Equidistribution theorems for holomorphic Siegel cusp forms of general degree: the level aspect

We prove equidistribution theorems for a family of holomorphic Siegel cusp forms of general degree in the level aspect. Our main contribution is to estimate unipotent contributions for general degree in the geometric side of Arthur's invariant trace formula in terms of Shintani zeta functions. Several applications including the vertical Sato-Tate theorem and low-lying zeros for standard $L$-functions of holomorphic Siegel cusp forms are discussed. We also show that the"non-genuine forms"which come from non-trivial endoscopic contributions by Langlands functoriality classified by Arthur are negligible.


Introduction
Let G be a connected reductive group over ‫ޑ‬ and ‫ށ‬ the ring of adeles of ‫.ޑ‬An equidistribution theorem for a family of automorphic representations of G.‫/ށ‬ is one of recent topics in number theory and automorphic representations.After Sauvageot's important results [1997], Shin [2012] proved a so-called limit multiplicity formula which shows that the limit of an automorphic counting measure is the Plancherel measure.It implies the equidistribution of Hecke eigenvalues or Satake parameters at a 994 Henry H. Kim, Satoshi Wakatsuki and Takuya Yamauchi fixed prime in a family of cohomological automorphic forms on G.‫./ށ‬ A quantitative version of Shin's result is given by Shin and Templier [2016].A different approach is discussed in [Finis et al. 2015] for G D GL n or SL n , treating more general automorphic forms which are not necessarily cohomological.Note that in the works of Shin and Shin and Templier, one needs to consider all cuspidal representations in the L-packets.Shin [2012, second paragraph on p. 88] suggested that one can isolate just holomorphic discrete series at infinity.In [Kim et al. 2020a;2020b], we carried out his suggestion and established equidistribution theorems for holomorphic Siegel cusp forms of degree 2. We should also mention Dalal's work [2022]; see Remark 3.12.See also the related works [Knightly and Li 2019;Kowalski et al. 2012].
In this paper we generalize several equidistribution theorems to holomorphic Siegel cusp forms of general degree.A main tool is Arthur's invariant trace formula, as used in the previous work, but we need a more careful analysis in the computation of unipotent contributions.Let us prepare some notations to explain our results.
Let G D Sp.2n/ be the symplectic group of rank n defined over ‫.ޑ‬For an n-tuple of integers k D .k 1 ; : : : ; k n / with k 1 k n > nC1, let D hol l D k be the holomorphic discrete series representation of G.‫/ޒ‬ with the Harish-Chandra parameter l D .k 1 1; : : : ; k n n/ or the Blattner parameter k.
Let ‫ށ‬ (respectively, ‫ށ‬ f ) be the ring of (respectively, finite) adeles of ‫,ޑ‬ and O ‫ޚ‬ be the profinite completion of ‫.ޚ‬For S 1 a finite set of rational primes, let S D f1g [ S 1 , ‫ޑ‬ S 1 D Q p2S 1 ‫ޑ‬ p , ‫ށ‬ S be the ring of adeles outside S and O ‫ޚ‬ S D Q p6 2S 1 ‫ޚ‬ p .We denote by 2 G.‫ޑ‬ S 1 / the unitary dual of G.‫ޑ‬ S 1 / D Q p2S 1 G.‫ޑ‬ p / equipped with the Fell topology.Fix a Haar measure S on G.‫ށ‬ S / so that S .G. O ‫ޚ‬ S // D 1, and let U be a compact open subgroup of G.‫ށ‬ S /.Consider the algebraic representation D k of the highest weight k so that it is isomorphic to the minimal K 1 -type of D hol l .Let h U denote the characteristic function of U .
Then we define a measure on 2 (1-2) where ….G.‫//ށ‬ 0 stands for the isomorphism classes of all irreducible unitary cuspidal representations of G.‫/ށ‬ and S D ˝0 p…S p .
To state the equidistribution theorem, we need to introduce the Hecke algebra C 1 c .G.‫ޑ‬ S 1 // which is dense under the map h 7 !O h, where O h. S 1 / D tr. S 1 .h// is in F. 2 G.‫ޑ‬ S 1 // consisting of suitable O pl S 1 -measurable functions on 2 G.‫ޑ‬ S 1 /.(See [Shin 2012, Section 2.3] for that space.)Let N be a positive integer.Put S N D fp prime W p j N g.We assume that S 1 \ S N D ∅.We denote by K p .N / the principal congruence subgroup of level N for G.‫ޚ‬ p / (see (2-3) for the definition), and set K S .N / D Q p…S K p .N /.For each rational prime p, let us consider the unramified Hecke algebra H ur .G.‫ޑ‬ p // C 1 c ‫ޑ.‬ p /, and for each Ä > 0, H ur .G.‫ޑ‬ p // Ä , the linear subspace of H ur .G.‫ޑ‬ p // consisting of all Hecke elements whose heights are less than Ä. (See (2-2).)Let H ur .G.‫ޑ‬ p // Ä Ä1 be the subset of H ur .G.‫ޑ‬ p // Ä consisting of all Hecke elements whose complex values have absolute values less than 1.Our first main result is Theorem 1.1.Fix k D .k 1 ; : : : ; k n / satisfying k 1 k n > n C 1. Fix a positive integer Ä.Then there exist constants a; b and c 0 > 0 depending only on G such that for each h 1 D ˝p2S 1 h 1;p , where h 1;p 2 H ur .G.‫ޑ‬ p // Ä Ä1 , we have Note that the implicit constant of the Landau O-notation is independent of S 1 , N and h 1 .
Let us apply this theorem to the vertical Sato-Tate theorem and higher level density theorem for standard L-functions of holomorphic Siegel cusp forms.
The principal congruence subgroup .N / of level N for G.‫/ޚ‬ is obtained by .N / D G.‫/ޑ‬ \ G.‫/ޒ‬K.N /; where K.N / D Q p<1 K p .N /.Let S k ..N // be the space of holomorphic Siegel cusp forms of weight k with respect to .N / (see the next section for a precise definition), and let HE k .N / be a basis consisting of all Hecke eigenforms outside N .We can identify HE k .N / with a basis of K.N /-fixed vectors in the set of cuspidal representations of G.‫/ށ‬ whose infinity component is (isomorphic to) D hol l .(See the next section for the details.)Put d k .N / D jHE k .N /j.Then we have [Wakatsuki 2018], for some constant where For each F 2 HE k .N /, we denote by F D 1 ˝˝0 p F ;p the corresponding automorphic cuspidal representation of G.‫./ށ‬ Henceforth, we assume that (1-4) Then the Ramanujan conjecture is true, namely, F ;p is tempered for any p; see Theorem 4.3.Unfortunately, this assumption forces us to exclude the scalar-valued Siegel cusp forms.
Let 2 G.‫ޑ‬ p / ur; temp be the subspace of 2 G.‫ޑ‬ p / consisting of all unramified tempered classes.We denote by .Â 1 .F ;p /; : : : ; Â n .F ;p // the element of corresponding to F ;p under the isomorphism 2 G.‫ޑ‬ p / ur; temp ' OE0; n =S n DW .Let p be the measure on defined in Section 7.
Theorem 1.2.Assume (1-4).Fix a prime p. Then the set ˚ Â 1 .F ;p /; : : : ; Â n .F ;p / 2 W F 2 HE k .N / « is p -equidistributed in , namely, for each continuous function f on , lim f Â 1 .F ;p /; : : : ; Â n .F ;p / D Z f .Â 1 ; : : : ; Â n / p : By using Arthur's endoscopic classification, we have a finer version of the above theorem.Under the assumption (1-4), the global A-parameter describing F , for F 2 HE k .N /, is always semisimple.(See Definition 4.1.)Let HE k .N / g be the subset of HE k .N / consisting of F such that the global A-packet containing F is associated to a simple global A-parameter.They are Siegel cusp forms which do not come from smaller groups by Langlands functoriality in Arthur's classification.In this paper, we call them genuine forms.Let HE k .N / ng be the subset of HE k .N / consisting of F such that the global A-packet containing F is associated to a nonsimple global A-parameter, i.e., they are Siegel cusp forms which come from smaller groups by Langlands functoriality in Arthur's classification.We call them nongenuine forms.We show that nongenuine forms are negligible.The following result is interesting in its own right.For this, we need some further assumptions on the level N .
Theorem 1.3.Assume (1-4).We also assume (1) N is an odd prime or (2) N is odd and all prime divisors p 1 ; : : : ; p r (r 2/ of N are congruent to 1 modulo 4 such that p i p j D 1 for i ¤ j , where denotes the Legendre symbol. Then (2) jHE k .N / ng j D O n;k; N 2n 2 Cn 1C for any > 0; (3) for a fixed prime p, the set ˚.Â 1 .F ;p /; : : : The above assumptions on the level N are necessary in order to estimate nongenuine forms related to nonsplit but quasisplit orthogonal groups in the Arthur's classification by using the transfer theorems for some Hecke elements in the quadratic base change in the ramified case [Yamauchi 2021].(See Proposition 4.12 for the details.) Next, we discuss `-level density (where `is a positive integer) for standard L-functions in the level aspect.Let us denote by ….GL n ‫//ށ.‬ 0the set of all isomorphism classes of irreducible unitary cuspidal representations of GL n ‫./ށ.‬Keep the assumption on k as in (1-4) and the above assumption on the level N .Then F can be described by a global A-parameter ).As a corollary, we obtain a result on the order of vanishing of L.s; F ; St/ at s D 1 2 , the center of symmetry of the L-function, by using the method of Iwaniec et al. [2000] for holomorphic cusp forms on GL 2 ‫/ށ.‬ (see also [Brumer 1995] for another formulation related to the Birch-Swinnerton-Dyer conjecture): Let r F be the order of vanishing of L.s; F ; St/ at s D 1 2 .Then we show that under the GRH (generalized Riemann hypothesis), This would be the first result of this kind in Siegel modular forms.We can also show a similar result for the degree 4 spinor L-functions of GSp.4/.
