{"id":87,"date":"2016-06-23T21:59:03","date_gmt":"2016-06-23T19:59:03","guid":{"rendered":"http:\/\/berndt-schwerdtfeger.de\/?page_id=87"},"modified":"2025-10-07T15:12:15","modified_gmt":"2025-10-07T13:12:15","slug":"arithmetic","status":"publish","type":"page","link":"https:\/\/berndt-schwerdtfeger.de\/?page_id=87","title":{"rendered":"Arithmetic"},"content":{"rendered":"<h3><a name=\"mfds\"><\/a>Modular Forms and Dirichlet series <small>by <em>Andrew OGG<\/em><\/small><\/h3>\n<ul>\n<li>version 1.0, 2018-08-29 : <a title=\"mfds.pdf\" href=\"wp-content\/uploads\/pdf\/mfds.pdf\"><img decoding=\"async\" src=\"wp-content\/uploads\/pdf.png\" alt=\"pdf\" \/><\/a><br \/>MSC Primary 11F11, Secondary 11F25, 11F66 <\/li>\n<\/ul>\n<p>This is a re-issue of Ogg\u2019s book published in 1969, typeset with TeX.<\/p>\n<h3><a name=\"cft\"><\/a>Class field theory<\/h3>\n<ul>\n<li>version 1.0, 2015-03-04 : <a title=\"cft.pdf\" href=\"wp-content\/uploads\/pdf\/cft.pdf\"><img decoding=\"async\" src=\"wp-content\/uploads\/pdf.png\" alt=\"pdf\" \/><\/a><br \/>first published: 2012-01-10<br \/> MSC Primary 11R37, Secondary 11S31<\/li>\n<\/ul>\n<p>These old notes summarize class field theory. As such they are not particularly well organized, as I realize now. They were created when studying the approach WEIL had chosen in his <em>Basic Number Theory<\/em>: the theory of simple algebras, in favor over group cohomology, which I was used to from Serre. So, in part, these notes follow his suggestion to translate between these two points of view. The emphasis here is on the local situation, as well as on the global function field case. Almost no proofs are given.<\/p>\n<h3><a name=\"cpc\"><\/a>Counting points on curves over finite fields <small>by <em>Enrico BOMBIERI<\/em><\/small><\/h3>\n<ul>\n<li>version 1.0, 2015-03-04 : <a title=\"bbk430.pdf\" href=\"wp-content\/uploads\/pdf\/bbk430.pdf\"><img decoding=\"async\" src=\"wp-content\/uploads\/pdf.png\" alt=\"pdf\" \/><\/a><br \/>first published: 2011-12-27 <\/li>\n<\/ul>\n<p>This is the <em>BOURBAKI<\/em> seminar talk n\u00b0 430 given by <em>Enrico BOMBIERI<\/em> in June 1973, set in TeX with the appropriate <em>BOURBAKI<\/em> class style. I only took advantage of the time since passed to complete the reference to <em>STARK<\/em>&#8216;s article, published later in 1973, and added a reference to <em>SCHMIDT<\/em>&#8216;s Lecture Notes that appeared in 1976. I corrected a minor typo in formula (2).<\/p>\n<h3><a name=\"lfun\"><\/a>Artin L-functions after Weil, Grothendieck<\/h3>\n<ul>\n<li>version 1.0, 2011-11-15 : <a title=\"lfun.pdf\" href=\"wp-content\/uploads\/pdf\/lfun.pdf\"><img decoding=\"async\" src=\"wp-content\/uploads\/pdf.png\" alt=\"pdf\" \/><\/a><br \/>first published: 2011-11-13<br \/>MSC Primary 11M41, Secondary 11R42<\/li>\n<\/ul>\n<p>This paper summarizes some notes taken when comparing Weil&#8217;s presentation of Artin&#8217;s L-functions with Grothendieck&#8217;s.