (cid:18) (cid:19) = O (1) (3.21)with the implied constant depending only on K .Now, Z ∞−∞ φ ( x ) U ′ U (cid:18) πix log ∆ (cid:19) dx = Z ∞−∞ φ ( x ) ℜ U ′ U (cid:18) πix log ∆ (cid:19) dx = Z ∞−∞ φ ( x ) ∞ X k =0 β k (∆) (cid:18) πix log ∆ (cid:19) k dx = b φ (0) β (∆) + Z ∞−∞ φ ( x ) ∞ X k =1 β k (∆) (cid:18) πix log ∆ (cid:19) k dx. (3.22)To estimate the integral, observe that β k (∆) = γ k − γ k (∆) + O (cid:18) (cid:19) (3.23)where γ k and γ k (∆) are the coefficients in the power series expansion of the logarithmicderivative of ζ K ( s ) and L ( s, χ ) , respectively, about s = 1 . The Riemann hypothesisfor L ( s, χ ) implies γ k (∆) ≪ (log log ∆) k +1 (3.24)and therefore β k (∆) ≪ (log log ∆) k +1 (3.25)with the implied constant depending on k and K . Hence, from (3.22), we obtain Z ∞−∞ φ ( x ) U ′ U (cid:18) πix log ∆ (cid:19) dx = b φ (0) β (∆) + Z ∞−∞ φ ( x ) ∞ X k =1 β k (∆) (cid:18) πix log ∆ (cid:19) k dx = b φ (0) β (∆) + ∞ X k =1 b φ (2 k ) (0) β k (∆) (cid:18) πi log ∆ (cid:19) k = b φ (0) β (∆) + O (cid:18) (log log ∆) (log ∆) (cid:19) (3.26)with the implied constant depending on φ and K . Finally, combining this with theexpression for the 1-level density given in 1.1, we obtain the full first lower-order term,completing the proof of Theorem 1.3. A PPENDIX A. ∆ - INDEPENDENCE IN P ROPOSITION ∆ -independence alluded to after Propo-sition 3.3. Namely, we have Proposition A.1.
Let K be a CM-field of odd class number such that K + has strictclass number 1, and let the values γ = γ ( K ) and M = M ( K ) associated to K bedefined as in Proposition 3.3 (note that γ ( K ) is distinct from the number field Eulerconstant γ K ). Then we may bound γ and M by constants depending only on K + . Thus, if we begin with a totally real field K of strict class number 1 and consider thefamily { K ∆ } of all CM-fields of odd class number for which K + = K , then γ ( K ∆ ) , M ( K ∆ ) = O (1) as ∆ → ∞ (A.1)with the implied constants depending on K . Actually, this is true even when K haseven class number, but that doesn’t matter for us since there may be too many ramifiedprimes. Proof.
Lemma 15 of [Ok] implies that if K is a totally real field of strict class number1, then for any CM-field K with K + = K , the Hasse unit index Q K satisfies Q K = [ O ∗ K : W K O ∗ K ] = 1 , (A.2)where W K is the group of roots of unity contained in K . Consequently, any indepen-dent set ǫ , ..., ǫ r of generators for O ∗ K modulo {± } also serves as independent set ofgenerators for O ∗ K modulo W K . This, together with the exact sequence → Gal(
K/K ) → Gal( K/ Q ) → Gal( K / Q ) → (A.3)implies that M ( K ) = max ≤ j ≤ rσ ∈ Gal ( K/ Q ) | log | σ ( ǫ j ) || depends only on K , as desired.To bound γ ( K ) , recall that O K = O K [ α ] , where α = (1 + √ β ) / for β ∈ O K atotally negative element. Thus, if x , ..., x N is an integral basis for O K over Q , then β j = ( x j if ≤ j ≤ Nαx j − N if N + 1 ≤ j ≤ N (A.4)is an integral basis for O K over Q . Consequently, the matrix ( β ( i ) j ) takes the block form ( β ( i ) j ) = (cid:18) X AXX AX (cid:19) (A.5)where X = ( x ( i ) j ) ≤ i,j ≤ N , A is the diagonal matrix A = α (1) α (2) . . . α ( N ) , (A.6) OW-LYING ZEROS OF NUMBER FIELD L -FUNCTIONS 25 and we’ve used the fact that x ( i ) j = x ( i + N ) j and α ( i ) = α ( i + N ) for ≤ i ≤ N since K ( i + N ) = K ( i ) and K is totally real. It is then straightforward to check that the inverseof ( β ( i ) j ) is given in block form by ( γ ij ) = (cid:18) X − A ( A − A ) − − X − A ( A − A ) − − X − ( A − A ) − X − ( A − A ) − (cid:19) . (A.7)Note that the invertibility of A − A follows from the fact that α ( i ) = α ( i ) for any i ;indeed, α ( i ) = (1 + √ β ( i ) ) / , and √ β ( i ) is purely imaginary since β is totally negative.Also, X is invertible since the integral basis x , ..., x N is linearly independent over Q .Consequently, to bound γ = max ≤ i,j ≤ N | γ ij | solely in terms of K , it suffices to sobound the entries of each of the matrices ( A − A ) − , A ( A − A ) − , and A ( A − A ) − .Recall from the beginning of Section 3 that | N K Q ( β ) | = ∆ D K / Q . (A.8)Moreover, | N K Q ( √ β ) | = | N K Q ( N KK ( √ β )) | = | N K Q ( β ) | . But by definition N K Q ( p β ) = Y K֒ → K ( i ) ≤ i ≤ N p β ( i ) (A.9)and since K is CM, we have (cid:12)(cid:12)(cid:12) √ β ( i ) (cid:12)(cid:12)(cid:12) = |√ β ( j ) | for all i, j (cf. [Wa], pg. 38). Therefore,since | N K Q ( √ β ) | = ∆ /D K / Q , we find that (cid:12)(cid:12)(cid:12)p β ( i ) (cid:12)(cid:12)(cid:12) = ∆ D K / Q ! / N (A.10)for any i . This in fact implies the desired bound on the entries of the matrices in ques-tion: we have ( A − A ) − = ( α (1) − α (1) ) − ( α (2) − α (2) ) − . . . ( α ( N ) − α ( N ) ) − (A.11)and for any i , we have (since α ( i ) = (1 + √ β ( i ) ) / ) | ( α ( i ) − α ( i ) ) − | = (cid:12)(cid:12)(cid:12)p β ( i ) (cid:12)(cid:12)(cid:12) − = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D K / Q ∆ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) / N ≤ | D K / Q | /N . (A.12) For the matrices A ( A − A ) − and A ( A − A ) − , we have for any i | α ( i ) ( α ( i ) − α ( i ) ) − | ≤ (cid:12)(cid:12)(cid:12) √ β ( i ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) √ β ( i ) (cid:12)(cid:12)(cid:12) ≤
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