Let us explain our strategy in comparison with the previous works.We choose a test function such that f is a pseudocoefficient of D hol l normalized as tr. 1 .f// D 1.A starting main equality is where I spec .f/ (respectively, I geom .f/) is the spectral (respectively, the geometric) side of Arthur's invariant trace I.f /.Under the assumption k n > n C 1, the spectral side becomes simple by the results of Arthur [1989] and Hiraga [1996], and it is directly related to S k ..N // because of the choice of a pseudocoefficient of D hol l .Now the geometric side is given by where Q S D f1g t S N t S 1 and .M.‫//ޑ‬ M; Q S denotes the set of .M; Q S /-equivalence classes in M.‫/ޑ‬ (see [Arthur 2005, p. 113]); for each M in a finite set L, we choose a parabolic subgroup P such that M is a Levi subgroup of P .(See loc.cit.for details.) Roughly speaking: If the test function f is fixed, the terms on (1-5) vanish except for a finite number of .M; Q S /equivalence classes.
The factor a M .Q S; / is called a global coefficient and it is almost the volume of the centralizer of in M if is semisimple.The general properties are unknown.
The factor I G M .; f / is called an invariant weighted orbital integral, and as the notation shows, it strongly depends on the weight k of D k .Therefore, it is negligible when we consider the level aspect.
The factor J M M .; h P / is an orbital integral of for h D S .K.N // 1 h 1 h K S .N / .
998 Henry H. Kim, Satoshi Wakatsuki and Takuya Yamauchi According to the types of conjugacy classes and M , the geometric side is divided into the terms where Therefore, everything is reduced to studying the unipotent contributions I 2 .f/ and I 3 .f/.An explicit bound for I 2 .f/ was given by [Shin and Templier 2016, proof of Theorem 9.16].However, as for I 3 .f/, since the number of .M; Q S /-equivalence classes in the geometric unipotent conjugacy class of each is increasing when N goes to infinity, it is difficult to estimate I 3 .f/ directly.In the case of GSp.4/, we computed unipotent contributions by using case-by-case analysis as in [Kim et al. 2020a].Here we give a new uniform way to estimate all the unipotent contributions.It is given by a sum of special values of zeta integrals with real characters for spaces of symmetric matrices; see Lemma 3.3 and Theorem 3.7.This formula is a generalization of the dimension formula (see [Shintani 1975;Wakatsuki 2018]) to the trace formula of Hecke operators.By using their explicit formulas [Saito 1999] and analyzing Shintani double zeta functions [Kim et al. 2022], we express the geometric side as a finite sum of products of local integrals and special values of the Hecke L functions with real characters, and then obtain the estimates of the geometric side; see Theorem 3.10.This paper is organized as follows.In Section 2, we set up some notations.In Section 3, we give key results (see Theorem 3.7 and Theorem 3.10) in estimating trace formulas of Hecke elements.In Section 4, we study Siegel modular forms in terms of Arthur's classification and show that nongenuine forms are negligible.In Section 5, we give a notion of newforms which is necessary to estimate conductors.Sections 6-10 are devoted to proving the main theorems.Finally, in the Appendix, we give an explicit computation of the convolution product of some Hecke elements, which is needed in the computation of `-level density of standard L-functions.

Preliminaries
A split symplectic group G D Sp.2n/ over the rational number field ‫ޑ‬ is defined by The compact subgroup k n > n, which is called the Blattner parameter.We write k for the holomorphic discrete series corresponding to the Blattner parameter k D .k 1 ; : : : ; k n /.We also write D hol l for one corresponding to the Harish-Chandra parameter l D .k 1 1; k 2 2; : : : ; k n n/ so that D hol l D k .Let H ur .G.‫ޑ‬ p // denote the unramified Hecke algebra over G.‫ޑ‬ p /, that is, Let T denote the maximal split ‫-ޑ‬torus of G consisting of diagonal matrices.We denote by X .T / the group of cocharacters on T over ‫.ޑ‬An element e j in X .T / is defined by e j .x/D diag.Choose a natural number N .We set We write S k ..N // for the space of Siegel cusp forms of weight k for .N /, i.e., S k ..N // consists of V k -valued smooth functions F on G.‫/ށ‬ satisfying the following conditions: where k denotes the finite dimensional irreducible polynomial representation of U.n/ corresponding to k together with the representation space V k and we set k .g;On the other hand, given a cuspidal representation of Sp.2n/ with a K.N /-fixed vector and whose infinity component is holomorphic discrete series of lowest weight k, there exists a holomorphic Siegel cusp form F of weight k with respect to .N / such that F D .(See [Schmidt 2017[Schmidt , p. 2409] ] for n D 2. One can generalize the contents there to n 3.) We define HE k .N/ to be a basis of K.N/-fixed vectors in the set of cuspidal representations of Sp.2n; ‫/ށ‬ whose infinity component is holomorphic discrete series of lowest weight k, and identify it with a basis consisting of all Hecke eigenforms outside N .In particular, each F 2 HE k .N / gives rise to an irreducible cuspidal representation F of Sp.2n/.Let F k .N / be the set of all isomorphism classes of cuspidal representations of Sp.2n/ such that K.N / ¤ 0 and 1 ' k .Consider the map ƒ W HE k .N / !F k .N /, given by F 7 !F .It is clearly surjective.For each D 1 ˝˝0 p p 2 F k .N /, set f D ˝0 p p .Then we get

Asymptotics of Hecke eigenvalues
For each function h2C Proof.This lemma immediately follows from Lemma 3.1.
Let V r denote the vector space of symmetric matrices of degree r, and define a rational representation of the group GL 1 GL r on V r by x .a;m/ D a t mxm, where x 2 V r and .a;m/ 2 GL 1 GL r .The kernel of is given by Ker D f.a 2 ; aI r / W a 2 GL 1 g, and we set Then, the pair .H r ; V r / is a prehomogeneous vector space over ‫.ޑ‬For 1 Ä r Ä n and 2n/, then we have where vol G D vol.G.‫/ޑ‬nG.‫,//ށ‬d k denotes the formal degree of k , and X .S 0 / denotes the set consisting of real characters D ˝v v on ‫ޒ‬ >0 ‫ޑ‬ n‫ށ‬ such that v is unramified for any v … S 0 t f1g.Note that S 0 may contain S 1 and all prime factors of N .
Proof.To study I unip .fk h S 0 h S 0 /, we need an additional zeta integral Q Z r .ˆQf ;r ; s/ defined by The zeta integral Q Z r .ˆQf ;r ; s/ is absolutely convergent for the range (3-2), and Q Z.ˆQ f ;r ; s/ is meromorphically continued to the whole s-plane; see [Shintani 1975;Wakatsuki 2018;Yukie 1993].Applying [Wakatsuki 2018, Propositions 3.8 and 3.11, Lemmas 5.10 and 5.16] to I unip .f/, we obtain , and this change is essentially required for the proof of (3-3).
By the same argument as in [Hoffmann and Wakatsuki 2018, (4.9)], we have where runs over all real characters on ‫ޒ‬ >0 ‫ޑ‬ n‫ށ‬ .Suppose that D ˝v v … X .S 0 /.Then, we can take a prime p … S 0 such that p is ramified and Hence, we get Z r .ˆQf ;r ; s; / Á 0, and the proof is completed.
Remark 3.5.The rational representation of H r on V r is faithful, but the representation x 7 !t mxm of GL r on V r is not.Hence, Z r .ˆQf ;r ; s; / is suitable for Saito's explicit formula [1999], which we use in the proof of Theorem 3.10, but Q Z r .ˆQf ;r ; s/ is not.This fact is also important for the study of global coefficients in the geometric side; see [Hoffmann and Wakatsuki 2018].
Let be a nontrivial additive character on ‫,ށ‪n‬ޑ‬ and a bilinear form h ; i on V r ‫/ށ.‬ is defined by hx; yi WD Tr.xy/.Let dx denote the self-dual measure on V r ‫/ށ.‬ for .h; i/.Then, a Fourier transform of ˆ2 C 1 .V r ‫//ށ.‬ is defined by where 1 denotes the trivial representation on ‫ޒ‬ >0 ‫ޑ‬ n‫ށ‬ .Take a test function where x D f 2 W x D xg.The zeta function r .ˆ0;s/ absolutely converges for Re.s/ > .rC 1/=2, and is meromorphically continued to the whole s-plane; see [Shintani 1975].Furthermore, r .ˆ0;s/ is holomorphic except for possible simple poles at s D 1; 3=2; : : : .rC 1/=2.
Then, there exists a rational function C n;r .x 1 ; : : : ; x n / over ‫ޒ‬ such that Proof.This can be proved by the functional equation (3-5) and the same argument as in [Wakatsuki 2018, proof of Lemma 5.16].
Note that r .b ˆh;r ; s/ is holomorphic in fs 2 ‫ރ‬ W Re.s/ Ä 0g, and C n;r .x 1 ; : : : ; x n / is explicitly expressed by the Gamma function and the partitions; see [Wakatsuki 2018, (5.17) and Lemma 5.16].We will use this lemma for the regularization of the range of k.The zeta integral Z r .ˆQf ;r ; n .r1/=2; 1/ was defined only for k n > 2n, but the right-hand side of the equality in Lemma 3.6 is available for any k.In addition, this lemma is necessary to estimate the growth of I unip .f/ with respect to S D S 1 t f1g.We later define a Dirichlet series D S m;u S .s/just before Proposition 3.9, and the series D S m;u S .s/appears in the explicit formula of Z r .ˆ;s; 1/ when r is even.For the case that r is even and 3 < r < n, it seems difficult to estimate the growth of its contribution to Z r .ˆQf ;r ; n .r1/=2; 1/, but we can avoid such difficulty by this lemma, since the part related to D S m;u S .s/ in Saito's formula [1999, Theorem 3.3] disappears in the special value r .b ˆh;r ; r n/.