<\/p>\n<p>Dedicated to my brother <em>Joachim<\/em>.<\/p>\n<h3><a name=\"eps\"><\/a>Local \u03b5-factors<\/h3>\n<ul>\n<li>version 1.0, 2014-03-21 : <a title=\"eps.pdf\" href=\"wp-content\/uploads\/pdf\/eps.pdf\"><img decoding=\"async\" src=\"wp-content\/uploads\/pdf.png\" alt=\"pdf\" \/><\/a><br \/>first published: 2011-01-30<br \/> MSC Primary 11S40, Secondary 11L05<\/li>\n<\/ul>\n<p>A short note on local L-series, functional equation, \u03b5-factors, Gauss sums and integrals.<\/p>\n<p>Dedicated to my sister <em>Petra<\/em>.<\/p>\n<h3><a name=\"mod\"><\/a>Modular fundamental domain<\/h3>\n<ul>\n<li><!--img src=\"wp-content\/uploads\/new.gif\" alt=\"new\" \/--> version 1.3, 2018-10-08 : <a title=\"mod.pdf\" href=\"wp-content\/uploads\/pdf\/mod.pdf\"><img decoding=\"async\" src=\"wp-content\/uploads\/pdf.png\" alt=\"pdf\" \/><\/a><br \/>first published: 2007-11-28<br \/>MSC Primary 11F06, Secondary 11F11 <\/li>\n<\/ul>\n<p>On drawings of modular fundamental domains with METAPOST or Java. In 2018 (v1.1) I added some remarks on Helena VERRILL&#8217;s <a title=\"Fundamental domains\" href=\"https:\/\/wstein.org\/Tables\/fundomain\/\" target=\"_blank\" rel=\"noopener noreferrer\">fundamental domain drawer<\/a>. In my v1.3 is a screenshot of the program for the modular curve X<sub>0<\/sub>(11). In particular this java applet of 2000 can be run as java application in the current java runtime environment, which is openjdk 13 (GA 2019-09-17).<\/p>\n<h3><a name=\"nf\"><\/a>Numbers of solutions of equations in finite fields <small>by <em>Andr\u00e9 WEIL<\/em><\/small><\/h3>\n<ul>\n<li>version 1.0, 2003-04-06 : <a title=\"nf.pdf\" href=\"wp-content\/uploads\/pdf\/nf.pdf\"><img decoding=\"async\" src=\"wp-content\/uploads\/pdf.png\" alt=\"pdf\" \/><\/a><br \/> MSC Primary 11G25, Secondary 14G15 <\/li>\n<\/ul>\n<p>I thank the <a title=\"American Mathematical Society\" href=\"http:\/\/www.ams.org\/\" target=\"_blank\" rel=\"noopener noreferrer\"> American Mathematical Society<\/a> for their kind permission to publish my TeX version here.<\/p>\n<p>This is the classical article by Andr\u00e9 Weil that contain his famous conjectures on the \u03b6-function of varieties over finite fields. It has since July 2007 again become available at the American Mathematical Society site: <a title=\"Numbers of solutions of equations in finite fields\" href=\"http:\/\/www.ams.org\/bull\/1949-55-05\/S0002-9904-1949-09219-4\/S0002-9904-1949-09219-4.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">Bull. Am. Math. Soc. <strong>55<\/strong> (1949), 497-508<\/a>.<\/p>\n<h3><a name=\"v4\"><\/a>Zeta-Function of <em>x<sup>4<\/sup> + y<sup>4<\/sup> + z<sup>4<\/sup> = 0<\/em><\/h3>\n<ul>\n<li>version 1.0, 2014-03-21 : <a title=\"v4.pdf\" href=\"wp-content\/uploads\/pdf\/v4.pdf\"><img decoding=\"async\" src=\"wp-content\/uploads\/pdf.png\" alt=\"pdf\" \/><\/a><br \/>first published: 2003-01-27<br \/>MSC Primary 11G20, Secondary 14G10, 11G05<\/li>\n<\/ul>\n<p>This note investigates the \u03b6-function of the Curve <em>C : x<sup>4<\/sup> + y<sup>4<\/sup> + z<sup>4<\/sup> = 0<\/em>, whose interesting factorization stimulated a closer look at its geometry and arithmetic. It was initiated by the question to find simple examples of curves with no rational points.