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The above four steps covered in this book then lead into a fifth step: The development of this fifth step will follow what we did in [JoL 94] and [JoL 96]. We comment on this next at greater length. Introduction 5 Zetas The theta inversion formula is indeed not the end. It is the beginning of the theory of zeta functions that can be developed from such a formula. There are immediately two possible integral transforms that can be applied to the formula. Taking the Mellin transform in the geometric context leads to what is called the spectral zeta function, and various analogues.
In both cases, there is also a fudge term. Using the Gauss transform instead of the Mellin transform gives the logarithmic derivative of the sine function, up to further fudge terms. Note that the Gauss transform of terms other than the main noncuspidal periodization of the heat kernel yields fudge terms in the functional equation.
It is relevant to note here a comment of Iwaniec [Iwa 95]. He writes down the functional equation of the Selberg zeta function in multiplicative form If you wish, the Selberg zeta-function satisfies an analogue of the Riemann hypothesis. However, the analogy with the Riemann zeta-function is superficial. Furthermore, the functional equation Gangolli—Warner handle some problems arising from the trace formula by using what is called the Maass—Selberg relations and the truncation method.
We avoid these by using the noncuspidal trace separately first to get the theta series proper; and second by using the Eisenstein—cuspidal affair to take care of other terms. Furthermore, Gangolli— Warner apply the trace formula to a test function out of which the Selberg zeta function arises, but this leads into other convergence problems.
We apply the Gauss transform as the fifth step to an existing theta inversion relation. These are three of the main differences between our procedure and a previous development of Selberglike functions. For further comments, see the end of Section Thus we have a sequence of geometric objects that can be displayed vertically as a ladder: The L notation suggests logarithmic derivative as well as classical L-functions. Other fudge factors will include higher-dimensional versions of gamma functions. Thus we have a zeta ladder in parallel to the ladder of spaces.
Thus classical Dedekind zeta functions will occur as fudge factors of geometric zetas. For example, the above term occurs as a fudge term for all levels above 2, i. In particular, zeros or poles at a given lower level belonging to the fudge factors are the main zeros or poles of such lower levels. More appropriately, they might be called fudge zeros for higher levels. Furthermore, the theory in the complex case is not just a poor relative of the theory in the real case, as it has often been conceived.
We do just what is necessary for present purposes, but a whole new area extending items of analytic number theory what is called classically approximate formulas to thetas coming from a geometric context, involving special functions, is arising. In other words, the possibility of extending the Hardy—Littlewood-type expansion to more general series, including generalized zeta and Eisenstein series, is opening up. Connections with Geometry Ladders should occur with certain types of spaces arising from geometry, in various ways.
For instance, one can already see the Siegel modular ladder corresponding to the groups Sp2g associated with abelian varieties of dimension g ; the ladder of moduli spaces for K3-surfaces corresponding to the group SO0 2, 19 , and Calabi—Yau manifolds with their more complicated moduli structure, which may involve b below; the moduli ladder of forms of higher degree as in a paper of Jordan [Jor ]; etc.
In any case, the geometric ladder and the ladder of zeta functions reflect each other, thereby interlocking the theory of spaces coming from algebraic and differential geometry, with analysis and a framework whose origins to a large extent stem from analytic number theory. On the other hand, for some purposes, and in any case as a necessary preliminary for everything else, the purely analytic aspects have to be systematically available.
Topologists have concentrated on the classification problem via connected sums, but we find indications that the stratification structure and its connection with eigenexpansion analysis deserves greater attention. Having a stratification as suggested above, one may then define an associated zeta function following the five steps listed previously, and relate the analytic properties of these zeta functions with the algebraic-differential geometry of the variety.
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We mention only a few papers: For a more complete bibliography, cf. These papers are partly directed toward index theorems and the connection with number-theoretic invariants, as in the proof of a conjecture of Hirzebruch in [AtDS 83] and [Mul 87]. In retrospect, we would interpret the Atiyah—Donnelly—Singer paper as working on two steps of a ladder, with the compactification of one space by another, and the index theorem being applied on this compact manifold.