Theorem 3.7.Suppose , and let h be a test function on G.‫ށ‬ f / given as (3-1).Then there exists a positive constant c 0 such that, if , it is sufficient to prove that the geometric side I geom .f/ equals the right-hand side of (3-6).If one uses the results in [Arthur 1989] and applies [Shin and Templier 2016, Lemma 8.4] by putting " W G GL m , m D 2n, B " D 1, c " D c 0 in their notations, then one gets I geom .f/ D I unip .f/.Hence, by Lemma 3.3 and putting h S 0 h S 0 D h, we have for sufficiently large k n .Let M .a/WD diag.1;: : : ; 1; a; : : : ; a/, where there are n entries of both 1 and a, for a 2 ‫ށ‬ .For any a p 2 ‫ޚ‬ p , b p 2 ‫ޑ‬ p , 2 X .T /, we have Hence, (3-4) holds for any p < 1, and so Z r .ˆQf ;r ; n .r1/=2; / vanishes for any ¤ 1.Therefore, by Lemma 3.6 we obtain the assertion (3-6) for sufficiently large k n .By the same argument as in [Wakatsuki 2018, proof of Theorem 5.17], we can prove that this equality (3-6) holds in the range k n > n C 1, because the both sides of (3-6) are rational functions of k in that range, see Lemma 3.6 and [Wakatsuki 2018, Proposition 5.3].Thus, the proof is completed.
Let S denote a finite subset of places of ‫,ޑ‬ and suppose 1 2 S .For each character D ˝v v on ‫ޑ‬ ‫ޒ‬ >0 n‫ށ‬ , we set . p s log p 1 p s : By partial summation, Here we use the prime number theorem: P pÄx log p x. Therefore, .S / 0 .s/2s .s/=.s 1/.
We need the Dirichlet series The following proposition is a generalization of [Ibukiyama and Saito 2012, Proposition 3.6]: Proposition 3.9.Let m 2 be an even integer.Suppose .
The Dirichlet series D S m;u S .s/ is meromorphically continued to ‫,ރ‬ and is holomorphic at any s 2 ‫ޚ‬ Ä0 .
Proof.See [Kim et al. 2022, Corollary 4.23] for the case m > 3.For m D 2, this statement can be proved by using [Hoffmann and Wakatsuki 2018;Yukie 1992].
Theorem 3.10.Fix a parameter k such that k n > nC1.Let h 1 2 H ur .G.‫ޑ‬ S 1 // Ä , and let h 2 C 1 c .G.‫ށ‬ f // be a test function on G.‫ށ‬ f / given as (3-1).Suppose sup x2G.‫ޑ‬ S 1 / jh 1 .x/jÄ 1.Then, there exist positive constants a, b, and Here the constants a and b do not depend on Ä, N 1 , or N .See Lemma 3.3 for vol G and d k . Proof.Set Let R be a finite set of places of ‫.ޑ‬ Take a Haar measure dx 1 on V r ‫,/ޒ.‬ and for each prime p, we write dx p for the Haar measure on V r ‫ޑ.‬ p / normalized by R where D x 0 0 0 2 V n .Note that this definition is compatible with ˆQ f ;r since h 1 is spherical for where N n / by using his results.
Case I. Assume r is odd and r < n.In the following, we set S D S 1 t f1g.By Saito's formula, we have Therefore, one has by using Lemma 3.8.
Case II.Assume r is even and 3 < r < n.By Saito's formula, Proposition 3.9, and Lemma 3.8, one can prove that jI.Q f ; r/j is bounded by up to a constant.Note that Proposition 3.9 was used for this estimate, since it is necessary to prove the vanishing of the term including D S r;u S .s/ in the explicit formula [Saito 1999, Theorem 3.3].Case III.Assume r D n.In this case, we should use a method different from Case I and Case II since Z r;S 1 . 1 ˆh1 ;r ; s; O S 1 / may have a simple pole at s D r n D 0. Take an n-tuple l D .l 1 ; : : : ; l n /, with l 1 l n > 2n, and let n.x/ D I n O n x I n 2 G where x 2 V n .Recall that Q f l satisfies the following two properties: Hence, by property (ii), we have where In the case s D .nC1/=2,we note that j det.x/j vanishes in the integral of Z n;1 .ˆQf l ;n ; .nC1/=2;O 1 /.Hence, it follows from property (ii) that and so we also find By Lemmas 3.6 and 3.11, the residue formula [Yukie 1993, Chapter 4] of Z n .ˆ;s; 1/ and the same argument as in [Hoffmann and Wakatsuki 2018, proof of Theorem 4.22], we obtain where From this, we have Case IV.Assume r D 2 < n.By Saito's formula [Hoffmann and Wakatsuki 2018, Theorem 4.15], we have Hence, it is enough to give an upper bound of jD S 2;u S .2n/j for u 1 < 0. Choose a representative element u S D .uv / v2S satisfying u p 2 ‫ޚ‬ p , with p 2 S 1 .Take a test function ˆD ˝vˆv such that the support of ˆ1 is contained in fx 2 V 0 2 ‫/ޒ.‬W det.x/ > 0g and ˆp is the characteristic function of diag.1;u p / C p 2 V 2 ‫ޚ.‬ p / (respectively, V 2 ‫ޚ.‬ p /) for each p 2 S 1 (respectively, p 6 2 S).Let and we set for some c 2 ‫.ގ‬By [Assem 1993, Lemma 2.1.1]and the assumption sup x2G.‫ޑ‬ S 1 / jh 1 .x/jÄ 1, we have jˆh 1 ;r j Ä ˆS1 ;r; Ä ; where ˆS1 ;r; Ä denotes the characteristic function of ˝p2S 1 p Ä V r ‫ޚ.‬ p /. Hence, by a change of variables, we get Z r;S 1 jˆh 1 ;r j; n r 1 It follows from classification theory of quadratic forms that #.V 0 r ‫ޑ.‬ S 1 /=H r ‫ޑ.‬ S 1 // N 1 .Therefore, we obtain a desired upper bound for Z r .S 1 ; h 1 /.Thus, we obtain Remark 3.12.We give some remarks on Shin and Templier's work [2016] and Dalal's work [2022].
In the setting of [Shin and Templier 2016], they considered "all" cohomological representations as a family which exhausts an L-packet at infinity since they chose the Euler-Poincaré pseudocoefficient at the infinite place.Then there is no contribution from nontrivial unipotent conjugacy classes.Therefore, our work is different from Shin-Templier's work in that we can consider only holomorphic forms in an L-packet.Shin suggested to consider a family of automorphic representations whose infinite type is any fixed discrete series representation.Dalal [2022] carried it out in the weight aspect by using the stable trace formula.The stabilization allows us to remove the contribution I 3 .f/ (see Section 1), but instead of I 3 .f/, the contributions of endoscopic groups have to enter.Dalal obtained a good bound for them by using the concept of hyperendoscopy introduced by Ferrari [2007].In studying the level aspect, it seems difficult to directly get a sufficient bound for the growth of the hyperendoscopic groups in question; since Sp.2n/ has infinitely many elliptic endoscopic groups where E runs over quadratic extensions of ‫ޑ‬ and SO.N 1 C 1; N 1 1; E=‫/ޑ‬ is the quasisplit orthogonal group attached to E=‫ޑ‬ (see [Arthur 2013, p. 13-14] and [Assem 1998, §4]), it is quite complicated to count the hyperendoscopic groups.(The referee pointed out to us that the essential difficulty in applying hyperendoscopy techniques is in computing endoscopic transfers of indicators of any level subgroup.In particular, answering the transfer problem is necessary to even know which set of groups we are counting.)We also observe the same complication coming from elliptic endoscopic groups in the unipotent terms of the (unstable) Arthur trace formula; see [Hoffmann and Wakatsuki 2018, p. 8].Assem's results [1993; 1998] make us expect that, for 1 Ä r Ä n, some parts of zeta integrals Z r .ˆQf ;r ; s; / probably correspond to the central contributions of the endoscopic groups SO.n r C1; n r C1/ Sp.2r 2/ and SO.n r C 2; n r; E=‫/ޑ‬ Sp.2r 2/.To avoid such complication, we have simplified the unipotent terms in several steps as follows: Our method showed the vanishing of a large part of the unipotent terms; see Lemma 3.3 and [Wakatsuki 2018].
The contributions of Z r .ˆQf ;r ; s; / vanish when is nontrivial; see Theorem 3.7.
Our careful analysis estimates upper bounds of the contributions of Z r .ˆQf ;r ; s; 1/ by using the functional equations; see the proof of Theorem 3.10.Analogous simplifications should be required even if we use the stable trace formula.