<\/p>\n<p>The first part develops the \u03b6-function formalism and its connection to rational points. In particular an isogeny for the Jacobian variety <em>J<\/em> of <em>C<\/em> is postulated: <em>J \u223c E x E x E<\/em>, where <em>E<\/em> is the elliptic curve <em>y<sup>2<\/sup> = x<sup>3<\/sup> + x<\/em>. The second part provides the proof, based on methods from the classical paper of Andr\u00e9 WEIL and a theorem of Gerd FALTINGS.<\/p>\n<h3><a name=\"alg\"><\/a>Algorithms for computer programs<\/h3>\n<ul>\n<li>v1.8, 2018-10-30 : <a title=\"alg.pdf\" href=\"wp-content\/uploads\/pdf\/alg.pdf\"><img decoding=\"async\" src=\"wp-content\/uploads\/pdf.png\" alt=\"pdf\" \/><\/a><br \/>first published: 2002-12-30<br \/>MSC Primary 11G05, Secondary 14H52<\/li>\n<\/ul>\n<p>This is a collection of routines for doing elementary arithmetic on a computer, like generating primes, factoring into primes and calculations on curves over finite fields.  They originated from some Algol programs of mine written in 1978-1980 at the Faculty of Mathematics in Bielefeld.  They are now implemented in C.<\/p>\n<p>These notes are meant to supplement the program logic, with emphasis put on the conceptual ideas behind those algorithms. These programs are made available under the <a title=\"Apache License 2.0\" href=\"http:\/\/www.apache.org\/licenses\/LICENSE-2.0\"  target=\"_blank\" rel=\"noopener noreferrer\">Apache<\/a> license. The only change in version 1.7 is to the factorisation routine. In version 1.8 the program listings make use of the package <em>listings<\/em>.<\/p>\n<p><!--table border=\"0\">\n\n\n<thead> \n\n\n<tr>\n\n\n<th> <\/th>\n\n \n\n<th> Program <\/th>\n\n \n\n<th> Language <\/th>\n\n \n\n<th> Revision <\/th>\n\n \n\n<th> Date <\/th>\n\n\n<\/tr>\n\n\n<\/thead>\n\n \n\n\n<tbody>\n\n\n<tr>\n\n\n<td><\/td>\n\n\n\n\n<td><a title=\"gcd,rex\" href=\"..\/alg\/gcd.rex\">GCD in Rexx<\/a><\/td>\n\n\n\n\n<td>REXX<\/td>\n\n\n\n\n<td>1.4<\/td>\n\n\n\n\n<td>2010-12-02<\/td>\n\n\n<\/tr>\n\n\n\n\n<tr>\n\n\n<td><\/td>\n\n\n\n\n<td><a title=\"xgcd.rex\" href=\"..\/alg\/xgcd.c\">Extended GCD in C<\/a><\/td>\n\n\n\n\n<td>C<\/td>\n\n\n\n\n<td>316<\/td>\n\n\n\n\n<td>2010-12-02<\/td>\n\n\n<\/tr>\n\n\n\n\n<tr>\n\n\n<td><\/td>\n\n\n\n\n<td><a title=\"fdiv.c\" href=\"..\/alg\/fdiv.c\">Factorisation into primes by division (original Knuth algorithm)<\/a><\/td>\n\n\n\n\n<td>C<\/td>\n\n\n\n\n<td>316<\/td>\n\n\n\n\n<td>2010-12-02<\/td>\n\n\n<\/tr>\n\n\n\n\n<tr>\n\n\n<td><\/td>\n\n\n\n\n<td><img decoding=\"async\" src=\"..\/icons\/new.gif\" alt=\"\" \/><a title=\"fact.c\" href=\"..