p-adic String Amplitudes and Local Zeta Functions (joint work with H. Compeán and W. Zúñiga)

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1 -adic String Amlitudes and Local Zeta Functions (joint work with H. Comeán and W. Zúñiga) Miriam Bocardo Gasar Advisor: Dr. Wilson A. Zúñiga Galindo Centro de Investigación y de Estudios Avanzados del I. P. N. Deartment of Mathematics 26 de mayo de 2017

2 -adic String Amlitudes and Local Zeta Functions (joint work with H. Comeán and W. Zúñiga) Overview These connections were established in our aer Regularization of - adic String Amlitudes, and Multivariate Local Zeta Functions, M. Bocardo-Gasar, H. García-Comeán, W. Zúñiga-Galindo. Results: *-adic string amlitudes are essentially local zeta functions. * Amlitudes are convergent integrals that admit meromorhic continuations as rational functions i.e. the roblem of regularization of -adic string amlitudes. Miriam Bocardo Gasar Advisor: Dr. Wilson-adic A. Zúñiga String GalindoCentro Amlitudes and de Investigación Local Zeta Functions y de Estudios (joint26 Avanzados work de mayo with del H. Comeán I P. N.Deartment and 2 / W. 23 Z

3 -adic String Amlitudes and Local Zeta Functions (joint work with H. Comeán and W. Zúñiga) Overview These connections were established in our aer Regularization of - adic String Amlitudes, and Multivariate Local Zeta Functions, M. Bocardo-Gasar, H. García-Comeán, W. Zúñiga-Galindo. Results: *-adic string amlitudes are essentially local zeta functions. * Amlitudes are convergent integrals that admit meromorhic continuations as rational functions i.e. the roblem of regularization of -adic string amlitudes. iriam Bocardo Gasar Advisor: Dr. Wilson-adic A. Zúñiga String GalindoCentro Amlitudes and de Investigación Local Zeta Functions y de Estudios (joint26 Avanzados work de mayo with del H. Comeán I P. N.Deartment and 2 / W. 23 Z

4 -adic String Amlitudes and Local Zeta Functions (joint work with H. Comeán and W. Zúñiga) Physicists have discovered that the limit when 1 allows to ass from the non-archimedean world to the Archimedean one. Hence -adic string theory could be useful to answer interesting questions in usual string theory. Paer in rearation aims to construct new toological string zeta functions. *This aroximation is based on the limit when 1 as in the classical toological zeta function introduced by Denef and Loeser. Miriam Bocardo Gasar Advisor: Dr. Wilson-adic A. Zúñiga String GalindoCentro Amlitudes and de Investigación Local Zeta Functions y de Estudios (joint26 Avanzados work de mayo with del H. Comeán I P. N.Deartment and 3 / W. 23 Z

5 -adic String Amlitudes and Local Zeta Functions (joint work with H. Comeán and W. Zúñiga) Physicists have discovered that the limit when 1 allows to ass from the non-archimedean world to the Archimedean one. Hence -adic string theory could be useful to answer interesting questions in usual string theory. Paer in rearation aims to construct new toological string zeta functions. *This aroximation is based on the limit when 1 as in the classical toological zeta function introduced by Denef and Loeser. Miriam Bocardo Gasar Advisor: Dr. Wilson-adic A. Zúñiga String GalindoCentro Amlitudes and de Investigación Local Zeta Functions y de Estudios (joint26 Avanzados work de mayo with del H. Comeán I P. N.Deartment and 3 / W. 23 Z

6 -adic String Amlitudes and Local Zeta Functions (joint work with H. Comeán and W. Zúñiga) Plan A brief introduction to Multivariate local zeta functions. -adic string amlitudes. Toological string zeta functions. Miriam Bocardo Gasar Advisor: Dr. Wilson-adic A. Zúñiga String GalindoCentro Amlitudes and de Investigación Local Zeta Functions y de Estudios (joint26 Avanzados work de mayo with del H. Comeán I P. N.Deartment and 4 / W. 23 Z

7 -adic String Amlitudes and Local Zeta Functions (joint work with H. Comeán and W. Zúñiga) Plan A brief introduction to Multivariate local zeta functions. -adic string amlitudes. Toological string zeta functions. Miriam Bocardo Gasar Advisor: Dr. Wilson-adic A. Zúñiga String GalindoCentro Amlitudes and de Investigación Local Zeta Functions y de Estudios (joint26 Avanzados work de mayo with del H. Comeán I P. N.Deartment and 4 / W. 23 Z