Arthur classification of Siegel modular forms
In this section, we study Siegel modular forms in terms of Arthur's classification [2013]; see §1.4 and §1.5 of loc.cit.. Recall G D Sp.2n/=‫.ޑ‬We call a Siegel cusp form which comes from smaller groups by Langlands functoriality "a nongenuine form".In this section, we estimate the dimension of the space of nongenuine forms and show that they are negligible.This result is interesting in its own right.Let F 2 HE k .N /, see Section 2, and D F be the corresponding automorphic representation of G.‫./ށ‬ According to Arthur's classification, can be described by using the global A-packets.Let us recall some notations.A (discrete) global A-parameter is a symbol satisfying the following conditions: (1) for each i , with 1 Ä i Ä r, i is an irreducible unitary cuspidal self-dual automorphic representation of GL m i ‫./ށ.‬In particular, the central character !i of i is trivial or quadratic; (2) for each i , d i 2 ‫ޚ‬ >0 and (3) if d i is odd, then i is orthogonal, i.e., L.s; i ; Sym 2 / has a pole at s D 1; (4) if d i is even, then i is symplectic, i.e., L.s; i ; ^2/ has a pole at s D 1; (5) We say that two global A-parameters r i D1 i OEd i and r 0 By [Arthur 2013, Theorem 1.5.2](though our formulation is slightly different from the original one), we have a following decomposition where m ; 2 f0; 1g; see [Atobe 2018, Theorem 2.2] for m ; .We have the following immediate consequence of (4-1): Proposition 4.2.Let 1 K.N / be the characteristic function of K.N / G.‫ށ‬ f /.Then Theorem 4.3.Assume (1-4).For a global A-parameter D r i D1 i OEd i , suppose that there exists 2 … with 1 ' k .Then is semisimple, i.e., d i D 1 for all i , and each i is regular algebraic and satisfies the Ramanujan conjecture, i.e., i;p is tempered for any p.Therefore, for each finite prime p, the local Langlands parameter at p of is described as one of the isobaric sum r i D1 i;p which is an admissible representation of GL 2nC1 ‫ޑ.‬ p /.
Definition 4.4.We denote by HE k .N / ng the subset of HE k .N / consisting of all forms which belong to ; under the isomorphism (4-1).A form in this space is called a nongenuine form.
Similarly, we denote by HE k .N / g the subset of HE k .N / consisting of all forms which belong to ; under the isomorphism (4-1).A form in this space is called a genuine form.

Proof
Here, if i is the central character of i and i;1 ' i , then dim.K GLm i .N / i;f / D l cusp;ort .mi ; N; i ; i /.Notice that the conductor of i is a divisor of N .Summing up, we have the claim.
Next we study l cusp;ort .n;N; ; / for 2 ….GL n ‫//ޒ.‬c and for n 2. Now if is a cuspidal representation of GL 2mC1 which is orthogonal, i.e., L.s; ; Sym 2 / has a pole at s D 1, then comes from a cuspidal representation on Sp.2m/.In this case, the central character ! of is trivial.
If is a cuspidal representation of GL 2m which is orthogonal, i.e., L.s; ; Sym 2 / has a pole at s D 1, then ! 2 D 1; If !D 1, comes from a cuspidal representation on the split orthogonal group SO.m; m/; If !¤ 1, then comes from a cuspidal representation on the quasisplit orthogonal group SO.m C 1; m 1/.
First we consider the case when is trivial in estimating l cusp;ort .2nC ı; N; ; /, where ı D 0 or 1.For a positive integer n, let We regard H as a twisted elliptic endoscopic subgroup G 0 .
Proposition 4.10.Let N be an odd positive integer.For the pair .G 0 ; H /, the characteristic function of vol.K H .N // 1 1 K H .N / as an element of C 1 c .H.‫ޑ‬ S N // is transferred to Proof Next we consider the case when is a quadratic character.In this case, a cuspidal representation contributing to L cusp;ort .K GL n .N /; 1 ; / comes from a cuspidal representation of the quasisplit orthogonal group SO.m C 1; m 1/ defined over the quadratic extension associated to .However any transfer theorem for Hecke elements in .GL 2m ; SO.m C 1; m 1// remains open.To get around this situation, we make use of the transfer theorems for some Hecke elements in the quadratic base change due to Yamauchi [2021].For this, we need the following assumptions on the level N : (1) N is an odd prime or (2) N is odd and all prime divisors p 1 ; : : : ; p r .r2/ of N are congruent to 1 modulo 4 and p i p j D 1 for i ¤ j , where denotes the Legendre symbol.In such a case, if m D 2, then D AI ‫ޑ‬ M for some cuspidal representation of GL 2 ‫ށ.‬ M /; an automorphic induction from GL 2 ‫ށ.‬ M / to GL 4 ‫ށ.‬‫ޑ‬ /.Since is cuspidal and orthogonal, has to be dihedral.Such are counted in [Kim et al. 2020b, Section 2.6] and it amounts to O.N 11=2C" / for any " > 0. This will be negligible because vol.K H .N // cN m.2m 1/ D cN 6 for some constant c > 0. Assume m 3. It is easy to see that the dimension of where Note that for any split reductive group G over ‫ޑ‬ and the principal congruence subgroup K G .N / of level N , we have that vol.K G .N // cN dimG for some constant c > 0 as N ! 1.Furthermore, !.N / log N=.log log N /.Hence 2 !.N / N , and A n .N / D O.N " / and C m i .N / D O.N " / for each 1 Ä i Ä r.
The second claim follows from the dimension formula (1-3).

A notion of newforms in S k ..N //
In this section, we introduce a notion of a newform in S k ..N // with respect to principal congruence subgroups.Since any local newform theory for Sp.2n/ is unavailable except for n D 1; 2, we need a notion of newforms so that we can control a lower bound of conductors for such newforms.This is needed in application to low lying zeros.(See Theorem 8.3 and Lemma 9.3.)Recall the description in terms of Arthur's classification.
Definition 5.1.The new part (space) of S k ..N // is defined by Lemma 5.3.Assume that (1-4) holds and N is squarefree.Then we have Proof.Let M j N .Take an automorphic representation D f ˝ k such that dim D 0 for any L j M , L < M .Under this condition, has an intersection with S new k ..M //, and also with S k ..N //.Let f D ˝p p .By the assumptions and Theorem 4.3, for any prime p − M , p is tempered spherical, and so p is an irreducible induced representation from a Borel subgroup B of G.‫ޑ‬ p /.So dim Thus, we obtain the assertion.
Theorem 5.4.Assume that (1-4) holds and N is squarefree.Then we have The Möbius inversion formula gives where denotes the Möbius function.Therefore, (5-1) By [Wakatsuki 2018, Corollary 1.2], there exist constants C k;r such that d k .N / D P n rD0 C k;r C N N f .r/if N > 2, where f .r/D 2n 2 C n C 1 2 r.r 1/ nr and C k;0 D C k .Further, we take two constants D 1 and D 2 so that d k .N / D P n rD0 C k;r C N N f .r/C D N for N D 1 or 2. Therefore, by (5-1), we obtain Since N is squarefree, From this, we obtain the assertion.Now, d p < 1. Hence Q pjN 1 d 1 p p n 2 f .r/< 1.Also d 1 p < p n since 1=.1 p 1 / < p.Therefore, which is the Artin constant.

Equidistribution theorem of Siegel cusp forms; proof of Theorem 1.1
By the definition in (1-1), we see that Notice that dim k D d k (under a suitable normalization of the measure).Applying Theorem 3.10 to S 1 , we have the claim by the Plancherel formula of Harish-Chandra: 7. Vertical Sato-Tate theorem for Siegel modular forms: proofs of Theorems 1.2 and 1.3 Suppose that k D .k 1 ; : : : ; k n / satisfies the condition (1-4).Put ‫ޔ‬ D fz 2 ‫ރ‬ W jzj D 1g.For F 2 HE k .N /, consider the cuspidal automorphic representation D F D 1 ˝˝0 p F ;p of G.‫/ށ‬ associated to F .As discussed in the previous section, under the condition (1-4), the A-parameter whose A-packet contains is semisimple and F ;p is tempered for all p.Then if p − N , F ;p is spherical, and we can write F ;p as F ;p D Ind G.‫ޑ‬ p / B.‫ޑ‬ p / p , where B D T U is the upper Borel subgroup and p is a unitary character on B.‫ޑ‬ p /.
By letting p ! 1, we recover the Sato-Tate measure Then Theorems 1.2 and 1.3 follow from Theorems 1.1 and 4.13.
8. Standard L-functions of Sp.2 n/ Let k D .k 1 ; : : : ; k n / and F 2 HE k .N /, and let F be a cuspidal representation of G.‫/ށ‬ associated to F .Assume (1-4) for k.By (4-1) and the observation there, the global A-packet … containing F is associated to a semisimple global A parameter D r i D1 i where i is an irreducible cuspidal representation of GL m i ‫./ށ.‬Then the isobaric sum … WD r i D1 i is an automorphic representation of GL 2nC1 ‫./ށ.‬Therefore, we may define .s; ip / be the local p-factor of L.s; F ; St/ for each rational prime p.
Let F D 1 ˝˝0 p p .For p − N , we have that p is the spherical representation of G.‫ޑ‬ p / with the Satake parameter .˛1p ; : : : ; ˛np ; 1; ˛ 1 1p ; : : : ; ˛ 1 np /.Then .1 ˛ip p s /.1 ˛ 1 ip p s /: We define the conductor q.F / of F to be the product of the conductors q. i / of i , for 1 Ä i Ä r.The epsilon factor .F / turns out to be always 1.
Proposition 8.2.Let F be associated to a semisimple A-parameter.Then .F / D 1.
Proof.Recall the global A-parameter D r i D1 i .Let !i be the central character of i .Since i is orthogonal, its epsilon factor is !i .1/ by [Lapid 2004, Theorem 1].Hence, by the condition on the central character.
If F 2 HE new k .N /, by Definition 5.1, it is not fixed by K GL 2nC1 .p e p 1 / for each p j N .By [Miyauchi and Yamauchi 2022, Theorem 1.2], we have q.… p / p m i .e p 1/ for some i .In particular, q.… p / p e p 1 for each p j N .Hence, q.F / N Q pjN p 1 .It is clear that q.… p / p if p j N .Hence, Now, q.F / 2 D q.F / q.F / N .Hence our result follows.
