Riemann's Existence Theorem
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QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday, October 24, 1995 (Day 1)
QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday, October 24, 1995 (Day 1) 1. Let K be a ¯eld of characteristic 0. a. Find three nonconstant polynomials x(t); y(t); z(t) K[t] such that 2 x2 + y2 = z2 b. Now let n be any integer, n 3. Show that there do not exist three nonconstant polynomials x(t); y(t); z(t) K[t] suc¸h that 2 xn + yn = zn: 2. For any integers k and n with 1 k n, let · · Sn = (x ; : : : ; x ) : x2 + : : : + x2 = 1 n+1 f 1 n+1 1 n+1 g ½ be the n-sphere, and let D n+1 be the closed disc k ½ D = (x ; : : : ; x ) : x2 + : : : + x2 1; x = : : : = x = 0 n+1: k f 1 n+1 1 k · k+1 n+1 g ½ n Let X = S D be their union. Calculate the cohomology ring H¤(X ; ¡ ). k;n [ k k;n 2 3. Let f : be any 1 map such that ! C @2f @2f + 0: @x2 @y2 ´ Show that if f is not surjective then it is constant. 4. Let G be a ¯nite group, and let ; ¿ G be two elements selected at random from G (with the uniform distribution). In terms2 of the order of G and the number of conjugacy classes of G, what is the probability that and ¿ commute? What is the probability if G is the symmetric group S5 on 5 letters? 1 5. Let be the region given by ½ = z : z 1 < 1 and z i < 1 : f j ¡ j j ¡ j g Find a conformal map f : ¢ of onto the unit disc ¢ = z : z < 1 . -
Patching and Galois Theory David Harbater∗ Dept. of Mathematics
Patching and Galois theory David Harbater∗ Dept. of Mathematics, University of Pennsylvania Abstract: Galois theory over (x) is well-understood as a consequence of Riemann's Existence Theorem, which classifies the algebraic branched covers of the complex projective line. The proof of that theorem uses analytic and topological methods, including the ability to construct covers locally and to patch them together on the overlaps. To study the Galois extensions of k(x) for other fields k, one would like to have an analog of Riemann's Existence Theorem for curves over k. Such a result remains out of reach, but partial results in this direction can be proven using patching methods that are analogous to complex patching, and which apply in more general contexts. One such method is formal patching, in which formal completions of schemes play the role of small open sets. Another such method is rigid patching, in which non-archimedean discs are used. Both methods yield the realization of arbitrary finite groups as Galois groups over k(x) for various classes of fields k, as well as more precise results concerning embedding problems and fundamental groups. This manuscript describes such patching methods and their relationships to the classical approach over , and shows how these methods provide results about Galois groups and fundamental groups. ∗ Supported in part by NSF Grants DMS9970481 and DMS0200045. (Version 3.5, Aug. 30, 2002.) 2000 Mathematics Subject Classification. Primary 14H30, 12F12, 14D15; Secondary 13B05, 13J05, 12E30. Key words and phrases: fundamental group, Galois covers, patching, formal scheme, rigid analytic space, affine curves, deformations, embedding problems. -
An Arithmetic Riemann-Roch Theorem for Pointed Stable Curves
An arithmetic Riemann-Roch theorem for pointed stable curves Gerard Freixas i Montplet Abstract.- Let ( , Σ, F∞) be an arithmetic ring of Krull dimension at most 1, O = Spec and (π : ; σ1,...,σ ) a n-pointed stable curve of genus g. Write S O X → S n = σ ( ). The invertible sheaf ωX S(σ1 + ... + σ ) inherits a hermitian struc- U X \ ∪j j S / n ture hyp from the dual of the hyperbolic metric on the Riemann surface ∞. In this articlek·k we prove an arithmetic Riemann-Roch type theorem that computes theU arith- metic self-intersection of ωX /S(σ1 + ... + σn)hyp. The theorem is applied to modular curves X(Γ), Γ = Γ0(p) or Γ1(p), p 11 prime, with sections given by the cusps. We ′ a b c ≥ show Z (Y (Γ), 1) e π Γ2(1/2) L(0, Γ), with p 11 mod 12 when Γ = Γ0(p). Here Z(Y (Γ),s) is the Selberg∼ zeta functionM of the open modular≡ curve Y (Γ), a,b,c are rational numbers, Γ is a suitable Chow motive and means equality up to algebraic unit. M ∼ Resum´ e.