8 -adic String Amlitudes and Local Zeta Functions (joint work with H. Comeán and W. Zúñiga) Plan A brief introduction to Multivariate local zeta functions. -adic string amlitudes. Toological string zeta functions. Miriam Bocardo Gasar Advisor: Dr. Wilson-adic A. Zúñiga String GalindoCentro Amlitudes and de Investigación Local Zeta Functions y de Estudios (joint26 Avanzados work de mayo with del H. Comeán I P. N.Deartment and 4 / W. 23 Z

9 -adic String Amlitudes and Local Zeta Functions (joint work with H. Comeán and W. Zúñiga) Multivariate local zeta functions Let f i : U Q n Q be non-constant analytic functions, Φ : Q n C be a locally constant function with comact suort on U, and s i C be comlex numbers. The multivariate local zeta function attached to ({s i }, {f i }, φ) is defined as l Z φ (s 1,..., s l ; f 1,..., f l ) = φ(x) f i (x) si dx. Q n i=1 Loeser studied these integrals and showed that they admit meromorhic continuations as rational functions in the variables si, i = 1,..., l. Loeser, F., Fonctions zêta locales d Igusa à lusieurs variables, intégration dans les fibres, et discriminants. Ann. Sc. Ec. Norm. Su. 22 (1989), no. 3, Miriam Bocardo Gasar Advisor: Dr. Wilson-adic A. Zúñiga String GalindoCentro Amlitudes and de Investigación Local Zeta Functions y de Estudios (joint26 Avanzados work de mayo with del H. Comeán I P. N.Deartment and 5 / W. 23 Z

10 -adic String Amlitudes and Local Zeta Functions (joint work with H. Comeán and W. Zúñiga) Local Zeta Functions What is the relation? -adic String Amlitudes Miriam Bocardo Gasar Advisor: Dr. Wilson-adic A. Zúñiga String GalindoCentro Amlitudes and de Investigación Local Zeta Functions y de Estudios (joint26 Avanzados work de mayo with del H. Comeán I P. N.Deartment and 6 / W. 23 Z

11 -adic String Amlitudes and Local Zeta Functions (joint work with H. Comeán and W. Zúñiga) II. -adic String Amlitudes The string theory was motivated by need of understanding asects of the strong interactions of elementary articles. Strong interactions are described by functions (amlitudes), which satisfy some hysical requirements. In 1968, G. Veneziano roosed a function for describing the interaction of four articles. The generalization of the Veneziano amlitude to case of interaction of N articles: N 2 A (N) (k) = x i k 1k i R 1 x i k N 1k i R x i x j k N 2 i k j R dx i. R N 3 2 i<j N 2 Miriam Bocardo Gasar Advisor: Dr. Wilson-adic A. Zúñiga String GalindoCentro Amlitudes and de Investigación Local Zeta Functions y de Estudios (joint26 Avanzados work de mayo with del H. Comeán I P. N.Deartment and 7 / W. 23 Z

12 -adic String Amlitudes and Local Zeta Functions (joint work with H. Comeán and W. Zúñiga) -adic String Amlitudes The -adic string theory started around 1987 with the work of three grous: Volovich, Freund and Witten, and Framton and Okada. Volovich noted that the integral exression for the Veneziano amlitude of the oen string can be generalized to a -adic integral. Miriam Bocardo Gasar Advisor: Dr. Wilson-adic A. Zúñiga String GalindoCentro Amlitudes and de Investigación Local Zeta Functions y de Estudios (joint26 Avanzados work de mayo with del H. Comeán I P. N.Deartment and 8 / W. 23 Z