Proposition 8.4.Keep the assumptions on N as in Proposition 4.12.Let F 2 HE k .N /.Then L.s; F ; St/ has a pole at s D 1 if and only if F is associated to a semisimple global A-parameter D 1 1 r where i is an orthogonal irreducible cuspidal representation of This proves [Shin and Templier 2016, Hypothesis 11.2] in our family.
Proof.This follows from the proof of Theorem 4.13, by noting that partitions m D .m 1 ; : : : ; m r / of 2n contribute to HE k .N / 0 .
Böcherer [1986] gave the relationship between Hecke operators and L-functions for level one and scalar-valued Siegel modular forms and it is extended by Shimura [1994a] to a more general setting.
Let a D .a 1 ; : : : ; a n /, 0 Ä a 1 Ä Ä a n , and D p;a D diag.p a 1 ; : : : ; p a n /.Let F be an eigenform in HE k .N / with respect to the Hecke operator T .D p;a / for all p − N , and let .F; D p;a / be the eigenvalue.
Then we have the following identity [Shimura 1994a, Theorem 2.9]: for some constant ˛.
Proof.Let S 1 be the set of all prime divisors of m n .Since m n > 1, S 1 is nonempty.The main term of right-hand side in Theorem 3.10 includes h 1 .1/.Clearly, h 1 .1/D 0 because the double coset defining the Hecke operator h 1 does not contain any central elements.Since the automorphic counting measure is supported on cuspidal representations, Theorem 3.10 implies the claim.Then from (8-1), we have, for p − N , More generally, for p − N , This proves [Kim et al. 2020b, Conjecture 6.1 in level aspect] for the Sp.4/ case.

`-level density of standard L-functions
In this section, we assume (1-4) and keep the assumptions on N in Proposition 4.12.Then we show unconditionally that the `-level density (`a positive integer) of the standard L-functions of the family HE k .N / has the symmetry type Sp in the level aspect.Shin and Templier [2016] showed it under several hypotheses with a family which includes nonholomorphic forms.
Under assumption (1-4), F satisfies the Ramanujan conjecture, namely, j˛i p j D 1 for each i .Let

Ã .N /:
By Theorem A.1, T .p; .0;: : : ; 0; 1// 2 , where there are n 1 entries of 0, is a linear combination of T p; .Therefore, by Theorem 8.7, if p − N , for some polynomial g 2 ‫ޚ‬OEx and c > 0.Here the main term 1 C p 1 g.p 1 / comes from the coefficient p i of T .p; n ‚ …" ƒ .0;: : : ; 0// in the linear combination.Here the explicit determination of the coefficient is necessary in our application.Hence, we have Proposition 9.1.For some ˛> 0 and p − N , for any > 0. Hence, the first error term O.p ˛N n / in Proposition 9.1 can be replaced (by taking Ä D 1) by The second error term O.p ˛N n / in Proposition 9.1 can be replaced (by taking We denote the nontrivial zeros of L.s; F ; St/ by F ;j D 1 2 C p 1 F ;j .Without assuming the GRH for L.s; F ; St/, we can order them as Ä Re.F ; 2 / Ä Re.F ; 1 / Ä 0 Ä Re.F ;1 / Ä Re.F ;2 / Ä : Let c.F / D q.F /.k 1 k n / 2 be the analytic conductor, and let From Theorems 5.4 and 8.3, we have Lemma 9.3.Let n > 1.We assume that N is squarefree.Then This proves [Shin and Templier 2016, Hypothesis 11.4] in our family.It is used in the proof of (9-1).
Proof.By Theorem 8.3, we have q.F / Ä N 2nC1 .It gives rise to the upper bound.If F 2 HE new k .N /, q.F / N 1=2 by Theorem 8.3.By Theorem 5.4, jHE new k .N /j .n 2 / 1 jHE k .N /j.Hence, Consider, for an even Paley-Wiener function , Then as in [Kim et al. 2020a, (9.1)], where HE k .N / 0 is in Proposition 8.4.(In [Kim et al. 2020a, (9.4)], the term O jHE k .N / 0 j=d k .N / was omitted.)By Proposition 9.1, we can show as in [Kim et al. 2020a] that for an even Paley-Wiener function such that the Fourier transform O of is supported in .ˇ; ˇ/, for some ˇ> 0, where P j 1 ;:::;j `is over j i D ˙1; ˙2; : : : with j a ¤ ˙jb for a ¤ b.Then as in [Kim et al. 2020b], using Theorem 8.7, we can show Theorem 9.4.We assume that N is squarefree.Let .x 1 ; : : : ; x `/ D 1 .x 1 / `.x `/, where each i is an even Paley-Wiener function and O .u 1 ; : : : ; u `/ D O 1 .u 1 / O `.u `/.Assume the Fourier transform O i of i is supported in .ˇ; ˇ/ for i D 1; ; `. (See (9-1) for the value of ˇ.) Then Remark 9.5.The above theorem is usually stated for Schwartz functions in the literature.But since Schwartz functions approximate any function in L 2 -space, the above theorem holds for Payley-Wiener functions, which are in L 2 ‫ޒ.‬ n /, and whose Fourier transforms have compact supports.
10.The order of vanishing of standard L-functions at s D 1 2 In this section, we show that the average order of vanishing of standard L-functions at s D 1 2 is bounded under GRH; see [Iwaniec et al. 2000;Brumer 1995].Under GRH on L.s; F ; St/, its zeros are Since .x/0 for x 2 ‫,ޒ‬ from (9-1), we have Hence, we have We can show a similar result for the spinor L-function of GSp.(1) (level aspect) Fix k 1 ; k 2 .Then for whose Fourier transform O has support in .u; u/ for some 0 < u < 1, as N ! 1 (See [Kim et al. 2020a, Proposition 9.1] for the value of u), (2) (weight aspect) Fix N .Then for whose Fourier transform O has support in .u; u/ for some 0 By a careful analysis, we can show that v 1 D 3; weight aspect:
Remark A.5.We would like to make corrections to [Kim et al. 2020a].
(3) On page 362, the coefficient of R p 2 should be p 4 C p 3 C p 2 C p D p P 3 i D0 p i which is the volume of Sp.4; ‫ޚ‬ p / diag.1;p 2 ; p 4 ; p 2 / Sp.4; ‫ޚ‬ p / explained in [Roberts and Schmidt 2007, p. 190].
(4) On page 403, Lemma 8.1, the inequality q.F / N is not valid.Similarly, on page 405, Lemma 8.3, the inequality q.F / N is not valid.We need to consider newforms as in Section 5 of this paper.
Then for a newform, we obtain the inequality q.F / N 1=2 and log c k;N log N is valid as in Lemma 9.3 of this paper.
(8) On page 409, line 10, we need to add 2 G 3 2 C G 1 2 , in order to account for the poles of ƒ.s; F ; Spin/, and the contour integral is over Re.s/ D 2. So, in (9.3), we need to add O jHE k .N / 0 j=jHE k .N /j .However, only CAP forms give rise to a pole, and the number of CAP forms in HE k .N / is O.N 8C /.So it is negligible.
In the case of standard L-functions, the non-CAP and nongenuine forms which give rise to poles are: 1 , where is an orthogonal cuspidal representation of GL.4/ with trivial central character, or 1 1 2 , where the i are dihedral cuspidal representations of GL.2/.In those cases, by   4.11 and [Kim et al. 2020b, Theorem 2.9], we can count such forms without extra conditions on N in Proposition 4.12.So our result is valid as it is written.
Remark A.6.The referee brought to our attention a possible gap in [Sauvageot 1997, p. 181]; see [Dalal 2022, p. 129] and[Nelson andVenkatesh 2021, p. 159].S.W. Shin communicated to us that the issue has not been fixed at this writing.However, we do not use the result in [Sauvageot 1997], nor any other later results [Dalal 2022;Shin 2012;Shin and Templier 2016] which depend on [Sauvageot 1997].
notion of newforms in S k ..N // 1017 6. Equidistribution theorem of Siegel cusp forms; proof of Theorem 1.1 1019 7. Vertical Sato-Tate theorem for Siegel modular forms: proofs of Theorems 1.2 and 1.3 1020 8. Standard L-functions of Sp.2n/ 1021 9. `-level density of standard L-functions 1024 10.The order of vanishing of standard L-functions at s D 1 2

m
delta measure supported at 0 S 1 , a unitary representation of G.‫ޑ‬ S 1 /, and m cusp . 0S 1 I U; ; D hol l cusp ./ tr.S .hU //; and D 1; I 2 .f/: M ¤ G and D 1; I 3 .f/: is unipotent, but ¤ 1; I 4 .f/: the other contributions.The first term I 1 .f/ is f .1/ up to constant factors, and the Plancherel formula O pl S 1 .O f / D f .1/yields the first term of the equality in Theorem 1.1.The condition N c 0 Q p2S 1 p 2nÄ in Theorem 1.1 implies that the nonunipotent contribution I 4 .f/ vanishes by [Shin and Templier 2016, Lemma 8.4].
For each d 2 D, we denote by d D Q v d;v the quadratic character on ‫ޑ‬ ‫ޒ‬ >0 n‫ށ‬ corresponding to the quadratic field ‫.ޑ‬ p d / via class field theory.If d D 1, then d means the trivial character 1.For each positive even integer m, we set OEd 0 i are equivalent if r D r 0 and there exists 2 S r such that d 0 i D d .i/ and 0 i D .i/ .Let ‰.G/ be the set of equivalent classes of global A-parameters.For each 2 ‰.G/, one can associate a set … of equivalent classes of simple admissible G.‫ށ‬ f / .g;K 1 /-modules; see [Arthur 2013].The set … is called a global A-packet for .Definition 4.1.Let D r i D1 i OEd i be a global A-parameter.is said to be semisimple if d 1 D D d r D 1; otherwise, is said to be nonsemisimple; is said to be simple if r D 1 and d 1 D 1.