-´ Soit ( , Σ, F∞) un anneau arithm´etique de dimension de Krull au plus 1, O = Spec et (π : ; σ1,...,σ ) une courbe stable n-point´ee de genre g. Posons S O X → S n = σ ( ). Le faisceau inversible ωX S(σ1+...+σ ) h´erite une structure hermitienne U X \∪j j S / n hyp du dual de la m´etrique hyperbolique sur la surface de Riemann ∞. Dans cet article nousk·k prouvons un th´eor`eme de Riemann-Roch arithm´etique qui calculeU l’auto-intersection arithm´etique de ωX /S(σ1 + ...+ σn)hyp. -
NNALES SCIEN IFIQUES SUPÉRIEU E D L ÉCOLE ORMALE
ISSN 0012-9593 ASENAH quatrième série - tome 42 fascicule 2 mars-avril 2009 NNALES SCIENIFIQUES d L ÉCOLE ORMALE SUPÉRIEUE Gérard FREIXAS i MONTPLET An arithmetic Riemann-Roch theorem for pointed stable curves SOCIÉTÉ MATHÉMATIQUE DE FRANCE Ann. Scient. Éc. Norm. Sup. 4 e série, t. 42, 2009, p. 335 à 369 AN ARITHMETIC RIEMANN-ROCH THEOREM FOR POINTED STABLE CURVES ʙʏ Gʀʀ FREIXAS ɪ MONTPLET Aʙʀ. – Let ( , Σ,F ) be an arithmetic ring of Krull dimension at most 1, =Spec and O ∞ S O (π : ; σ1,...,σn) an n-pointed stable curve of genus g. Write = j σj ( ). The invertible X→S U X\∪ S sheaf ω / (σ1 + + σn) inherits a hermitian structure hyp from the dual of the hyperbolic met- X S ··· · ric on the Riemann surface . In this article we prove an arithmetic Riemann-Roch type theorem U∞ that computes the arithmetic self-intersection of ω / (σ1 + + σn)hyp. The theorem is applied to X S ··· modular curves X(Γ), Γ=Γ0(p) or Γ1(p), p 11 prime, with sections given by the cusps. We show a b c ≥ Z(Y (Γ), 1) e π Γ2(1/2) L(0, Γ), with p 11 mod 12 when Γ=Γ0(p). Here Z(Y (Γ),s) is ∼ M ≡ the Selberg zeta function of the open modular curve Y (Γ), a, b, c are rational numbers, Γ is a suitable M Chow motive and means equality up to algebraic unit. ∼ R. – Soient ( , Σ,F ) un anneau arithmétique de dimension de Krull au plus 1, O ∞ =Spec et (π : ; σ1,...,σn) une courbe stable n-pointée de genre g. -
Automorphy of Mod 2 Galois Representations Associated to Certain Genus 2 Curves Over Totally Real Fields
AUTOMORPHY OF MOD 2 GALOIS REPRESENTATIONS ASSOCIATED TO CERTAIN GENUS 2 CURVES OVER TOTALLY REAL FIELDS ALEXANDRU GHITZA AND TAKUYA YAMAUCHI Abstract. Let C be a genus two hyperelliptic curve over a totally real field F . We show that the mod 2 Galois representation ρC;2 : Gal(F =F ) −! GSp4(F2) attached to C is automorphic when the image of ρC;2 is isomorphic to S5 and it is also a transitive subgroup under a fixed isomorphism GSp4(F2) ' S6. To be more precise, there exists a Hilbert{Siegel Hecke eigen cusp form h on GSp4(AF ) of parallel weight two whose mod 2 Galois representation ρh;2 is isomorphic to ρC;2. 1. Introduction Let K be a number field in an algebraic closure Q of Q and p be a prime number. Fix an embedding Q ,! C and an isomorphism Qp ' C. Let ι = ιp : Q −! Qp be an embedding that is compatible with the maps Q ,! C and Qp ' C fixed above. In proving automorphy of a given geometric p-adic Galois representation ρ : GK := Gal(Q=K) −! GLn(Qp); a standard way is to observe automorphy of its residual representation ρ : GK −! GLn(Fp) after choosing a suitable integral lattice. However, proving automorphy of ρ is hard unless the image of ρ is small. If the image is solvable, one can use the Langlands base change argument to find an automorphic cuspidal representation as is done in many known cases. Beyond this, one can ask that p and the image of ρ be small enough that one can apply the known cases of automorphy of Artin representations (cf. -
