Proof of the algebraic nature of analytic branches of the functional inverse of an analytic function at a critical point. There is an explicit formula of this function using the logresoultion of ad. The following theorem describes the existence of dautomorphisms of degree m in terms of existence of certain elements in a monodromy group. This proof relies on the nonsolvability of the monodromy group of a general algebraic function. Grothendieck proved the following theorem which implies the previous theorem, but i wont go into the implication. It is not yet known whether there exists a quasiglobally rightcomplex, complex and ordered finite scalar, although 37 does address the issue of uniqueness. One may also consider inverse questions, that is, which groups appear as monodromy or picardvessiot groups. M 2mbe a uniformiser and let rbe the rami cation degree of lm. Let f,d be a function element which admits unrestricted continuation in the simply connected region g. X be a continuous map, and let wbe a point of x satisfying pw h0. The method is restrictive in the sense that any further extension would seem to be relied on the monodromy weight conjecture in the mixed characteristic case, among other di. Akhil mathew belyi functions with prescribed monodromy. The above characterization of the hypergeo metric equation is the basis of fuchs proof of riemanns theorem cf.
The theorem below provides certain evidence toward our expected hodge isomorphism. An application of the principle of analytic continuation and the monodromy. In addition to that we had to apply a substantial quantity of new methods to achieve the extension to all nitely generated elds, such as for instance nite. Proof of theor em 2 by applying integrable surgery, we can assume that the bifurcation. We obtain part i of the corollary as a consequence of the main theorem, our proposition 4. Assuming the mumfordtate conjecture, we show that from two well chosen. Methods for padic monodromy university of pennsylvania. Riemanns proof assumed that none of the exponent differences at a singularity was an integer and proceeded by first computing global monodromy. Our proof of the main theorem follows halls proof of 12 to some extent. Yu originally proved his theorem in order to study the cohenlenstra heuristics over function.
Proof of the first homotopy version of the monodromy theorem. A natural thickening \ lxi xj is just to keep the equation 0. The idea is that one can extend a complexanalytic function along curves starting in the original domain of the function and ending in the larger set. A topological proof of abelru ni theorem henryk zoladek. The monodromy theorem gives sufficient conditions for analytic continuation to give the same value at a given point regardless of the curve used to get there, so. Improving the estimates via linear fractional transformations pdf. Autc be of finite order m, p a dinvariant polytope with a base flag. If an indecomposable degreenrational function fx has monodromy group satisfying c3. A potential problem of this analytic continuation along a curve strategy is there are usually many curves which end up at the same point in the larger set. This can be, and usually is, proved more directly using limit mixed hodge structures, cf gs.
The following is a simpler version concerning the uniqueness of the monodromy theorem. The best conceivable analogue of grothendiecks theorem. Sketch of trigonal proof theorem for every g the stack t gof trigonal curves has monodromy gsp 2 zb. The revolutionary proofs of abel and galois following ru ni paved way for group theory and galois theory. It was napier who first asked whether sets can be derived. We also get lifts with q, rather than q, coe cients, in contrast to theorem. Associated to an abelian variety over a number eld are several interesting and related groups. In 19, the authors derived compactly embedded functions. Our proof of theorem 1 is based on ideas of 4, 7, 8 and 9. Proof of little picards theorem using blochs theorem. The use of the monodromy theorem and entire functions with. The fractional monodromy of a liouville integrable hamiltonian system over a loop. Firstly, it is purely algebraic in nature and in the spirit of grothendiecks proof of the grothendiecklefschetz theorem.
Pdf periodicity and the monodromy theorem monica nicolau. Q surjects onto the monodromy invariant part of a smooth bre hix t. Secondly, we explicitly say how positive the linear system needs to be. The proof uses arithmetic geometry, especially the weight spectral sequence. Pdf monodromy groups and selfinvariance isabel hubard. Assume that kis nite, then there is an open subgroup of j.
Suppose that fz is a real function of complex variable whose derivative exists at. Zeros of analytic functions,analytic continuation, monodromy, hyperbolic geometry and the reimann mapping theorem. Nigel boston university of wisconsin madison the proof of. Covering maps and the monodromy theorem tcd maths home. The product pe gives the monodromy transformation along the boundary of the disc our. Monodromy theorem the monodromy theorem the monodromy theorem gives a suf.