Proof.
By the proof of [Chenevier and Lannes 2019, Corollary 8.5.4], we see that d 1 D D d r D 1. Hence, is semisimple.Further, by comparing infinitesimal characters c. 1 /, c. 1 / of 1 , 1 respectively, we see that each i is regular algebraic by [Chenevier and Lannes 2019, Corollary 6.3.6 and Proposition 8.2.10].It follows from [Caraiani 2012;2014] that i;p is tempered for any p.
Let P .2nC 1/ be the set of all partitions of 2n C 1 and P m be the standard parabolic subgroup of GL 2nC1 associated to a partition 2n C 1 D m 1 C C m r , and m D .m 1 ; : : : ; m r /.In order to apply the formula (4-2), it is necessary to study the transfer of Hecke elements in the local Langlands correspondence established by [Arthur 2013, Theorem 1.5.1].We regard G D Sp.2n/ as a twisted elliptic endoscopic subgroup of GL 2nC1 ; see [Ganapathy and Varma 2017] or [Oi 2023].
the 1 of m 2 1 in the exponent of left-hand side in the above equation is inserted because of the fixed central character.Since dim SO.m; m/ D m.2m 1/ and m 3, spaces … K GL 2m M .N/ f for which … is noncuspidal are negligible in the estimation.Further, … is orthogonal with trivial central character.(The central character of … is ı N M=‫ޑ‬ D 1.) Therefore, we can bound l cusp;ort .2m;N; ; / by l cusp;ort .2m;N; BC M 1 ‫ޒ=‬ ./; 1/; which is similarly defined for cuspidal representations of GL 2m ‫ށ.‬ M /.Applying the argument of the proof of Proposition 4.11 to .GL 2m =M; SO.m; m/=M /, the quantity l cusp;ort .2m;N; ; / is bounded by 2 2m!.N/ vol.K H M .N// 1 , where H M WD SO.m; m/=M and !.N/ denotes the number of prime ideals dividing N. The claim follows from O M =N ' ‫=ޚ‬N ‫ޚ‬ since vol.K H M .N// D vol.K H .N // and clearly !.N/ D !.N /.
1) where a D .a 1 ; : : : ; a n / runs over 0 Ä a 1 Ä Ä a n .Let m D .m 1 ; : : : ; m n /, m 1 jm 2 j jm n , and D m D diag.m 1 ; : : : ; m n /, and let .F; D m / be the eigenvalue of the Hecke operator T .D m /.Let L N .s;F / D X m; .mn ;N /D1.F; D m / det.D m / s : Then L N .s;F / D Y p−N L.s; F / p ; L.s; F / p D X a .F; D p;a / det.D p;a / s : It converges for Re.s/ > 2n C .k 1 C C k n /=n C 1. Hence, we have N .s/Ä n Y i D1 N .2s2i / L N .s;F / D L N .sn; F ; St/; where L N .s;F ; St/ D Q p−N L p .s;F ; St/, and N .s/D Q p−N .1 p s / 1 .The central value of L N .s;F / is at s D nC 1 2 , and L N .s;F / has a zero at s D nC 1 2 since L N .s;F ; St/ is holomorphic at s D 1 2 .Theorem 3.10 implies Theorem 8.5.For m D .m 1 ; : : : ; m n /, m 1 jm 2 j jm n with m n > 1 and .m

FFF
N /D1 a F .m/m s and L.s; F / p D 1 X kD0 a F .p k /p ks for each prime p − N .Here a F .p k / D P a .F; D p;a /, where the sum is over all a D .a 1 ; : :: ; a n / such that 0 Ä a 1 Ä Ä a n , a 1 C C a n D k.Hence, for k > 0 and p − N , 1 d k .N / X 2HE k .N / a F .p k / D O. p ka N n /:More generally: Corollary 8.6.For m > 1, with .m;N / D 12HE k .N / a F .m/ D O.m ˛N n /: Proof.We have a F .m/ D P m .F; D m /, where the sum is over all m D .m 1 ; : : : ; m n /, m 1 jm 2 j jm n , m 1 m n D m.Our assertion follows from Theorem 8.5.Write L N .s;F ; St/ D .m/ms : : : : ; 0/ D .N /I 2n .N /: 4/. Recall the following from [Kim et al. 2020a]: Proposition 10.2.Assume .N; 11Š/ D 1.
w 1 D 6 in [Kim et al. 2020a, Proposition 8.2] in the level aspect.Hence u D 1 40 in the level aspect.As in Theorem 10.1, we have Theorem 10.3.Let G D GSp.4/.Assume the GRH, and let r F D ord sD 1 2 L.s; F ; Spin/.Then

Proposition
Since any local newform theory for Sp.2n/ is unavailable except for n D 1; 2, we define the old space S old k ..N // to be the intersection of S k ..N // with the smallest G.‫ށ‬ f /-invariant space of functions on G.‫/ޑ‬nG.‫/ށ‬ containing S k ..M // for all proper divisors M of N .The new space S new k ..N // is the orthogonal complement of S old k ..N // in S k ..N // with respect to the Petersson inner product.Then if F 2 S new k ..N //, q.F / N 1=2 (Theorem 8.3), and if N is squarefree, we can show that dim S new k ..N // .n 2 / 1 d k .N / if n 2 (Theorem 5.4 Then we may define the standard L-function of F 2 HE k .N / by L.s; F ; St/ WD r Y iD1 L.s; i /;which coincides with the classical definition in terms of Satake parameters of F outside N .Then we show unconditionally that the `-level density of the standard L-functions of the family HE k .N / has the symmetry type Sp in the level aspect.(SeeSection9 for the precise statement.Shin and Templier[2016]showed it under several hypotheses for a family which includes nonholomorphic forms.)Here, in order to obtain lower bounds for conductors, it is necessary to introduce a concept of newforms.This may be of independent interest.
is isomorphic to the unitary group U.n/ via the mapping A For each rational prime p, we also set K p D G.‫ޚ‬ p / and put K D Q pÄ1 K p .The compact groups K v and K are maximal in G.‫ޑ‬ v / and G.‫,/ށ‬ respectively, Holomorphic discrete series of G.‫/ޒ‬ are parameterized by n-tuples k D .k 1 ; : : : ; k n / 2 ‫ޚ‬ n such that k 1 By the Cartan decomposition, any function in H ur .G.‫ޑ‬ p // is expressed by a linear combination of characteristic functions of double cosets K p .p/K p . 2 X .T //.A height function k k on X .T / is defined by Let m D .m 1 ; : : : ; m n /, m 1 jm 2 j jm n , and D m D diag.m 1 ; : : : ; m n /.Let T .D m / be the Hecke operator defined by the double coset Specifically, for each prime p, let D p;a D diag.p a 1 ; : : : ; p a n /, with a D .a 1 ; : : : ; a n / and 0 Ä a 1 Ä Ä a n .Let F be a Hecke eigenform in S k ..N // with respect to the Hecke operator T .D p;a / for all p − N .(See[Kimetal. 2020a, Section 2.2]for Hecke eigenforms in the case of n D 2. One can generalize the contents there to n 3.) Then F gives rise to an adelic automorphic form F on Sp.2n; ‫/ޑ‬n Sp.2n; ‫,/ށ‬ and F gives rise to a cuspidal representation F which is a direct sum F D 1 ˚ ˚ r , where the i are irreducible cuspidal representations of Sp.2n/.Since F is an eigenform, the i are all near-equivalent to each other.Since we do not have the strong multiplicity one theorem for Sp.2n/, we cannot conclude that F is irreducible.However, the strong multiplicity one theorem for GL n implies that there exists a global A-parameter 2 ‰.G/ such that i 2 … for all i [Schmidt 2018, p. 3088].(See Section 4 for the definition of the global A-packet.) [Clozel and Delorme 1990]15-16]for the relationship between the classical Hecke operators and adelic Hecke operators for n D 2. One can generalize the contents there to n 3 easily.Let f k denote a pseudocoefficient of k with tr k .fk/D1;see[ClozelandDelorme 1990].Suppose k n > n C 1 and h 2 C 1 c .K.N /nG.‫ށ‬ f /=K.N //.The spectral side I spec .fkh/ of the invariant trace formula is given byI spec .fkh/DX D k ˝ f ; auto.rep. of G.‫/ށ‬m Tr. f .h//DTrT h j S k ..N // ; C 1 that we obtain Tr. 1 .fk//D 0 for any unitary representation1 .6Šk / of G.‫./ޒ‬We choose two natural numbers N 1 and N , which are mutually coprime.Suppose that N 1 is squarefree.Set S 1 D fp W p j N 1 g.We write h N for the characteristic function of Q p…S 1 tf1g K p .N /.For each automorphic representation D 1 ˝˝0 p p , we set S 1 D ˝p2S 1 p .