Life and Work of Friedrich Hirzebruch
Jahresber Dtsch Math-Ver (2015) 117:93–132 DOI 10.1365/s13291-015-0114-1 HISTORICAL ARTICLE Life and Work of Friedrich Hirzebruch Don Zagier1 Published online: 27 May 2015 © Deutsche Mathematiker-Vereinigung and Springer-Verlag Berlin Heidelberg 2015 Abstract Friedrich Hirzebruch, who died in 2012 at the age of 84, was one of the most important German mathematicians of the twentieth century. In this article we try to give a fairly detailed picture of his life and of his many mathematical achievements, as well as of his role in reshaping German mathematics after the Second World War. Mathematics Subject Classification (2010) 01A70 · 01A60 · 11-03 · 14-03 · 19-03 · 33-03 · 55-03 · 57-03 Friedrich Hirzebruch, who passed away on May 27, 2012, at the age of 84, was the outstanding German mathematician of the second half of the twentieth century, not only because of his beautiful and influential discoveries within mathematics itself, but also, and perhaps even more importantly, for his role in reshaping German math- ematics and restoring the country’s image after the devastations of the Nazi years. The field of his scientific work can best be summed up as “Topological methods in algebraic geometry,” this being both the title of his now classic book and the aptest de- scription of an activity that ranged from the signature and Hirzebruch-Riemann-Roch theorems to the creation of the modern theory of Hilbert modular varieties. Highlights of his activity as a leader and shaper of mathematics inside and outside Germany in- clude his creation of the Arbeitstagung, -
Life and Work of Egbert Brieskorn (1936-2013)
Life and work of Egbert Brieskorn (1936 – 2013)1 Gert-Martin Greuel, Walter Purkert Brieskorn 2007 Egbert Brieskorn died on July 11, 2013, a few days after his 77th birthday. He was an impressive personality who left a lasting impression on anyone who knew him, be it in or out of mathematics. Brieskorn was a great mathematician, but his interests, knowledge, and activities went far beyond mathematics. In the following article, which is strongly influenced by the authors’ many years of personal ties with Brieskorn, we try to give a deeper insight into the life and work of Brieskorn. In doing so, we highlight both his personal commitment to peace and the environment as well as his long–standing exploration of the life and work of Felix Hausdorff and the publication of Hausdorff ’s Collected Works. The focus of the article, however, is on the presentation of his remarkable and influential mathematical work. The first author (GMG) has spent significant parts of his scientific career as a arXiv:1711.09600v1 [math.AG] 27 Nov 2017 graduate and doctoral student with Brieskorn in Göttingen and later as his assistant in Bonn. He describes in the first two parts, partly from the memory of personal cooperation, aspects of Brieskorn’s life and of his political and social commitment. In addition, in the section on Brieskorn’s mathematical work, he explains in detail 1Translation of the German article ”Leben und Werk von Egbert Brieskorn (1936 – 2013)”, Jahresber. Dtsch. Math.–Ver. 118, No. 3, 143-178 (2016). 1 the main scientific results of his publications. -
Preface Issue 1-2013
Jahresber Dtsch Math-Ver (2013) 115:1–2 DOI 10.1365/s13291-013-0060-8 PREFACE Preface Issue 1-2013 Hans-Christoph Grunau © Deutsche Mathematiker-Vereinigung and Springer-Verlag Berlin Heidelberg 2013 Annette Werner provides an introduction to the research area of “Non-Archimedean analytic spaces”, i.e. analytic geometry over complete non-Archimedean fields. These fields arise in algebra and number theory and the corresponding geometry looks com- pletely different from the usual analytic, projective or differential geometry. The au- thor presents a new approach to this kind