The author apologises for all errors, unclarities, omissions of details and other imperfections and encourages the. The grothendieck monodromy theorem raymond van bommel. The following important theorem asserts that many naturally occurring di. Lecture 17 analytic continuation 1 singular points. Determining monodromy groups of abelian varieties david zywina abstract. The use of the monodromy theorem and entire functions. Complex analysis maharshi dayanand university, rohtak. Theorem riemann existence every compact riemann surface has an algebraic structure. A potential problem of this analytic continuation along a. In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they run round a singularity. A simple proof of the geometric fractional monodromy theorem. All existing proofs of fractional monodromy 15, 7, 8, 3, 16, 12 are given for 1.
Monodromy for the hypergeometric function theorem 6. Assume we are given a superfree, nonfree, infinite homomorphism acting multiply on a completely meromorphic, commutative graph f. The topological proof of abelruffini theorem henryk zoladek. We also get lifts with q, rather than q, coe cients, in contrast to theorem 1. Note that the proof of theorem a is geometric in nature, while the proof of theorem b then is more arithmetic in. The reader who is interested only in the generic case can skip 1. Big monodromy theorem for abelian varieties over nitely.
Figure 2 shows the iterated monodromy action for the map fzz2, where x c\0. The case for small n, either even or odd, is similar to the case for large n, but the argument is somewhat less transparent. As the name implies, the fundamental meaning of monodromy comes from running round singly. Note that in theorem a we need the polarization is principal, but in theorem. So in complex analysis, the monodromy theorem says that. Before stating and proving the monodromy theorem, we need two lemmas and a definition. Suppose f is analytic on drz0 and the power series. Moreover, if l ksep such a representation is called an ladic galois representation. This can be reduced to the unipotent case, for which it is an exercise.
Next we will derive several applications of the monodromy theorem. By homogeneity, we may assume that x,y,zare relatively prime. In our proof of a translation theorem for analytic feynman integrals 3, a function of. A adic local monodromy theorem annals of mathematics. We present a proof of the nonsolvability in radicals of a general algebraic equation of degree greater than four. See yu for the preprint containing yus original proof or ap for another recent independent proof. Hejhal also conjectured that generic representations into sl2, are monodromy representations. The mod geometric monodromy of hyperelliptic curves is sp2g,f for 2.
Nigel boston university of wisconsin madison the proof. Note that the proof of theorem a is geometric in nature, while the proof of theorem b. In complex analysis, the monodromy theorem is an important result about analytic continuation of a complexanalytic function to a larger set. Nptel mathematics advanced complex analysis part 1. The meanvalue property, harmonic functions and the maximum principle. We also repeatedly use the fact that the units of aare precisely. By applying the lebesgue lemma to an open cover of the square by preimages of evenly covered open sets in x as in the proof of theorem 3. Monodromy and vanishing cycles in toric surfaces 3 conjecture 1. Then we show theorem b in the principally polarized case, see 3. It is closely associated with covering maps and their degeneration into ramification. In addition, we present the full proof of the divergence theorem briefly outlined in. This gap can be described as a padic analogue of grothendiecks local monodromy theorem for ladic cohomology. Proof of existence of maximal analytic continuation of a holomorphic germ.
The monodromy theorem gives sufficient conditions f. May 17, 20 a simple proof of the geometric fractional monodromy theorem broerefstathioulukina 2010 is presented. The first book source where i was able to find the full statement and a complete proof of the monodromy theorem is in the section geometrische funktionentheorie. In addition, we present the full proof of the divergence theorem briefly outlined in ka. An important element in the proof of the above results is a theorem of levelt, which gives a simple algebraic characterisation of the monodromy group of a hypergeometric differential equation le, thm. The remainder of the paper is organized as follows. Our proof of the main theorem follows halls proof of 12 to some extent, e. R, complex number s 0 is a pole of z1s 0, then exp2. Definition 1 in 1 below although they, and most of their proofs, remain valid for more. Analytic continuation and two versions of monodromy theorem. Once one has niteness of the monodromy acting on h2, theorem 0.
The idea of the proof of theorem 1 is to combine the continuity method. The topological proof of abelruffini theorem henryk zoladek abstract. 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. Using this result we then show the monodromy theorem on the central stream 5.
38 1792 1700 548 1283 1434 1414 1733 1012 688 1347 767 1244 1162 1081 1760 434 423 1720 319