V r W det.x/ ¤ 0g and dg is a Haar measure on H r ‫./ށ.‬The zeta integral Z r .ˆQf ;r ; s; / is absolutely convergent for the range Let S 0 be a finite set of finite places of ‫.ޑ‬ Take a test function h S 0 2 C 1 c .G.‫ޑ‬ S 0 //, and let h S 0 denote the characteristic function of we define its Fourier transform c ˆ0 in the same manner.The zeta function Z [Shintani 1975;1/ satisfies the functional equation[Shintani 1975; Yukie 1993] Lemma 3.8.Let s 2 ‫.ޒ‬For s > 1, 1 p s / 1 ; and .s/D L.s; 1/; where L p .s; p / D .1 p .p/p s / 1 if p is unramified, and L p .s; p / D 1 if p is ramified.where .S / 0 .s/D d d s S .s/.For s Ä 1, j S .s/jÄ .N S / s j .s/j;where N S D Q p2S nf1g p. Proof.First of all, .1 p s / 1 1 for p 2 S. Hence S .s/Ä .s/.Let log S .s/D P p6 2S log.1 p s / 1 .Then .S / 0 .s/S .s/D X p6 2S and dx D Q v2R dx v .It is known that Z r;R .ˆR;s; O R / absolutely converges for Re.s/ rC1 2 , and is meromorphically continued to the whole s-plane.Suppose that R does not contain 1, that is, R consists of primes.Write Á p .x/ for the Clifford invariant of x 2 V 0 r ‫ޑ.‬ p /, see [Ikeda 2017, Definition 2.1], and set Á R ..x p / p2R / D Q p2R Á p .x p /.For D 1 R (trivial) or Á R , we put .ˆR/.x/ D ˆR.x/ .x/.It follows from the local functional equation [Ikeda 2017, Theorems 2.1 and 2.2] over ‫ޑ‬ p .R D fpg/ that Z r;p .ˆp; s; O p / is holomorphic in the range Re.s/ < 0, and Z r;p .ˆp; s; O p / possibly has a simple pole at s D 0. Hence, for any R, Z r;R .ˆR; s; O R / does not have any pole in the area Re.s/ < 0, but it may have a pole at s [Sweet 1995]1975, Lemma 1], one obtains Z 2;1 .bˆ1;n 1 2 ; O 1 / D 0 for any orbit O 1 in V 0 2 ‫./ޒ.‬Therefore, from the functional equation[Yukie 1992, Corollary (4.3)], one deduces Using the local functional equations in [Ikeda 2017, Theorem 2.1] (see also[Sweet 1995]), one gets T .ˆ;s/ D d d s 1 T .ˆ;s; s 1 / ˇs1 D0 and T .ˆ;s; s 1 / D Z ‫ށ‬ Z ‫ށ‬ jy 2 j s k.1; u/k s 1 ‰.y; yu/ du d y: r .S 1 ; h 1 / N aÄCb 1 for some a and b.
Remark 4.7.Keep the notation in the previous proposition.If … is the twisted endoscopic transfer of , then the claim immediately implies Ä dim ‫ރ‬ … K GL 2nC1 .N / : In fact, we have dim ‫ރ‬ K.N / D tr.I Â W … K GL 2nC1 .N / !…K GL 2nC1 .N / /, where I Â W …!… is the intertwining operator defining the twisted trace.Since I Â is of finite order, we have the above inequality; see the argument for [Yamauchi 2021, Theorem 1.6].
. Let D 1 ˝˝0 p p be an element of … for D r i D1 i .Let … p be the local Langlands correspondence of p to GL 2nC1 ‫ޑ.‬ p / established by [Arthur 2013, Theorem 1.5.1], and let L.… p / W L ‫ޑ‬ p ! GL 2nC1 ‫/ރ.‬ be the local L-parameter of … p , where L ‫ޑ‬ p D W ‫ޑ‬ p for each p < 1 and L ‫ޒ‬ D W ‫ޒ‬ SL 2 ‫./ރ.‬Since the localization p of the global A-parameter at p is tempered by Theorem 4.3, we see that L.… p / is equivalent to p .Since L.… p / is independent of 2 … and multiplicity one for GL 2nC1 ‫/ށ.‬ holds, the isobaric sum D r i D1 i as an automorphic representation of GL 2nC1 ‫/ށ.‬ gives rise to a unique global L-parameter on … .On the other hand, it follows from [Arthur 2013, Theorem 1.5.1] that j… p j Ä 2 2nC1 for the local A-packet … p at p if p j N , and … p is a singleton if p − N .It yields that j… j Ä 2 .2nC1/!.N / .Since the local Langlands correspondence by f D ˝0 p<1 p the finite part of the cuspidal representation .Plugging this into Proposition 4.2, we have jHE k .N / ng j D vol.K.N // 1 X D vol.K.N // 1 tr..1 K.N / // Ä vol.K GL 2nC1 .N // 1 tr .r i D1 i /.1 K GL 2nC1 .N / / D dim .r i D1 i / K GL 2nC1 .N / f ; where we denote f / in terms of the data .mi ; N; i ; i / with 1 Ä i Ä r.Since P m ‫ށ.‬ f /n GL 2nC1 ‫ށ.‬ f /=K.N / ' P m .O ‫/ޚ‬n GL 2nC1 .O ‫=/ޚ‬K.N / ' P m .‫=ޚ‬N‫/ޚ‬n GL 2nC1 .‫=ޚ‬N‫/ޚ‬ and a complete system of the representatives can be taken from elements in GL 2nC1 .O ‫,/ޚ‬ and therefore, they normalize K.N /.Then a standard method for fixed vectors of an induced representation shows that dim . It follows from[Ganapathy and Varma 2017, Lemma 8.2.1 (i)].Each cuspidal representation of G 0 ‫/ށ.‬contributing to l cusp;ort .N; ; 1/ can be regarded as a simple A-parameter.Also as a cuspidal representation, it strongly descends to a generic cuspidal representation … of H.‫/ށ‬ whose L-parameter L.… / at infinity of … is same as one of 1 .In this setting, by[Arthur 2013, Proposition 8.3.2(b)], the problem is reduced to estimate L cusp;generic;ort .H.‫/ޑ‬nH.‫;/ށ‬L.…/;1/m./KH .N / ; m. / 2 f0; 1; 2g;where runs over all irreducible unitary, cohomological orthogonal cuspidal automorphic representations of H.‫/ށ‬ whose L-parameter at infinity is isomorphic to L.… / with the central character D 1.Proposition 4.11.Keep the notations as above.Then l cusp;ort .2nCı;N;;1/ Ä C n .N /dim.L cusp;gen .H; N; L.… /; 1//, where C n .N / WD 2 .2nCı/!.N / andı D 0 if G 0 D GL 2n ; 1 if G 0 D GL 2nC1 :dim.L cusp;gen .H; N; L.… /; 1// c vol.K H .N // 1 cN dim.H / for some c > 0, when the infinitesimal character of L.… / is fixed and N ! 1.
These conditions are needed in order that for any quadratic extension M=‫ޑ‬ with the conductor d M dividing N , there exists an integral ideal N of M such that NN Â D .dM/where Â is the generator of Gal.M=‫./ޑ‬Proposition4.12.Keep the assumptions on N as above.Then Let M=‫ޑ‬ be the quadratic extension associated to and O M the ring of integers of M .Let Â be the generator of Gal.M=‫./ޑ‬LetK GL 2m M .N/ be the principal congruence subgroup of GL 2m .O ‫ޚ‬ ‫ޚ˝‬ O M / of the level N. Clearly, the Â -fixed part of K GL 2m M .N/ is K GL 2m .dM/where d M is the conductor of M=‫ޑ‬ and it contains K GL 2m .N / since d M jN .Applying [Yamauchi 2021, Theorem 1.6], we have for a cuspidal representation of GL 2m ‫/ށ.‬ and its base change … WD BC M=‫ޑ‬ ./ to GL 2m ‫ށ.‬ M /, vol K GL 2m .N / 1 tr .1 K GL 2m .N / / Ä vol K GL 2m If … is noncuspidal, then by Arthur and Clozel [1989], there exists a cuspidal representation of GL m ‫ށ.‬ M / such that … D M.N/ 1 tr ….1 K GL 2m M .N/ / : Recall that our contributing to L cusp;ort .2m;N; ; / is orthogonal, namely, L.s; ; Sym 2 / has a pole at s D 1.Note that L.s; …; Sym 2 / D L.s; ; Sym 2 /L.s; ; Sym 2 ˝ /.Now, L.s; .˝ // D L.s; ; ^2 ˝ /L.s; ; Sym 2 ˝ /.Suppose … is cuspidal.Then 6 ' ˝ .So the left-hand side has no zero at s D 1, and L.s; ; Sym 2 ˝ / has no zero at s D 1. Therefore, L.s; …; Sym 2 / has a pole at s D 1. Â : for any d jN; d ¤N Remark 5.2.As the referee pointed out, S old k ..N // is the intersection of S k ..N // with the smallest G.‫ށ‬ f /-invariant space of functions on G.‫/ޑ‬nG.‫/ށ‬ containing S k ..M // for all proper divisors M :The orthogonal complement S old k ..N // of S new k ..N // in S k ..N // with respect to Petersson inner product is said to be the old space.Let HE new k .N / be a subset of HE k .N / which is a basis of S new k ..N //.