13 -adic String Amlitudes and Local Zeta Functions (joint work with H. Comeán and W. Zúñiga) The -adic string amlitudes for N articles have the form: N 2 N 2 A (N) (k) := x i k1ki 1 x i k N 1k i ki kj x i x j dx i, where N 2 Q N 3 2 i<j N 2 dx i is the normalize Haar measure of Q N 3, k = (k 1,..., k N ), k i = (k 0,i,..., k 25,i ), i = 1,..., N, N 4, ( with Minkowski roduct k i k j = k 0,i k 0,j + k 1,i k 1,j + + k 25,i k 25,j ) satisfying N k i = 0, k i k i = 2 for i = 1,..., N 1. i=1 Brekke, Freund and Witten (Non-Archimedean string dynamics) estudied these -adic string amlitudes and work out in exlicit forms. A central roblem is to know whether or not integrals of tye (1) converge for some values k i k j C. Miriam Bocardo Gasar Advisor: Dr. Wilson-adic A. Zúñiga String GalindoCentro Amlitudes and de Investigación Local Zeta Functions y de Estudios (joint26 Avanzados work de mayo with del H. Comeán I P. N.Deartment and 9 / W. 23 Z (1)

14 -adic String Amlitudes and Local Zeta Functions (joint work with H. Comeán and W. Zúñiga) The -adic string amlitudes for N articles have the form: N 2 N 2 A (N) (k) := x i k1ki 1 x i k N 1k i ki kj x i x j dx i, where N 2 Q N 3 2 i<j N 2 dx i is the normalize Haar measure of Q N 3, k = (k 1,..., k N ), k i = (k 0,i,..., k 25,i ), i = 1,..., N, N 4, ( with Minkowski roduct k i k j = k 0,i k 0,j + k 1,i k 1,j + + k 25,i k 25,j ) satisfying N k i = 0, k i k i = 2 for i = 1,..., N 1. i=1 Brekke, Freund and Witten (Non-Archimedean string dynamics) estudied these -adic string amlitudes and work out in exlicit forms. A central roblem is to know whether or not integrals of tye (1) converge for some values k i k j C. Miriam Bocardo Gasar Advisor: Dr. Wilson-adic A. Zúñiga String GalindoCentro Amlitudes and de Investigación Local Zeta Functions y de Estudios (joint26 Avanzados work de mayo with del H. Comeán I P. N.Deartment and 9 / W. 23 Z (1)

15 -adic String Amlitudes and Local Zeta Functions (joint work with H. Comeán and W. Zúñiga) -adic String Amlitudes A (N) (k) = Q N 3 N 2 x i k 1k i 1 x i k N 1k i 2 i<j N 2 x i x j k i k j N 2 dx i. Z φ (k) = Q N 3 N 2 φ (x) x i k 1k i 1 x i k N 1k i 2 i<j N 2 x i x j k i k j N 2 dx i. Miriam Bocardo Gasar Advisor: Dr. Wilson-adic A. Zúñiga String GalindoCentro Amlitudes and de Investigación Local Zeta Functions y de Estudios (joint 26Avanzados work de mayo with del H Comeán I. P. N.Deartment 10 and/ W. 23 Z

16 -adic String Amlitudes and Local Zeta Functions (joint work with H. Comeán and W. Zúñiga) -adic String Amlitudes A (N) (k) = Q N 3 N 2 x i k 1k i 1 x i k N 1k i 2 i<j N 2 x i x j k i k j N 2 dx i. Z φ (k) = Q N 3 N 2 φ (x) x i k 1k i 1 x i k N 1k i 2 i<j N 2 x i x j k i k j N 2 dx i. Miriam Bocardo Gasar Advisor: Dr. Wilson-adic A. Zúñiga String GalindoCentro Amlitudes and de Investigación Local Zeta Functions y de Estudios (joint 26Avanzados work de mayo with del H Comeán I. P. N.Deartment 10 and/ W. 23 Z

17 -adic String Amlitudes and Local Zeta Functions (joint work with H. Comeán and W. Zúñiga) Consider the following integral: Z(s) = Q s x dx. Assume that this integral exists for some s 0 R. Then s J 0 (s 0 ) = x 0 dx and J 1(s 0 ) = s x 0 dx Z Q \Z exist. But converges for s 0 > 1, J 1 (s 0 ) = converges for s 0 < 1. J 0 (s 0 ) = j=1 j Z s0. x s0 dx = j(1+s0) < Miriam Bocardo Gasar Advisor: Dr. Wilson-adic A. Zúñiga String GalindoCentro Amlitudes and de Investigación Local Zeta Functions y de Estudios (joint 26Avanzados work de mayo with del H Comeán I. P. N.Deartment 11 and/ W. 23 Z j=1