of arithmetic geometry with “better” topo- logical properties. Hans Grauert passed away in September 2011. Alan Huckleberry has written a very illustrative and informative obituary in which he reviews some of the most in- fluential and fundamental works of “one of the greatest giants of the second half of the 20th century” in complex geometry. We learn about the—often quite classical— complex analytic background, the key achievements, the main implications and some of the really modern ideas of these works. While the precise formulation of the mathematics requires some abstract and deep concepts, the present article focusses on the underlying analytical intuition and geometrical imagination. Moreover, Hans Grauert’s interactions with his advisors, friends, coworkers, mathematical compan- ions and students are outlined in some detail. Recently released books on nonlinear potential theory on metric spaces, on statis- tical regression for specific kinds of data, and on random matrices in mathematical physics are extensively discussed and reviewed. Every four years the (managing) editor of the Jahresbericht is elected by the mem- bers of the DMV. -
Friedrich Hirzebruch a Handful of Reminiscences∗
Friedrich Hirzebruch a handful of reminiscences∗ Piotr Pragacz∗∗ The idea of dedicating an IMPANGA volume to Friedrich Hirzebruch arose a few years ago. A lot has changed since then. First of all, Friedrich Hirzebruch passed away on May 27, 2012. Following his death many conferences, lectures and articles [2, 4, 3, 22] were dedicated to him. The articles [1, 5, 7, 9] and the video- interview [13] appeared while he was still alive. A book by Winfried Scharlau is in preparation. These publications accurately describe the life and work of Professor Friedrich Hirzebruch from the point of view of his close colleagues and coworkers. Therefore, though initially I intended to write an article about him similar to Notes on the life and work of Alexander Grothendieck [16] (1st volume of IMPANGA Lecture Notes) or the one on Józef Maria Hoene-Wro«ski [17] (2nd volume of IMPANGA Lecture Notes), I decided to change my plans, so this essay will be of a dierent nature. I would like to share a collection of reminiscences about Professor Friedrich Hirzebruch from the vantage of a person, for whom he was a mentor in the years 19932006. In addition, I would like to highlight his relations with IMPANGA. F. Hirzebruch during the Leonhard Euler Congress in Saint Petersburg in 2007 I met Friedrich Hirzebruch for the rst time in 1988 during the algebraic semester at the Banach Center, at the old residence on Mokotowska street. He came for the conference in algebraic geometry organized as a part of that semester. The ∗This article was translated by Masha Vlasenko with the editorial assistance of the author. -
Jahrbuch Der Akademie Der Wissenschaften Zu Göttingen 2012
Jahrbuch der Akademie der Wissenschaften zu Göttingen DerMuse Polyhymnia wirdnachgesagt, dass sie Schreibenden Ruhm bringt, deren Werkesie für unsterblich hält. Aufsuchen kann man die Göt- tin der Hymnendichtung, des Tanzes, der Pantomime und der Geometrie im Akademiesaal in der Aula der Universität Göttingen. DieOrdentlichen Mitglieder der Akademie der Wissenschaften zu Göttingen begeben sich re- gelmäßig zu ihren Plenarsitzungen dorthin. Polyhymnia ist unter den neun Musen, die den Saal als Wandbemalung zieren, als jene zu erkennen, die sich mitdem Ellenbogen auf einen Baumstamm stützt. Wiedie anderen Gemälde des Sitzungszimmers geht auch die Abbildung der Polyhymnia auf ein antikes Vorbild zurück. DasVorbildfür die Göttinger Polyhymnia findet man in dem Musenrelief eines römischen Sarkophags im Louvre. JAHRBUCH DER AKADEMIE DER WISSENSCHAFTEN ZU GÖTTINGEN 2012 De Gruyter Akademie der Wissenschaften zu Göttingen Theaterstraße 7 37073 Göttingen Telefon: 0551-39-5424 Fax: 0551-39-5365 E-Mail: [email protected] http://www.adw-goe.de Verantwortlich: DerPräsidentder Akademie der Wissenschaften ISSN 0373-9767 LibraryofCongress Cataloging-in-Publication Data: ACIP catalog recordfor this book has been applied for at the LibraryofCongress Bibliografische Information der Deutschen Nationalbibliothek DieDeutsche Nationalbibliothek verzeichnet diesePublikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http://dnb.dnb.de abrufbar. ©2014 Walter de Gruyter GmbH &Co. KG,Berlin/Boston Satz: PTP-Berlin Protago -