Fix k D .k 1 ; : : : ; k n /, and let m D Q pjm np when p is spherical.For F 2 HE k .N /, let … be the Langlands transfer of F to GL 2nC1 .If F 2 HE k .N / g , then L.s; …; ^2/ has no pole at s D 1, and L.s; …; Sym 2 / has a simple pole at s D 1. Let Then F .p 2 / D Sym 2 .…/ .p/ and F .p/ 2 D … … .p/ D ^2.…/ .p/ C Sym 2 .…/ .p/.Note that F .p/ D b F .p/, and b F .p 2 / D 2 F .p 2 / F .p/ 2 .Let p ˛N n /; for N p 4n : Remark 9.2.By a more careful analysis, we can replace the error term O.N aÄCb is the number of prime factors of N and W .Sp/.x/ D 1 .sin 2 x/=.2 x/.(When we exchange two sums, if p − N , we use Proposition 9.1.If p j N , by the Ramanujan bound, jb F .p/j Ä n and jb F .p 2 /j Ä n.Hence by the trivial bound, we would obtain P pjN b F .p/ log p= where each i is an even Paley-Wiener function and O .u 1 ; : :: ; u `/ D O 1 .u 1 / O `.u `/.We assume that the Fourier transform O i of i is supported in .ˇ; ˇ/ for i D 1; : : : ; `.The `-level density function is Note that the coefficient of R p 2 there should be replaced with p 4 C p 3 C p 2 C p.. Let us first compute the coset decomposition.Put ƒ D GL n ‫ޚ.‬ p / where the identity element is denoted by 1 n .For any ring R, let S n .R/ be the set of all symmetric matrices of size n defined over R and M m n .R/ be the set of matrices of size m n defined over R. PutM n .R/ D M n n .R/ for simplicity.For each D 2 M n ‫ޚ.‬ p /, we define B.D/ WD fB 2 M n ‫ޚ.‬ p / j t BD D t DBg: For each B 1 ; B 2 2 B.D/, we write B 1 B 2 if there exists M 2 M n ‫ޚ.‬ p / such that B 1 B 2 D MD.We denote by B.D/= the set of all equivalence classes of B.D/ by the relation .We regard ‫ކ‬ p (respectively, ‫=ޚ‬p 2 ‫)ޚ‬ as the subset f0; 1; : : : ; p 1g (respectively, f0; 1; : : : ; p 2 1g) of ‫.ޚ‬ Let D I be the set of the following matrices in M n ‫ޚ.‬ p /:where we fill out zeros in the blank blocks.The cardinality of D I is 1Cp C Cp n 1 D .p n 1/=.p 1/ which is equal to that of ƒnƒd n 1 ƒ, where d n 1 D diag.1; p; : : : ; p/ containing n 1 entries of p.Similarly, let D II be the set of the following matrices: Further, for each D I s , B can be taken over if s ¤ 0, then x ¤ 0 and B D 0; if s D 0, then x D 0 and B D 0.If s ¤ 0, for D II s .y/,y 2 M s 1 ‫ކ.‬ p /,where B 21 ; B 22 and B 23 run over M 1 s ‫ކ.‬ p /; ‫=ޚ‬p 2 ‫,ޚ‬ and M 1 t ‫ކ.‬ p /, respectively.whereBrunsoverSn ‫ކ.‬ p / with r p .B/ D 1.The number of such B's is p n 1.Proof.We just apply the formula[Andrianov 2009, (3.94)].First we need to compute a complete system of representatives of ƒnƒtƒ ' .t 1 ƒt /\ƒnƒ for each t 2 fd n 1 ; d 1 ; p1 II ).For t D p 1 n , it is obviously a singleton.As for the computation of B.D/= , we give details only for D 2 D I , and the case of D II is similarly handled.For each D D D I s n ‚ …" ƒ .0;:::;0//: This agrees with [Kim et al. 2020a, (2.7)] when n D 2. [] Since p − N , we work on K D Sp.2n; ‫ޚ‬ p / instead of .N /.Put T p;n 1 WD pT .p;.0;:: : ; 0; 1// D K diag.1;1CA ; 1 Ä s Ä n 1; y 2 M s 1 ‫ކ.‬ p /: The cardinality of D II is 1 C p C C p n 1 D .p n 1/=.p 1/ which is equal to that of ƒnƒd 1 ƒ, where d 1 D diag.1; : : : ; 1; p/ containing n 1 entries of 1. Finally for each M 2 M n ‫ޚ.‬ p / we denote by r p .M / the rank of M mod p‫ޚ‬ p .Lemma A.2. Assume p is odd.The right coset decomposition T p;n 1 D `˛2J K˛consists of where D runs over the set D I and B runs over complete representatives of B.D/= such that r p .˛/D 1.Ã ;where B 22 runs over ‫=ޚ‬p 2 ‫ޚ‬ and B 23 runs over M 1 .n1/ ‫ކ.‬ p /; n g where d n 1 D diag.1; p; : : : ; p/ and d 1 D diag.1; : : : ; 1; p/ containing n 1 entries of p and 1, respectively.By direct computation, for t D d n 1 (respectively, t D d 1 ), it is given by D I (respectively, D Therefore, we may compute B.DA s /= and convert them by multiplying A 1 s on the right.We write B 2 B.DA s / as a block matrix B 12 B 12 B 13 p t B 12 B 22 p t B 32 t B 13 B 32 B 33 where B 11 2 S s ‫ޚ.‬ p /, B 22 2 ‫ޚ‬ p , and B 33 2 S n 1 s ‫ޚ.‬ p /.We write X 2 M n ‫ޚ.‬ p / as Our matrix B in B.DA s /= is considered by taking modulo XDA s for any X 2 M n ‫ޚ.‬ p /. Hence B can be, up to equivalence, of the form pX 11 p 2 X 12 pX 13 pX 21 p 2 X 22 pX 23 pX 31 p 2 X 32 pX 33 Therefore, B 11 ; B 13 ; B 21 ; B 22 ; B 23 , and B 33 run over M s ‫ކ.‬ p /; M s .n 1 s/.‫ކ‬ p / ; M 1 s.‫ކ‬ p / ; ‫=ޚ‬p 2 ‫;ޚ‬ M 1 .n 1 s/.‫ކ‬ p / ; and M n 1 s ‫ކ.‬ p /; respectively.The claim now follows from the rank condition r p .˛II .D; B// D 1 and the modulo K on the left again.As for D D p1 n in the case of type III, it is easy to see that S.D/= is naturally identified with S n ‫ކ.‬ p /. Recall p is an odd prime by assumption.The number of matrices in S n ‫ކ.‬ p / of rank 1 is given in [MacWilliams 1969, Theorem 2]. Then for each D D D I s , we have a bijection B.D/= !B.DA s /= ; B 7 !BA s : with respect to the partition s C 1 C .n 1 s/ of n where the column is also decomposed as in the row.where B 11 , B 33 , and B 13 belong to S s ‫ކ.‬ p /, S n 1 s ‫ކ.‬ p /, and M s .n 1 s/ ‫ކ.‬ p /, respectively.Further, to multiply A 1 s on the right never change anything.Therefore, (A-1) gives a complete system of representatives of B.D/= for D D D I s .The condition r p .˛I .D; B// D 1 and the modulo K on the left yield the desired result.For each D 2 D II s , a similar computation shows any element of S. p D/= is given by s ‚…"ƒ ]. Let us compute m. i / for each i .Let J I be the subset of J consisting of the elements Ä s Ä n 2; x 2 M 1 .n 1 s/ ‫ކ.‬ p / and ˛n 1 I D diag.p 2 ; p; : : : ; p; 1; p; : : : ; p/ containing n 1 entries of p both times.Similarly, let J II be the subset of J consisting of the elements ˛s II .y;B21;B22;B33/DÄsÄn 1, y 2 M s 1 ‫ކ.‬ p /, and B 21 ; B 23 , and B 22 run over M 1 s ‫ކ.‬ p /; M 1 .n 1 s/ ‫ކ.‬ p /, and ‫=ޚ‬p 2 ‫,ޚ‬ respectively.In addition,˛0 II .C 22 ; C 23 / D ; C 22 2 ‫=ޚ‬p 2 ‫;ޚ‬ C 23 2 M 1 .n1/‫ކ.‬ p /:Finally, let J III be the subset of J consisting of the elements S n ‫ކ.‬ p / with r p .B/ D 1:where the measure is normalized as vol.K/ D 1.Proof.Except for the case of type III, it follows from elementary divisor theory.For type III, it follows from[MacWilliams 1969] that the action of GL n ‫ކ.‬ p / on the set of all matrices of rank 1 in S n ‫ކ.‬ p / given by B 7 !tXBX;X 2 GL n ‫ކ.‬ p / and such a symmetric matrix B has two orbits O.diag.1; 0; : : : ; 0// and O.diag.g; 0; : : : ; 0//, both containing n 1 entries of 0, where g is a generator of ‫ކ‬ p .The claim follows from this and elementary divisor theorem again.For the latter claim, it is nothing but jJ j, and we may compute the number of each type.Remark A.4.Since K D Sp 2n ‫ޚ.‬ p / contains Weyl elements, Notice that Kd n 1 .p/KDK.p 2 d n 1 .p/ 1 /K; where d n 1 .p/WDdiag.1;p;:::;p;p 2 ; p; : : : ; p/ with n 1 entries of p both times..By definition and Lemma A.3 with Remark A.4, it is easy to see thatm.i/Dˇfˇ2JW i ˇ 1 2 Kd n 1 .p/KgĎShimura1994b,p. 52]for the first equality.We are now ready to compute the coefficients.For m. 1 /, we observe the p-integrality.We see that only ˛0 II .C 22 ; C 23 / with C 22 D 0 and C 23 D 0 1 .n1/cancontributethere.Hence, m. 1 / D 1.For m. 2 /, we observe the p-integrality and the rank condition.Then only ˛0 II .0;0 1 .n1//and ˛1 II .y;0;0;0 1 .n2//,with y 2 ‫ކ‬ p , can do there.Hence m. 2 / D 1Cp.For m. 3 /, only ˛III .B/, where B 2 S n ‫ކ.‬ p / with r p .B/ D 1 contribute.By Lemma A.2-(3), we have m. 3 / D p n 1.Finally, we compute m. 4 /.Since p 2 4 D I 4 , the condition is checked easily.All members of J D J I [ J II [ J III can contribute there.Therefore, we have only to count the number of each type.Hence, we have m. 4 / D as desired.Note that m. 4 / is nothing but the volume of Kd n 1 .p/K;see Lemma A.3.Recalling T p;n 1 WD pT .p; .0;: : : ; 0; 1//, we have n 1 ‚ …" ƒ p; : : : ; p/K D K diag.i ‚ …" ƒ p; : : : ; p; 1; i ‚ …" ƒ p; : : : ; p; p 2 ; i ‚ …" ƒ p; : : : ; p; p 2 ; i ‚ …" ƒ p; : : : ; p; 1; n ‚ …" ƒ .0;: : : ; 0// D KI 2n K: