# Zeta Function 2012

Interactive Research Center of Science, Graduate School of Science and Engineering, Tokyo Institute of Technology,

### Organizing Committee:

Nobushige Kurokawa (TITECH), Shuji Saito (TITECH), Masatoshi Suzuki (TITECH)

### Speakers

Anton Deitmar (Tübingen)

Ivan Fesenko (Nottingham)

Yasuro Gon (Kyushu)

Kazuya Kato (Chicago)

Christophe Soulé (IHÉS)

Masatoshi Suzuki (TITECH)

Michael Tsfasman (Moscow)

Frank Thorne (University of South Carolina)

Masao Tsuzuki (Sophia)

Shuji Yamamoto (Keio)

Takashi Nakamura (Tokyo University of Science)

Hirotaka Akatsuka (Kyushu)

### This workshop is partially supported by

JSPS Grant-in-Aid for Young Scientists (B) #21740004 representative Masatoshi Suzuki,

### 24(Mon)/Sep:

9:30-10:30 Kato, K., "Some generalizations of the conjectures of Sharifi, I."

10:45-11:45 Kato, K., "Some generalizations of the conjectures of Sharifi, II."

14:00-15:00 Fesenko, I., "Can higher number theory become the mainstream development in number theory of the 21st century?."

15:30-16:30 Fesenko, I., " Analytic and geometric adelic structures on arithmetic surfaces, and the zeta function."

16:45-17:30 Nakamura, T., "Zeros of Epstein zeta-functions (joint work with Łukasz Pańkowski)."

### 25(Tue)/Sep:

9:30-10:30 Soulé, C., "Zeta functions of varieties over the field with one element, I."

10:45-11:45 Soulé, C., "Zeta functions of varieties over the field with one element, II."

15:30-16:30 Deitmar, A., "Non-additive geometry plus epsilon."

16:45-17:30 Yamamoto, S., "Sum formula for certain multiple L-values and polylogarithms."

### 26(Wed)/Sep:

9:30-10:30 Thorne, F., "Counting fields using geometry and zeta functions, I."

10:45-11:45 Thorne, F., "Counting fields using geometry and zeta functions, II."

Free afternoon

### 27(Thu)/Sep:

9:30-10:30 Gon, Y., "Zeta functions defined by class numbers of binary quadratic forms over rings of algebraic integers."

10:45-11:45 Gon, Y., "Zeta functions of Ruelle and Selberg types for Hilbert modular groups."

14:00-15:00 Suzuki, M., "Zeta functions as a variant of the cosine function."

15:30-16:30 Suzuki, M., "Zeta functions and canonical systems of linear differential equations."

16:45-17:30 Akatsuka, H., "The Euler product for the Riemann zetafunction on the critical line."

### 28(Fri)/Sep:

9:30-10:30 Tsuzuki, M., "Relative trace formulas and spectral average of period integrals."

10:45-11:45 Tsuzuki, M., "A certain spectral average of central values of automorphic L-functions."

14:00-15:00 Tsfasman, M., "Sphere packings and asymptotic properties of zeta-functions, I."

15:30-16:30 Tsfasman, M., "Sphere packings and asymptotic properties of zeta-functions, II."

### Kato: Some generalizations of the conjectures of Sharifi

Romyar Sharifi formulated conjectures which relate ideal class groups of cyclotomic fields to modular curves. In this talk, I explain some attempts to generalize his conjectures.

### Fesenko: I. Can higher number theory become the mainstream development in number theory of the 21st century?

Higher number theory is a relatively recent development in mathematics, it is about 40 years old. It uses a higher dimensional perspective to study fundamental issues about arithmetic objects. One natural example comes from elliptic curves over global fields, e.g. rational numbers. Thousands of previous papers study various deep problems about these objects using noncommutative one-dimensional theories which involve extensions of the global field generated by torsion points of the object. However, the most famous results and methods can work over small number fields only, and cannot be extended to a functorial and general theory. Higher number theory point of view is that one can use two-dimensional class field theory which describes the maximal abelian extension of the function field of the object in purely commutative way, using Milnor K_2-theory. Since arithmetic is much better compatible with commutative methods than with noncommutative, higher number theory opens a large number of perspectives to achieve important progress and to develop functorial and general theories. This work is highly interdisciplinary, it uses higher structures coming from arithmetic, geometry, analysis, algebra. As another confirmation of the great unity of mathematics, there are similar developments in other area, one of which, categorification, goes to dimension n+1 in order to study objects in dimension n as shadows of richer structures in dimension n+1.