Algebraic Models of the Line in the Real Affine Plane
ALGEBRAIC MODELS OF THE LINE IN THE REAL AFFINE PLANE ADRIEN DUBOULOZ AND FRÉDÉRIC MANGOLTE Abstract. We study smooth rational closed embeddings of the real affine line into the real affine plane, that is algebraic rational maps from the real affine line to the real affine plane which induce smooth closed embeddings of the real euclidean line into the real euclidean plane. We consider these up to equivalence under the group of birational automorphisms of the real affine plane which are diffeomorphisms of its real locus. We show that in contrast with the situation in the categories of smooth manifolds with smooth maps and of real algebraic varieties with regular maps where there is only one equivalence class up to isomorphism, there are non-equivalent smooth rational closed embeddings up to such birational diffeomorphisms. Introduction It is a consequence of the Jordan–Schoenflies theorem (see Remark 11 below) that every two smooth proper embeddings of R into R2 are equivalent up to composition by diffeomorphisms of R2. Every algebraic closed embedding of the real affine line 1 into the real affine plane 2 induces a smooth AR AR proper embedding of the real locus of 1 into the real locus 2 of 2 . Given two such algebraic R AR R AR embeddings f; g : 1 ,! 2 , the famous Abhyankar-Moh Theorem [1], which is valid over any field of AR AR characteristic zero [23, x 5.4], asserts the existence of a polynomial automorphism φ of 2 such that AR f = φ ◦ g. This implies in particular that the smooth proper embeddings of R into R2 induced by f and g are equivalent under composition by a polynomial diffeomorphism of R2. -
The Fine Structure of the Moduli Space of Abelian Differentials in Genus 3
THE FINE STRUCTURE OF THE MODULI SPACE OF ABELIAN DIFFERENTIALS IN GENUS 3 EDUARD LOOIJENGA AND GABRIELE MONDELLO ABSTRACT. The moduli space of curves endowed with a nonzero abelian differential admits a natural stratification according to the configuration of its zeroes. We give a description of these strata for genus 3 in terms of root system data. For each non-open stratum we obtain a presentation of its orbifold fundamental group. INTRODUCTION The moduli space of pairs (C,ϕ) with C a complex connected smooth projective curve of genus g ≥ 2 and ϕ a nonzero abelian differential on C, and denoted here by Hg, comes with the structure of a Deligne-Mumford stack, but we will just regard it as an orbifold. The forgetful morphism Hg Mg exhibits Hg as the complement of the zero section of the Hodge bundle over Mg. A partition of Hg into suborbifolds is defined by looking at the→ multiplicities of the zeros of the abelian differential: for any numerical partition k := (k1 ≥ k2 ≥ ··· ≥ kn > 0) of 2g − 2, we have a subvariety ′ H (k) ⊂Hg which parameterizes the pairs (C,ϕ) for which the zero divisor Zϕ of ϕ is of type k. We call a stratum of Hg a connected component of some H ′(k). (NB: our terminology slightly differs from that of [5]). Classification of strata. Kontsevich and Zorich [5] characterized the strata of Hg in rather simple terms. First consider the case when C is hyperelliptic. Then an effective divisor of degree 2g−2 on C is canonical if and only if it is invariant under the hyperelliptic involution.