By using sheaftheoretical methods such as constructible sheaves, we generalize the formula of libgobersperber concerning the zeta functions of monodromy at infinity of polynomial maps into various directions. The zeta function is an important function in mathematics. We offer an equivariant version of the classical monodromy zeta function of a singularity as a series with coefficients from the grothendieck ring of finite gsets tensored by the field of. A window into zeta and modular physics msri publications volume 57, 2010 basic zeta functions and some applications in physics klaus kirsten 1. Through geometrical arguments, they showed that the monodromy conjecture holds for candidate poles of the topological zeta function of order 1 that are poles. The riemann zeta function is defined as the analytic continuation of the function defined for. Pdf we offer an equivariant version of the classical monodromy zeta function of a singularity as a series with coefficients from the grothendieck ring. Integration over spaces of nonparametrized arcs and. Deformations of singularities, monodromy, zetafunction, newton diagram. The large sieve, monodromy and zeta functions of curves. Thus the euler characteristic of the milnor ber is. For a complex polynomial or analytic function f, there is a strong correspondence between poles of the socalled local zeta functions or complex power. Zeta functions and monodromy for a toric idealistic cluster functions zr top,f r.
Monodromy zetafunctions of deformations and newton. For a oneparameter deformation of an analytic complex function germ of several variables, there is defined its monodromy zeta function. In this paper, i will demonstrate an important fact about the zeros of the zeta function, and how it relates to the prime number theorem. For a oneparameter deformation of an analytic complex function germ of several variables, there is defined its monodromy zetafunction. Wan, newton polygons of zeta functions and lfunctions, ann.
Analytic continuation of the riemann zeta function 9. One can show that the equivariant zeta function of the monodromy transformation h on a neighbourhood of an intersection of the irreducible components of e 0 is equal to one. The monodromy conjecture predicts that the poles of the topological zeta function and related zeta functions associated to a polynomial f induce monodromy eigenvalues of f. We give a survey on what is known about this operator. We give a varchenko type formula for this zetafunction if the deformation is nondegenerate with respect to its newton diagram. The actual configuration may depend on the choice of the branch cuts, but the group generated by the branch cycles is always the same. Z0 that are variants of the topological zeta function and that play a role in the holomorphy conjecture, as stated by denef. The monodromy functions giving the multivaluedness are computable. Monodromy conjecture, igusas zeta function, motivic zeta function, surface singularity, nondegenerate, lattice polytope view full volume pdf view other years and numbers. Max planck institute for mathematics, max planck society. However, the formula 2 cannot be applied anymore if the real part. On an equivariant analogue of the monodromy zeta function. Finally, in x5 it is exempli ed the di erent behavior of acampos formula using nonabelian groups, showing that \double points in an embedded resolution may contribute to the monodromy zeta function. Using techniques similar to those of riemann, it is shown how to locate and count nontrivial zeros of s.
Finally we dis cuss orbifold versions of the lefschetz number and of the monodromy zeta function. Cameron franc special values of riemanns zeta function. The monodromy group of an algebraic function wolfram. Monodromy zeta function formula for embedded qresolutions. Furthermore, we will introduce some other zeta function which are closly related to the riemann zeta function.
The local topological zeta function is also an invariant of. For all except a finite number ofp we have that ifs is a pole ozps, then e5 is an eigenvalue of the monodromy acting on the cohomology in some dimension of the milnor fiber off associated to some point of the hypersurface f 0. For all except a finite number ofp we have that ifs is a pole ozps, then e5 is an eigenvalue of the monodromy acting on the cohomology in some dimension of the milnor fiber off associated. Monodromy eigenvalues and zeta functions with differential. This thesis is an exposition of the riemann zeta function. Then the zeta function of monodromy at oo is given by where the product is taken over all faces a of a.
The usual differentiation rules apply for analytic functions. Z0 does not divide the order of any eigenvalue of the local monodromy of f at any point of f. We also give a short and new proof of the monodromy conjecture for plane curves. Moreover, in 1859 riemann gave a formula for a unique the socalled holomorphic extension of the function onto the entire complex plane c except s 1. A potential problem of this analytic continuation along a. Fundamental in our work is a detailed study of the formula for the zeta function of monodromy by varchenko and the study of the candidate poles of the topological zeta function yielded by what we call b1facets.
That is, we treat in a conceptual unity the poles of the generalized topological zeta function and the monodromy eigenvalues associated with an. Monodromy zeta function of a qnormal crossing divisor lemma the monodromy zeta function of a normal crossing divisor given by xm1 1 x m k k. On an equivariant analogue of the monodromy zeta function mpsauthors guseinzade, sabir m. Weil, numbers of solutions of equations over finite fields, bull. Dec 09, 2016 interestingly, that vertical line where the convergent portion of the function appears to abruptly stop corresponds to numbers whose real part is eulers constant, 0. We offer an equivariant version of the classical monodromy zeta function of a singularity as a series with coefficients from the grothendieck ring of finite gsets tensored by the field of rational numbers. In what follows, we will view f as a combination of characters we seek a resolution of the base points of the pencil of hypersurfaces which are compactifications of the hypersurfaces in gnm. Some generalizations of the riemann zeta function are outlined, as well as the calculation of zeta constants and the development of some identities. These are analytic objects complex functions that are intimately related to the global elds. The idea is that one can extend a complexanalytic function from here on called simply analytic function along curves starting in the original domain of the function and ending in the larger set. On the zeta function of monodromy of a polynomial map. Varchenko v computing the zeta function of monodromy at an isolated singular point say pe c of a holomorphic map f. There are more conjectures relating the poles of the topological zeta function igusa zeta function and the eigenvalues of monodromy.