A reconsideration of the above-mentioned papers in light of the present perspective is now in order. Working as we do in the complex case, where it is possible to use an explicit Gaussian representation for the heat kernel, and using the Gauss transform rather than the Mellin transform, puts a very different slant on the whole subject, and allows us to go in a very different direction, starting with the explicit theta inversion relation and its Gauss transform.
The present book simply provides the very first step in the simplest ladder we could think of, in a manner that is adapted to its extension to the other steps of the ladder, as well as other ladders. Towers of Ladders The structure goes still further. However, one may consider an arbitrary number field, and the Hilbert—Asai symmetric space associated with it [Asa 70], [Jol 99]. Going up a tower ipso facto introduces questions of number theory. However, the existence of a ladder over a fixed base such as Spec Z or Spec Z[i] has its own numbertheoretic relevance, because the Riemann—Dedekind zeta should turn out to be a common factor of the zetas in all steps of the ladder, using Eisenstein series twisted by the heat kernel as in [Jol 02].
Our treatment prepares the ground for a continuation of [Jol 02], in what we view as an open-ended development. U is the subgroup of upper triangular unipotent matrices; A is the subgroup of diagonal matrices with positive diagonal components; K is the unitary subgroup. After giving the proof for this decomposition, we discuss characters on A, and then tabulate systematically integral formulas related to Haar measure on G, and various other decompositions e.
In the general case of semisimple Lie groups, these are mostly due to Harish-Chandra [Har 58].
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This heat Gaussian controls the general expansion of functions in various spaces. We choose the space of all Gaussians as a natural space of test functions for which we can prove the inversion formulas explicitly and very simply, at the level of elementary calculus. These test functions immediately provide the appropriate background for the heat kernel, which is characterized among them by simple conditions.
We are motivated differently, namely by the development of zeta functions from theta inversion relations, including regularized products, regularized series, and explicit formulas. The spherical inversion then provides an essential background for this direction. The essential feature here is the explicit form of the various formulas, notably our formulas for the orbital integral in Theorems 1. The first explicit connection of this chapter with the trace formula will be found at the end of Chapter 5.
In turn, the explicit version of the trace formula yields theta inversion formulas, whence zeta functions via Mellin and especially Gauss transforms as in [JoL 94], when applied to the heat kernel. One may summarize an essential relationship by saying that the t in the Poisson inversion formula is the same t as in the heat kernel or heat Gaussian. We note that the literature on SL2 C or R uses mostly if not entirely what is called the Selberg transform.
We find it much more valuable to plug in right away with the transforms used by Harish-Chandra. Gelfand—Naimark first treated the representation theory in the complex case of the classical groups [GeN 50], and Harish-Chandra completed this for all complex groups, and then all real groups [Har 54], [Har 58], by means of the Harish-Chandra series, taking his motivation from linear differential equations.
However, in Chapter XII of [JoL 01], for spherical inversion we suggested the possibility of an entirely different approach to the general case, having its origins in the Flensted—Jensen method of reduction to the complex case [FlJ 78], [FlJ 86]. This program is in the process of being carried out, using the normal transform and its relation to spherical inversion and the heat Gaussian, as described in the paper with A.
This includes the real semisimple symmetric spaces embedded in their complexification as a special case. In any case, [JoLS 03] shows that SLn C is not only a particular even significant example for the theory of semisimple or reductive groups. It is a dominant object in this theory, controlling the others. Let e1 , e2 be the standard vertical unit vectors of C2.
This corresponds to dividing by the norm. Then k is unitary. The element b is uniquely determined up to permutation of diagonal components. By the bijection in Theorem 1. More generally, an arbitrary square matrix is called regular if it can be conjugated to a regular diagonal matrix. Regularity will become relevant for measures in Section 1. A K-bi-invariant function is necessarily even.
Characters will arise naturally in connection with conjugation. This action induces what we also call the conjugation action on various functors, one of which is now discussed naively and directly. Action on the Lie algebra. Then E12 and iE12 are eigenvectors for the conjugation action by elements of A.
Until Chapter 6, it will be most convenient to work with the above decompositions. Readers may, however, already look at Section 6. See also Section 6. Left invariance by A is immediate. Let G be a locally compact group with Haar measure dg. Notation is as in Proposition 1. Assume that G, K are unimodular. Let dg, dp, dk be given Haar measures on G, P, K respectively. The second integral formula simply comes from plugging in Proposition 1. All four of G, K, U, A are unimodular, so all the hypotheses in the previous propositions are satisfied.