### Fesenko: II. Analytic and geometric adelic structures on arithmetic surfaces, and the zeta function

Some of deepest conjectures in modern mathematics relate two invariants, one defined analytically and another defined geometrically. An example is the original conjecture of Birch and Swinnerton-Dyer published in 1965. It is one of the Millennium Problems. In its more general form it proposes the equality of the analytic rank of the Hasse-Weil zeta function of an elliptic curve over a global field, i.e. the order of the pole of the zeta function at the central point, minus 1 or 2, and the arithmetic rank of the curve over the field. Geometrically, elliptic curves over global fields are best studied using their regular proper models viewed as arithmetic surfaces. Then this conjecture can be reformulated as a similar relation between the analytic rank of the zeta function of the model at the central point and the geometric Picard rank of the surface. Developments in higher adelic analysis and geometry by Beilinson, Parshin, the speaker and his school have resulted in understanding of a remarkable fact: unlike the classical number theory where there is only one adelic structure associated to a global field, there are two different adelic structures on arithmetic surfaces. One of them is essentially based on 1-cycles and is of geometric origin. This adelic structure knows the geometric rank of the surface. The second adelic structure is an analytic one, it is rather based on 0-cycles and is very useful in the speaker's two-dimensional generalization of the famous Iwasawa-Tate method of zeta integrals. In particular, the analytic adelic structure knows the analytic rank of the surface. The Birch and Swinnerton-Dyer conjecture can then be conceptually understood and deduced, modulo an auxiliary property, from a relation between the two adelic structures on surfaces. It appears that higher global class field theory, developed by K. Kato and Sh. Saito, in a new explicit adelic form, supplies such a relation between the two structures.

### Nakamura: Zeros of Epstein zeta-functions (joint work with Łukasz Pańkowski)

Let ${\mathcal{Q}}$ be a positive definite $n\times n$ matrix and $\zeta (s; {\mathcal{Q}})$ be the Epstein zeta-function associated with ${\mathcal{Q}}$. In this talk, we prove $\zeta (s; {\mathcal{Q}})$ has at least $cT$ zeros in the region $\Re s > (n-1)/2$ when $n\ge 4$ is even and ${\mathcal{Q}}$ satisfies some conditions. Moreover we show that $\zeta (s; {\mathcal{Q}})$ has at most $CT$ zeros in the region $\Re s > (n-1)/2$.

### Soulé: Zeta functions of varieties over the field with one element

In recent years, several definitions have been proposed for algebraic varieties over the "field with one element". For some of them there exists a reasonable notion of zeta function. We shall review these constructions, starting with those of Tits, Manin, and the speaker in the first hour, and ending with work of Connes and Consani in the second hour.

Non-additive geometry is an approach to geometry over the elusive "field of one element". In this talk, I will give a survey over recent activities which have been undertaken by several authors in recent years. In particular, I will speak about monoidal schemes and a non-additive cohomology theory.

### Deitmar: II. Non-additive geometry plus epsilon

This talk describes an attempt to add more structure to non-addititve geometry by allowing at least partial addition. This additive structure is best described by saying that one replaces monoids by monoidal embeddings into rings. In geometry this corresponds to replacing toric varieties by toric embeddings.

### Yamamoto: Sum formula for certain multiple L-values and polylogarithms

The sum formula is one of the most classical relations among multiple zeta values, which is due to Euler in the double zeta case. It expresses a Riemann zeta value as a sum of multiple zeta values. In this talk, we generalize it to a relation between single and multiple L-values. We also give an equivalent formula, which expresses single polylogarithm as in terms of certain multiple polylogarithms.

### Thorne : I. Counting fields using geometry and zeta functions

In the first lecture, I will give an overview of some of the methods that have been developed to count fields, class groups, and objects of related interest. It is often the case that such objects can be naturally parameterized in terms of groups acting on lattices. To give a classical example, class groups of quadratic fields can be parameterized by GL_2(Z)-orbits on the lattice of binary quadratic forms. Gauss, Mertens, and Siegel studied these GL_2(Z)-orbits using geometry, and determined the average size of the class group of these quadratic fields. (In the real case, these class numbers are weighted by the regulator.) This is an example of a prehomogeneous vector space, and we will discuss what prehomogeneous vector spaces are, and how their study leads to arithmetic density results. We will say something about Gauss's geometric approach, as further developed by Davenport-Heilbronn, Bhargava, and many others. We will briefly describe an analytic approach using zeta functions, which will feature more prominentlyin the second talk.

### Thorne: II. Counting fields using geometry and zeta functions (II)

In the second lecture, I will discuss my own work, joint with Takashi Taniguchi (one part also joint with Manjul Bhargava). Davenport and Heilbronn proved an asymptotic formula for the counting function of cubic fields, by parameterizing them by binary cubic forms. Shintani associated a zeta function to this space, and by refining and applying Shintani's theory we found a curious secondary term in this counting function, and also proved that they are not equidistributed in certain arithmetic progressions. We can also use our zeta function approach to count sextic fields with Galois group S_3, but I will highlight differences in this case which go beyond what we have successfully explained. I will also say a bit about connections to ongoing work of others (notably Bob Hough and Yongqiang Zhao, who each have very different explanations for the secondary term), and aboutthe many directions in which we hope to proceed further.