18 -adic String Amlitudes and Local Zeta Functions (joint work with H. Comeán and W. Zúñiga) Regularization of -adic String Amlitudes Let N 4 and s ij C with s ij = s ji for 1 i < j N 1. We define the -adic oen string N-oint zeta function as Z (N) (s) = Q N 3 N 2 x i s 1i where s = (s ij ) C D, N 2 1 x i s (N 1)i D = (N 4)(N 3)/2 + 2(N 3). 2 i<j N 2 x i x j s ij N 2 dx i, (2) dx i is the normalized Haar measure of Q N 3, Miriam Bocardo Gasar Advisor: Dr. Wilson-adic A. Zúñiga String GalindoCentro Amlitudes and de Investigación Local Zeta Functions y de Estudios (joint 26Avanzados work de mayo with del H Comeán I. P. N.Deartment 12 and/ W. 23 Z

19 -adic String Amlitudes and Local Zeta Functions (joint work with H. Comeán and W. Zúñiga) We use an aroach insired in the calculations resented in Brekke Lee, Freund, Peter G. O., Olson Mark, Witten Edward, Non-Archimedean string dynamics, Nuclear Phys. B 302 (1988), no. 3, and in the Igusa s -adic stationary hase formula. We define for I T = {2,..., N 2}, the sector attached to I as { } Sect(I) = (x 2,..., x N 2 ) Q N 3 ; x i 1 i I and Then Z (N) (s; I) := Sect(I) N 2 Q N 3 x i s1i = I T Sect(I) 1 x i s (N 1)i 2 i<j N 2 Z (N) (s) = I T Z (N) (s; I). x i x j sij N 2 dx i. Miriam Bocardo Gasar Advisor: Dr. Wilson-adic A. Zúñiga String GalindoCentro Amlitudes and de Investigación Local Zeta Functions y de Estudios (joint 26Avanzados work de mayo with del H Comeán I. P. N.Deartment 13 and/ W. 23 Z

20 -adic String Amlitudes and Local Zeta Functions (joint work with H. Comeán and W. Zúñiga) We use an aroach insired in the calculations resented in Brekke Lee, Freund, Peter G. O., Olson Mark, Witten Edward, Non-Archimedean string dynamics, Nuclear Phys. B 302 (1988), no. 3, and in the Igusa s -adic stationary hase formula. We define for I T = {2,..., N 2}, the sector attached to I as { } Sect(I) = (x 2,..., x N 2 ) Q N 3 ; x i 1 i I and Then Z (N) (s; I) := Sect(I) N 2 Q N 3 x i s1i = I T Sect(I) 1 x i s (N 1)i 2 i<j N 2 Z (N) (s) = I T Z (N) (s; I). x i x j sij N 2 dx i. Miriam Bocardo Gasar Advisor: Dr. Wilson-adic A. Zúñiga String GalindoCentro Amlitudes and de Investigación Local Zeta Functions y de Estudios (joint 26Avanzados work de mayo with del H Comeán I. P. N.Deartment 13 and/ W. 23 Z

21 -adic String Amlitudes and Local Zeta Functions (joint work with H. Comeán and W. Zúñiga) Z (N) (s; I) = Z (N) (s; I, 0) Z (N) (s; I, 1). Z (N) (s; I, 0) = Z I i I x i s 1i 1 x i s (N 1)i 2 i<j N 2 i,j I x i x j s ij dx i i I Z (N) (s; I, 1) = M(s) T \I Z i T \I y i y i y j s ij 2 i<j N 2 i,j T \I 2+s 1i +s (N 1)i + s ij 2 j N 2,j i i T \I dx i where M(s) = T \I + i T \I (s 1i + s (N 1)i ) + 2 i<j N 2 i T \I,j T s ij + 2 i<j N 2 s ij. i I,j T \I Miriam Bocardo Gasar Advisor: Dr. Wilson-adic A. Zúñiga String GalindoCentro Amlitudes and de Investigación Local Zeta Functions y de Estudios (joint 26Avanzados work de mayo with del H Comeán I. P. N.Deartment 14 and/ W. 23 Z