Unless otherwise specified, all integral formulas are with respect to the Iwasawa measure. The Iwasawa measure is canonically determined by the Iwasawa decomposition. Thus the basic Iwasawa measure integration formula can be written INT 1: K A U Remark. Next comes the Jacobian for the commutator map. Remember that the Jacobian is taken on C viewed as 2-dimensional over R, so in the change of variables formula, the absolute value of the complex derivative gets squared.
The next relation is the main one for this section. The statement about compact support is immediate from the matrix U-coordinate. We factorize J z for diagonal z as follows. The equality on the right comes from INT 3 and factorization 1. The first formula is applicable to all diagonal z, but the second, with the orbital integral, is applicable only to regular z.
As usual in measure-integration theory, the equality extends to functions for which the integrals are absolutely convergent, by continuity. This will be important for us, and certain convergence properties will be dealt with at the appropriate time and place. Harish-Chandra uses the notation Ff for what we call the Harish transform. He applies it to both expressions. See [Har 58], and further historical and notational comments in [JoL 01], p. Gelfand Let G be a locally compact unimodular group, and let K be a compact subgroup.
Then the convolution is commutative whenever the convolution integral is absolutely convergent. We introduce a new space, the space of Gaussians, defined below in Section 6. We shall then see in Theorem 1. This Gaussian space is admirably suited for getting explicit formulas, especially leading to the construction of the heat kernel, carried out in Chapter 2.
In [JoL 03b], it is shown that the Gauss space is dense in anything one wants. A We use the same Haar measure da on A as before, with the coordinate character y. If F does not have compact support, the integral may converge only in a restricted domain of s, usually some half-plane where the transform is analytic.
As in Section 1. Let f be measurable and K-bi-invariant on G. Using the fact that M, H preserve invariance under the Weyl group, we get the following corollary: We combine this with Theorem 1. The multiplicative property of Theorem 1. In particular, they hold for the space of Gaussians defined in the next section.
The Mellin transform is just a Fourier transform with a change of variables, with its usual formalism. The Harish transform will be developed further as in the next section, which gives another expression for the orbital integral, allowing us to compute this integral explicitly for a more explicit space of test functions, the Gaussian functions, which are introduced in Section 1. From the conjugation action on matrices, one sees that the centralizer in G of a regular diagonal element is the set of diagonal matrices.
We now go on with a theorem that allows for a more explicit determination of the orbital integral, and therefore of the Harish transform. We shall deal with the x, y coordinates, polar A-coordinates, and ordinary polar coordinates in C. The final expression will be given most simply in terms of the y-coordinate. The end result is the following theorem: Let the Haar measure on G be the Iwasawa measure as in Section 1. The proof will be mostly in computing the Jacobians for the various coordinates, to express the orbital integrand in a simple form, and we shall also determine the limits of integration in terms of the y-variable.
In Lie theory, the natural normalization of measures is to give all compact groups except points measure 1. This normalization will be seen to be the one that also fits polar integration, in terms of the polar decomposition of G. This will be carried out in Section 1. The computation of Theorem 1. There exist a number of such spaces in current use, such as the Schwartz space, or functions with compact support.
For our purposes, we define the space of Gaussians on G so that under the spherical transform, they correspond to ordinary Gaussian functions on R. This measure is also the measure that causes Fourier inversion to be true without any extraneous constant factor. We say that this measure is Fourier normalized. The main point is that there is such a factor having this desirable effect.
The Heat Kernel and Theta Inversion on SL2
However, the heat differential equation will be irrelevant for a while, so we may as well deal with any Gaussian. As usual, the Haar measure on G is the Iwasawa measure. We then use the definition of the Harish transform as a product of the orbital integral times the D a factor. This factor cancels, giving the stated answer. Next comes the Mellin transform. Sometimes we deal with the imaginary axis iR, and Gauss iR is the space generated by the functions having the above value at ir. We let Gauss G be the G-Gauss space, i. Immediate from the linear independence of the exponential functions with different constants c.