### Gon: I. Zeta functions defined by class numbers of binary quadratic forms over rings of algebraic integers

We consider certain zeta functions defined as Euler products in terms of "fundamental units" and class numbers of binary quadratic forms over the integer ring of a given algebraic number field. In the case of quadratic forms over the rational integer ring, it is known that this zeta function is described by the Selberg zeta function for the modular group SL(2,Z). We review basic facts on the Selberg zeta function for the modular group and introduce "Selberg type zeta functions" in other cases.

### Gon: II. Zeta functions of Ruelle and Selberg types for Hilbert modular groups

We present an example of Selberg type zeta functions for noncompact higher rank locally symmetric spaces. We study certain Selberg type zeta functions and　Ruelle type zeta functions attached to the Hilbert modular group of a real quadratic field. We show that they have meromorphic extensions to the whole complex plane and satisfy functional equations. Besides as an application, we have an asymptotic average of the class numbers of indefinite binary quadratic forms over the real quadratic integer ring.

### Suzuki: I. Zeta functions as a variant of the cosine function

Cosine and sine functions are one of the most simple entire functions such that all their zeros lie on a line. In this talk, we try to interpret zeta functions as a variant of the cosine function and present a usefulness of this idea starting from approximate functional equations for zeta functions and Levinson's method for the critical zeros of the Riemann zeta function. In addition, we review the theory of entire functions of Hermite-Biehler type as a counterpart of the exponential function. A goal of this talk is conjectural formulae of zeta functions that express them as sums of two entire functions of Hermite-Biehler type.

### Suzuki: II. Zeta functions and canonical systems of linear differential equations

A canonical system of linear differential equations is a generalization of Sturm-Liouville and Schrödinger equations. The theory of de Branges Hilbert space relates entire functions of Hermite-Biehler type with canonical systems. As a consequence, zeta functions should relate with canonical systems according to the first talk. Moreover, the Riemann hypothesis is described in terms of a canonical system. However, an explicit construction of a canonical system is quite difficult in general if we start from a Hermite-Biehler type entire function, because it is a kind of an inverse scattering problem. In this talk, we construct canonical systems attached to zeta functions by considering some integral operators endowed with kernels consisting of quotients of zeta functions. We note that Euler products is necessary for our construction.

### Akatsuka: The Euler product for the Riemann zetafunction on the critical line

In this talk we discuss the Euler product for the Riemann zeta-function on the critical line. In particular we formulate a conjectural asymptotic formula of the Euler product on the critical line. We will relate it with a condition on the distribution of prime numbers, which is a little deeper than the Riemann hypothesis. We will also give some numerical evidence of the asymptotic formula.

### Tsuzuki: I. Relative trace formulas and spectral average of period integrals

The relative trace formula is an apparatus originally introduced by H. Jacquet to establish functorial transfer of distingushed automorphic representations on different adele groups by comparing the period integrals of automorphic forms. In this talk, after we explain the basic idea of the relative trace formula in a general setting, we forcus on a quite simple but interesting example in the classical setting of elliptic cusp forms. We compute the spectral side and the geometric side of a relative trace formula as explicit as they can be; we apply it to estimate the asymptotic size of the average of the period integrals of (Maass) cusp forms.

### Tsuzuki: II. A certain spectral average of central values of automorphic L-functions

In this talk, we reintroduce the relative trace fromula in the setting of automorphic representations of $GL(2)$ over a totally real number field. We consider the relative trace formulas involving the following two kinds of periods of automorphic forms: (1) the toral periods (related to the Jacquet-Langlands $L$-functions) and (2) the Rankin-Selberg periods (related to the Asai $L$-functions). As an application, we give a spectral equidistribution result of central automorphic $L$-values and a possible application to subconvexity estimates of automorphic $L$-functions.

### Tsfasman: Sphere packings and asymptotic properties of zeta-functions

The classical problem how to pack equal spheres densely in a eucleadian space of high dimension can be looked at from the point of view of a number theorist or an algebraic geometer. In the first talk I shall describe the problem and give several simple constructions of dense sphere packings from number and function fields. Their parameters depend on the zeta-function of the field. If we want to know what happens in large dimensions, we need a tool to study the behaviour of zeta-functions of such fields when the field grows (e.g. in a tower). This question will be treated in the second talk: how to define a zeta-function of a tower of fields if we want it to reflect the asymptotic properties of zeta-functions of the fields of the tower? what can be said on the behaviour of such parameters as the regulator, the number of places of a given degree, and so on, when the discriminant (genus) grows? how do the zeroes of the zeta-function behave in a tower? what is a natural asymptotic statement we need instead of the generalized Riemann hypothesis?