22 -adic String Amlitudes and Local Zeta Functions (joint work with H. Comeán and W. Zúñiga) Problem: Z (N) (s) has a meromorhic continuation to the whole C D as a rational function in the variables s ij. We showed that the functions Z (N) (s; I, 0) and Z (N) (s; I, 1) - Meromorhic continuations to the whole C D as rational functions in the variables s ij. - Holomorhic on certain domain. - The intersection of all these domains contains an oen and connected subset of C D. The algorithms comute recursively the integrals. These results are still valid if we relace Q by F q ((t)), the field of formal Laurent series over a finite field F q. Miriam Bocardo Gasar Advisor: Dr. Wilson-adic A. Zúñiga String GalindoCentro Amlitudes and de Investigación Local Zeta Functions y de Estudios (joint 26Avanzados work de mayo with del H Comeán I. P. N.Deartment 15 and/ W. 23 Z

23 -adic String Amlitudes and Local Zeta Functions (joint work with H. Comeán and W. Zúñiga) Problem: Z (N) (s) has a meromorhic continuation to the whole C D as a rational function in the variables s ij. We showed that the functions Z (N) (s; I, 0) and Z (N) (s; I, 1) - Meromorhic continuations to the whole C D as rational functions in the variables s ij. - Holomorhic on certain domain. - The intersection of all these domains contains an oen and connected subset of C D. The algorithms comute recursively the integrals. These results are still valid if we relace Q by F q ((t)), the field of formal Laurent series over a finite field F q. Miriam Bocardo Gasar Advisor: Dr. Wilson-adic A. Zúñiga String GalindoCentro Amlitudes and de Investigación Local Zeta Functions y de Estudios (joint 26Avanzados work de mayo with del H Comeán I. P. N.Deartment 15 and/ W. 23 Z

24 -adic String Amlitudes and Local Zeta Functions (joint work with H. Comeán and W. Zúñiga) Problem: Z (N) (s) has a meromorhic continuation to the whole C D as a rational function in the variables s ij. We showed that the functions Z (N) (s; I, 0) and Z (N) (s; I, 1) - Meromorhic continuations to the whole C D as rational functions in the variables s ij. - Holomorhic on certain domain. - The intersection of all these domains contains an oen and connected subset of C D. The algorithms comute recursively the integrals. These results are still valid if we relace Q by F q ((t)), the field of formal Laurent series over a finite field F q. Miriam Bocardo Gasar Advisor: Dr. Wilson-adic A. Zúñiga String GalindoCentro Amlitudes and de Investigación Local Zeta Functions y de Estudios (joint 26Avanzados work de mayo with del H Comeán I. P. N.Deartment 15 and/ W. 23 Z

25 -adic String Amlitudes and Local Zeta Functions (joint work with H. Comeán and W. Zúñiga) Conditions of convergence (C1) J + i J(Re(s 1i ) + Re(s (N 1)i )) + (C2) K 1 + (C3) 1 + Re(s ij ) > 0 2 i<j N 2 i,i K (C4) M + Re(s ti ) + i M Re(s ij ) > 0 2 i<j N 2 i,j M where J,K, M T, i, j T {1, N 1}. 2 i<j N 2 i J Re(s ij ) + Re(s ij ) > 0, t {1, N 1} 2 i<j N 2 i T \J,j J Re(s ij ) < 0 Miriam Bocardo Gasar Advisor: Dr. Wilson-adic A. Zúñiga String GalindoCentro Amlitudes and de Investigación Local Zeta Functions y de Estudios (joint 26Avanzados work de mayo with del H Comeán I. P. N.Deartment 16 and/ W. 23 Z

26 -adic String Amlitudes and Local Zeta Functions (joint work with H. Comeán and W. Zúñiga) Main Results [Bocardo-Gasar, García-Comeán, Zúñiga-Galindo] Theorem (1), Bocardo-Gasar, García-Comeán, Zúñiga-Galindo) (1) The -adic oen string N-oint zeta function, Z (N) (s), gives rise to a holomorhic function on H(C), which contains an oen and connected subset of C D. Furthermore, Z (N) (s) admits an analytic continuation to C D, denoted also as Z (N) (s), as a rational function in the variables s ij, i, j {1,..., N 1}. The real arts of the oles of Z (N) (s) belong to a finite union of hyerlanes, the equations of these hyerlanes have the form C1-C4 with the symbols <, > relaced by =. (2) If s = (s ij ) ij C D, with Re(s ij ) 0 i, j {1,..., N 1}, then Z (N) (s) = +. Miriam Bocardo Gasar Advisor: Dr. Wilson-adic A. Zúñiga String GalindoCentro Amlitudes and de Investigación Local Zeta Functions y de Estudios (joint 26Avanzados work de mayo with del H Comeán I. P. N.Deartment 17 and/ W. 23 Z