We tabulate one more useful integral. The world is made up so that this normalization is also the normalization that makes Fourier inversion come out without an extra constant. We defined Gaussians using this coordinate. Note that log y 2 is uniquely determined by g, and is real analytic on G. Properties of the polar height will be given in Section 1. We first go straight for spherical inversion. Even here, we have the natural scaling factor 2. We discuss scaling the form systematically in Sections 2. The symmetric way will be relevant in the next section, so we add some comments concerning it.
We shall use the Gauss space in a natural way as we develop the polar Haar measure. We want an integration formula in terms of this A-component. For K-bi-invariant functions, one does not need the two K integrations in the above formula.
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The total dk-measure of K is assumed to be normalized to 1. The question arises how the polar Haar measure is related to the Iwasawa measure. The answer is the following theorem: Since the formula holds with constant factor 1 on the Gauss space, it holds with constant factor 1 for any test function for which the integrals are absolutely convergent.
We may call Sp the polar normalized spherical transform, or simply the polar spherical transform. We shall exhibit an integral kernel for Sp quite different from the one used in the definition of spherical transform. It was the K-invariantization of a character; it will now be related to polar coordinates. The computation is at the same level of calculus as in the proof of Theorem 1.
We observe that the polar spherical kernel has a product structure. We do this as follows. We plug this in formula 2. As to the second statement, we first make explicit the L2 -scalar product on G, for the polar measure. We have now concluded the basic analysis of spherical inversion. The next chapter will give a major application. This property becomes especially significant in the higher-rank case. In higher dimensions, one has to assume the evenness condition in addition to the K-bi-invariance. Hence a function on A that is invariant under permutations extends to a K-bi-invariant function on G.
We first compare the polar height and the height of the A-Iwasawa component. The polar height is bigger. In general reductive Lie group theory, this result is due to Harish-Chandra. Let e1 , e2 be the unit column vectors of C2. Clearing denominators, the inequality falls out. The following property is deeper, and is due to Cartan—Harish-Chandra. See [Lan 99], on Bruhat—Tits spaces, which provides an elementary self-contained treatment sufficient for our purposes.
See also [JoL 01a], Chapter This is an immediate consequence of the triangle inequality in Theorem 1. A full treatment on Posn R is in [Lan 99], mentioned above. See also [JoL 01], Section The polar height will be seen in the next chapter as the function that comes most basically into the definition of a normalized Gaussian, the heat Gaussian, giving rise to its point-pair invariant, the heat kernel.
See especially the items on scaling in Section 2. Chapter 2 The Heat Gaussian and Heat Kernel The Gaussian space is very useful when computing certain explicit simple formulas which come up in spherical inversion theory. One also wants to know that an identity for all Gaussian functions implies a similar identity for more general test functions usually occurring in analysis.
This section provides a background for the systematic approach in [JoL 03b]. For some purposes, one wants to normalize the Gaussian functions more precisely. The ultimate normalization is that of the heat kernel. The structure of the heat kernel was discovered by Gangolli in his fundamental paper [Gan 68], including the fact that it is the inverse image of a normalized Gaussian on Euclidean space under the spherical transform.
One then gets a formula for the heat kernel in terms of a Gaussian, which exhibits a quadratic exponential decay. The existence of such a formula in closed form holds only on complex groups G. On real groups, it is given in integral form. The general formalism and structure of this integral and its connection with spherical inversion is explained in [JLS 03], as a special case of totally geodesic embeddings of symmetric spaces into each other.
This general framework also provides a more extensive context for the Flensted— Jensen transform, going from the complex to the real case [FlJ 78]. The first two conditions are obvious from the definitions. As to WD 3, it suffices to prove it for one choice of Haar measure, say the polar measure. By Chapter 1, Corollary 1. We formalize these notions below. The Iwasawa measure gives rise to the Iwasawa Dirac family, obtained from 2. Since we shall scale measures, we must now index this transform to indicate the dependence on the measure, and thus write SIw.
For the polar measure, 2. For example, we can say that formula 2. For the moment, repeating 2. Further comments on scaling will be made in Sections 2. By scaling the form B, we shall see that we can get rid of a factor 2. In light of Theorem 2. Writing gt by itself, it is understood that a Haar measure has been fixed. From the general properties of Section 1.
It is symmetric and G-invariant. The same remarks as above apply to the heat kernel.