Complex analysis 7 is analytic at each point of the entire finite plane, then fz is called an entire function. In particular, some formulas for the zeta functions of global monodromy along the fibers of bifurcation points of polynomial maps will be obtained. Monodromy eigenvalues and poles of zeta functions cauwbergs. A motivic version of the monodromy zeta function can be constructed as the exponent of the generating series of lefschetz numbers of iterates of the monodromy transformation of a singularity see 9.
Monodromy zetafunctions of deformations andnewtondiagrams. The aim of the article is an extension of the monodromy conjecture of denef and loeser in dimension two, incorporating zeta functions with differential forms and targeting all monodromy eigenvalues, and also considering singular ambient spaces. Calculation of the monodromy at oo the purpose of this section is to prove theorems 1 and 2 calculating explicitly the zeta function of monodromy at oo. Np c in a neighborhood np of p in terms of analogous invariants associated with 0394pf, the newton polyhedron of f at p. Pdf monodromy zetafunctions of deformations and newton. Dec 24, 2008 through geometrical arguments, they showed that the monodromy conjecture holds for candidate poles of the topological zeta function of order 1 that are poles. In complex analysis, the monodromy theorem is an important result about analytic continuation of a complexanalytic function to a larger set. The authors start from their work and obtain the same result for igusas \p\adic and the motivic zeta function. There are more conjectures relating the poles of the topological zeta function igusa zeta function and the eigenvalues of. More precisely, we prove that for every tamely ramified abelian variety a over a complete discretely valued field with algebraically closed residue field, its motivic zeta function has a unique pole at chai. Introduction it is the aim of these lectures to introduce some basic zeta functions and their uses in the areas of. We give a varchenko type formula for this zeta function if the deformation is nondegenerate with respect to its newton diagram.
Pdf an equivariant version of the monodromy zeta function. Lagarias, university of michigan ann arbor, mi, usa. Motivic zeta functions of abelian varieties, and the. Monodromy zeta functions at infinity, newton polyhedra and. The zeta function and its relation to the prime number theorem ben rifferreinert abstract. The riemannsiegel formula and large scale computations of the riemann zeta function by glendon ralph pugh b. Eudml on the zeta function of monodromy of a polynomial map. Pdf for a oneparameter deformation of an analytic complex function germ of several variables, there is defined its monodromy zetafunction. Ams, american mathematical society, the tricolored ams logo, and advancing research, creating connections, are trademarks and services marks of the american mathematical society and registered in the u. We prove for abelian varieties a global form of denef and loeser. Included are techniques of analytic continuation and relationships to special functions. We consider toric modifications with respect to the initial expansions in the tschirnhausen coordinate systems of a complex plane curve singularity. In particular, new cases among the nondegenerate surface singularities for which the monodromy conjecture is proven now, are the nonisolated singularities, the. The zeta function of the monodromy transformation can also be extracted from the motivic milnor.
The argument is formulated in hilbert spaces of entire functions 1. The set class of functions holomorphic in g is denoted by hg. Then, the lefschetz numbers and the complex monodromy zeta function are given by hk xr i1. However, not every monodromy eigenvalue can be recovered from a pole. Monodromy conjecture for nondegenerate surface singularities. Basic zeta functions and some applications in physics. This demonstration shows the structure of the branches of a multivalued function defined by a polynomial equation, illustrating the transitions between the branches along paths going around a branch point. The bernsteinsato polynomial, or the bfunction, is an important invariant of hypersurface singularities. Zeta functions and monodromy for surfaces that are general. Monodromy zetafunctions of deformations and newton diagrams. This function is going to be used in the last section where we are going to talk about the symmetric zeta function, which is based on a linear combination of lfunctions and zeta functions. Kudryavtseva 1 filip saidak peter zvengrowski abstract an exposition is given, partly historical and partly mathematical, of the riemann zeta function s and the associated riemann hypothesis. In 2011 lemahieu and van proeyen proved the monodromy conjecture for the local topological zeta function of a nondegenerate surface singularity. A point where the function fails to be analytic, is called a singular point or singularity of the function.
The numerical data on these toric modifications give rise to a graph which allows to describe the topological zeta function of the singularity. Integration over spaces of nonparametrized arcs and motivic. We prove the monodromy conjecture for the topological zeta function for all nondegenerate surface singularities. An equivariant version of the monodromy zeta function e. The polylogarithm function is equivalent to the hurwitz zeta function either function can be expressed in terms of the other and both functions are special cases of the lerch transcendent. Polylogarithms should not be confused with polylogarithmic functions nor with the offset logarithmic integral which has a similar notation.
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