27 -adic String Amlitudes and Local Zeta Functions (joint work with H. Comeán and W. Zúñiga) Regularization of -adic String Amlitudes Regularization: We take the -adic N-oints Z (N) (s) as regularizations of the amlitudes A (N) (k) and define A (N) (k) = Z (N) (s) sij =k i k j with i {1, N 1}, j T or i, j T. By Theorem 1), A (N) (k) are well-defined rational functions of the variables s ij, i, j {1,..., N 1}, which agree with the original -adic amlitudes when they converge. Miriam Bocardo Gasar Advisor: Dr. Wilson-adic A. Zúñiga String GalindoCentro Amlitudes and de Investigación Local Zeta Functions y de Estudios (joint 26Avanzados work de mayo with del H Comeán I. P. N.Deartment 18 and/ W. 23 Z

28 -adic String Amlitudes and Local Zeta Functions (joint work with H. Comeán and W. Zúñiga) Parte II Toological string zeta functions Miriam Bocardo Gasar Advisor: Dr. Wilson-adic A. Zúñiga String GalindoCentro Amlitudes and de Investigación Local Zeta Functions y de Estudios (joint 26Avanzados work de mayo with del H Comeán I. P. N.Deartment 19 and/ W. 23 Z

29 -adic String Amlitudes and Local Zeta Functions (joint work with H. Comeán and W. Zúñiga) Recall that Z (N) (s) = I T Z (N) (s; I, 0) Z (N) (s; I, 1). The integrals Z (N) (s; I, 0), Z (N) (s; I, 1) are local zeta functions. As a consequence of our method if we relace Q by K e the unique unramified extension of Q of degree e we obtain similar exlicit formulas for Z (N) (s; I, 0, K e ), Z (N) (s; I, 1, K e ) in the variables es ij. By using the theory of Denef and Loeser, we obtain Z (N) to (s; I, 0) = ĺım e 0 Z (N) (s; I, 0, K e ) Z (N) to (s; I, 1) = ĺım e 0 Z (N) (s; I, 1, K e ) Denef and F. Loeser, Caractéristiques d Euler Poincaré, fonctions zêta locales et modifications analytiques J. Amer. Math. Soc. 5 (1992), no. 4, Miriam Bocardo Gasar Advisor: Dr. Wilson-adic A. Zúñiga String GalindoCentro Amlitudes and de Investigación Local Zeta Functions y de Estudios (joint 26Avanzados work de mayo with del H Comeán I. P. N.Deartment 20 and/ W. 23 Z

30 -adic String Amlitudes and Local Zeta Functions (joint work with H. Comeán and W. Zúñiga) We define the N-oints toological string amlitude as Z (N) to (s) := Z (N) to (s; I, 0) Z (N) to (s; I, 1). I T and the toological string amlitude as A (N) to (k) = Z (N) to (s) sij =k i k j with i {1,.., N 1}, j T or i, j T where T = {e,..., N 2}, which are rational functions of the variables k i k j, k i k j, i, j {1,...N 1}. Miriam Bocardo Gasar Advisor: Dr. Wilson-adic A. Zúñiga String GalindoCentro Amlitudes and de Investigación Local Zeta Functions y de Estudios (joint 26Avanzados work de mayo with del H Comeán I. P. N.Deartment 21 and/ W. 23 Z

31 -adic String Amlitudes and Local Zeta Functions (joint work with H. Comeán and W. Zúñiga) -adic string amlitudes. Local zeta functions. R, C, Q, C((t)) string amlitudes. Miriam Bocardo Gasar Advisor: Dr. Wilson-adic A. Zúñiga String GalindoCentro Amlitudes and de Investigación Local Zeta Functions y de Estudios (joint 26Avanzados work de mayo with del H Comeán I. P. N.Deartment 22 and/ W. 23 Z

32 -adic String Amlitudes and Local Zeta Functions (joint work with H. Comeán and W. Zúñiga) Thank you for your attention. Miriam Bocardo Gasar Advisor: Dr. Wilson-adic A. Zúñiga String GalindoCentro Amlitudes and de Investigación Local Zeta Functions y de Estudios (joint 26Avanzados work de mayo with del H Comeán I. P. N.Deartment 23 and/ W. 23 Z