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The next section will give essentially general consequences of the Weierstrass— Dirac property in the context of convolutions. Note that a bounded function is in LEG. Functions will be assumed measurable. Uniformly for z in a compact set and 0 2. The choice of such a measure also determines the convolution operation. The next theorems are valid with these compatible choices. The heat Gaussian and kernel satisfy the semigroup property: The space of Gaussians Gauss G is closed under the convolution product.
The only difficulty in light of Chapter 1, Theorem 4. A priori it might lie in a bigger space. One might actually compute the convolution integral, but we argue another way, avoiding another computation, say for the polar heat kernel. We know from Chapter 1, Theorem 1. Since the heat Gaussians range over all Gaussians up to a constant factor, the last statement is also proved. If the function f in Theorem 2. In the present case of Gaussian convolution, one can deal with test functions having a nontrivial rate of growth such as characters, which have linear exponential growth, as we shall see in Section 3.
We start by repeating Proposition 2. The heat kernel satisfies the conditions defining a WD family of kernels. These are the conditions that define a WD family on a Riemannian manifold when there is no group that allows one to pass from the one-variable function gt to the two-variable function Kt. One formulation of an approximation theorem associated with Weierstrass—Dirac families first manifested itself in Weierstrass [Wei 85].
For textbook treatments, see for instance [Lan 93] Chapter 8 or [Lan 99] on approximation theorems. The most basic approximation theorem valid for all WD families is the following: However, with the quadratic exponential decay, we can do better than that. The pattern of proof is the same as that of the standard case just quoted, but we have to use the faster decay of the heat kernel. We want the analogous result for approximation in L1 and L2. We need standard basic inequalities from integration theory on a unimodular locally compact group as follows.
Readers will find proofs in standard texts on integration and measure theory, e. The proofs are usually given on Euclidean space, but they go over to the general group setting. Proof of Proposition 2. G G Interchanging the order of integration, we have absolute convergence. Hence we can apply Fubini to see that the defining integral converges absolutely. Then we note that 2. This concludes the proof.
For further approximation properties of Gaussians, see [JoL 03]. Gaussians can be used to approximate more general functions. One can verify formulas first for Gaussians, and then by continuity extend the formulas to more general spaces. Even more importantly, Gaussians can be used as the natural functions giving rise to explicit theta inversion formulas in the context of semisimple Lie groups. For some other applications, one has to let t go to the right rather than the left.
We are thus led to complexify t as follows. The formula for the Iwasawa gt from Section 2. This is immediate from 2. Of course, Proposition 2. This proves the theorem. The final conclusion depending on approximation applies to the cases in which we have proved the approximation, namely: The above theorem is the first concerning convolution eigenvalues, in the present case eigenvalue 0.
In Chapter 3, we shall consider further theorems concerning eigenvalues. See Chapter 3, Proposition 2. We show how one Gaussian can be normalized in a significant way. We have to discuss invariant differential operators, because one of the two normalization conditions will be a differential equation. We recall briefly such operators in the special case at hand. We view g as a 6-dimensional vector space over R. Then g has subspaces: This form is positive definite on a, and G-invariant for the conjugation action of G on g.
We shall deal with the Casimir operator, whose definition is based on a universal property in multilinear algebra as follows. Let B be a nonsingular symmetric bilinear form on V.
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This element is called the universal Casimir element depending on B. For higher dimensions, cf. Note that Casimir depends on the choice of B, but is independent of the choice of Haar measure. Now we come to the lemmas proving 2. We reproduce here a standard computation. This proves the lemma.
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We shall deal with the notion of direct image in the special case that concerns us here. As to the second equality, let f be a function on A. The normalization property shows that this expression is constant in t1 , t2. This concludes the proof of Proposition 2. So ys is an eigenfunction of Casimir on G. This will be used in connection with Eisenstein series in Chapter We may rewrite 2. We are then in the situation of Proposition 2. In Chapter 11 we apply this to convolution between the heat kernel and Eisenstein series.
These can be viewed as a continuation of the present considerations. Note that the eigenvalue of Proposition 2. It will be systematically used all the way, in cases in which it becomes more elaborate to justify its legitimacy. In the next section we meet a slightly more involved case, using the inverse spherical transform rather than S itself. Then having defined the heat kernel, we take convolutions with the heat kernel, and DUTIS with the heat kernel forever after.
For the short proof, see if necessary [Lan 93] Chapter 8, Lemma 8. Let V be an open subset of Rn. The factor 2 disappears in the formula for the eigenvalues if one uses the Laplacian instead of Casimir. Quite generally, the Casimir element depends on the original choice of the scalar product on the Lie algebra. As in Section 2. Thus the question arises, how does Casimir change under scaling, i. Hence the Casimir operator scaling formula becomes the following: Thus if we use 2B instead of B for the basic bilinear form, then Casimir with respect to 2B is the above Laplacian.
R] in the complex case with which we are dealing. These are the normalizations that are used in a standard way by the differential geometers on H3. We are not doing physics; this is just a code name for historical reasons. Mathematically, this heat equation and its solutions have a controlling effect in many, if not all, areas of mathematics. For the fundamental properties, especially the characterization of the heat kernel by initial conditions, see Dodziuk [Dod 83].
For our purposes, we need only adopt a direct definition, based on the preceding sections, as follows. The significant aspect of this expression is that the coefficient of t in the exponent is the eigenvalue of Casimir on the spherical kernel Proposition 2. We now come to the second fundamental condition pertaining to the heat equation.
Then gt satisfies the heat equation. Changing the Haar measure by a constant changes the heat Gaussian by a constant, so it suffices to prove the theorem for one measure, which we take as the polar measure. By Chapter 1, Theorem 7. Let Kt be the point-pair invariant associated with gt. Then Kt satisfies the heat equation in each variable. The last stage of the construction of the heat kernel in Theorem 2.
Then F t, z satisfies the heat equation. We shall meet further examples of Theorem 2. Actually, on a Lie group, we have the possibility of using jointly or separately the normalization of the Haar measure, and the scalar product used to define the dual basis in the Lie algebra, whence the Casimir element, whence the heat equation.
Scaling the Haar measure, i. In any case, the construction we have followed gives ipso facto the properties relating the heat kernel to the spherical transform. The two conditions can therefore be taken ab ovo as the conditions defining the heat family. We arrived at its existence and other properties by using the specific context of Gaussians in addition to the spherical inversion map to get immediately an explicit determination of this heat family.
The arguments given here are expected to work for complex reductive groups in general. To get the real case in addition, cf. Originally Gangolli did the general real case including the complex case directly, using the Harish-Chandra spherical inversion theory [Gan 68], and we followed this main idea. To show that such a solution satisfies the Dirac condition as we formulated it takes some additional arguments. Gangolli told us that one can use a lemma of Gording [Gar 60] for this purpose. Instead of using the same function for the Jacobian factor and for the metric, we keep these functions separate.
As before, let B be the trace form on a. We now view the above formula as a function of B. This is immediate from Sc 3 and the formula. If a function satisfies the heat equation, then any scalar multiple of the function satisfies the heat equation. The others are trivially valid under scaling. In this case, we have the normalization of the geometers.
Back cover copy The present monograph develops the fundamental ideas and results surrounding heat kernels, spectral theory, and regularized traces associated to the full modular group acting on SL2 C. The authors begin with the realization of the heat kernel on SL2 C through spherical transform, from which one manifestation of the heat kernel on quotient spaces is obtained through group periodization. From a different point of view, one constructs the heat kernel on the group space using an eigenfunction, or spectral, expansion, which then leads to a theta function and a theta inversion formula by equating the two realizations of the heat kernel on the quotient space.
The trace of the heat kernel diverges, which naturally leads to a regularization of the trace by studying Eisenstein series on the eigenfunction side and the cuspidal elements on the group periodization side. By focusing on the case of SL2 Z[i] acting on SL2 C , the authors are able to emphasize the importance of specific examples of the general theory of the general Selberg trace formula and uncover the second step in their envisioned "ladder" of geometrically defined zeta functions, where each conjectured step would include lower level zeta functions as factors in functional equations.
Table of contents Introduction. Review Text From the reviews: A well-prepared graduate student would do well with this book. More experienced analytic number theorists will find it enjoyable and spellbinding. I heartily recommend to other analytic number theorists of a similar disposition.
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