\/alg\/fact.c\">Factorisation into primes (modified algorithm with exponents)<sup>1<\/sup><\/a><\/td>\n\n\n\n\n<td>C<\/td>\n\n\n\n\n<td>316<\/td>\n\n\n\n\n<td>2012-12-02<\/td>\n\n\n<\/tr>\n\n\n\n\n<tr>\n\n\n<td><\/td>\n\n\n\n\n<td><a title=\"prim.c\" href=\"..\/alg\/prim.c\">Sieve of Eratosthenes (prime table generation)<\/a><\/td>\n\n\n\n\n<td>C<\/td>\n\n\n\n\n<td>1.4<\/td>\n\n\n\n\n<td>2010-12-03<\/td>\n\n\n<\/tr>\n\n\n\n\n<tr>\n\n\n<td><\/td>\n\n\n\n\n<td><a title=\"primx.c\" href=\"..\/alg\/primx.c\">Sieve of Eratosthenes (modified using bit fields)<\/a><\/td>\n\n\n\n\n<td>C<\/td>\n\n\n\n\n<td>1.4<\/td>\n\n\n\n\n<td>2010-12-03<\/td>\n\n\n<\/tr>\n\n\n\n\n<tr>\n\n\n<td><\/td>\n\n\n\n\n<td><a title=\"jac1.algol\" href=\"..\/alg\/jac1.algol\">Number of rational points<\/a> over <strong>F<sub>p<\/sub><\/strong> of <em>y<sup>2<\/sup> = x<sup>3<\/sup> + x<\/em> and <em>x<sup>4<\/sup> + y<sup>4<\/sup> + z<sup>4<\/sup> = 0<\/em><\/td>\n\n\n\n\n<td>Algol<\/td>\n\n\n\n\n<td>1.1<\/td>\n\n\n\n\n<td>2003-04-05<\/td>\n\n\n<\/tr>\n\n\n\n\n<tr>\n\n\n<td><\/td>\n\n\n\n\n<td><a title=\"jac2.algol\" href=\"..\/alg\/jac2.algol\">Number of rational points<\/a> over <strong>F<sub>p<sup>2<\/sup><\/sub><\/strong> of <em>y<sup>2<\/sup> = x<sup>3<\/sup> + x<\/em> and <em>x<sup>4<\/sup> + y<sup>4<\/sup> + z<sup>4<\/sup> = 0<\/em><\/td>\n\n\n\n\n<td>Algol<\/td>\n\n\n\n\n<td>1.1<\/td>\n\n\n\n\n<td>2003-04-05<\/td>\n\n\n<\/tr>\n\n\n\n\n<tr>\n\n\n<td><\/td>\n\n\n\n\n<td><a title=\"jac2a.algol\" href=\"..\/alg\/jac2a.algol\">Number of rational points<\/a> over <strong>F<sub>p<sup>2<\/sup><\/sub><\/strong> the same simplified for <em>p = 3 (mod 4)<\/em><\/td>\n\n\n\n\n<td>Algol<\/td>\n\n\n\n\n<td>1.1<\/td>\n\n\n\n\n<td>2003-04-05<\/td>\n\n\n<\/tr>\n\n\n\n\n<tr>\n\n\n<td><\/td>\n\n\n\n\n<td><a title=\"jac3.algol\" href=\"..\/alg\/jac3.algol\">Number of rational points<\/a> over <strong>F<sub>p<sup>3<\/sup><\/sub><\/strong> of <em>y<sup>2<\/sup> = x<sup>3<\/sup> + x<\/em> and <em>x<sup>4<\/sup> + y<sup>4<\/sup> + z<sup>4<\/sup> = 0<\/em><\/td>\n\n\n\n\n<td>Algol<\/td>\n\n\n\n\n<td>1.1<\/td>\n\n\n\n\n<td>2003-04-05<\/td>\n\n\n<\/tr>\n\n\n\n\n<tr>\n\n\n<td><\/td>\n\n\n\n\n<td><a title=\"ellip.algol\" href=\"..\/alg\/ellip.algol\">The elliptic program in Algol<\/a><\/td>\n\n\n\n\n<td>Algol<\/td>\n\n\n\n\n<td>1.1<\/td>\n\n\n\n\n<td>2003-04-05<\/td>\n\n\n<\/tr>\n\n\n\n\n<tr>\n\n\n<td><\/td>\n\n\n\n\n<td><a title=\"ell.c\" href=\"..\/alg\/ell.c\">The elliptic Code in C<\/a> (Weierstrass models of elliptic curves over <strong>Z<\/strong>)<\/td>\n\n\n\n\n<td>C<\/td>\n\n\n\n\n<td>1.7<\/td>\n\n\n\n\n<td>2010-12-02<\/td>\n\n\n<\/tr>\n\n\n<\/tbody>\n\n\n<\/table-->\n<!--p><sup>1<\/sup> : I have updated this program to fix an error for large prime factors (>2<sup>21<\/sup>), the <em>prime<\/em> array was changed to long long.<\/p-->\n<h3><a name=\"zeta\"><\/a>Zeta-function and Bernoulli numbers<\/h3>\n<ul>\n<li>version 1.1, 2015-03-04 : <a title=\"zeta.pdf\" href=\"wp-content\/uploads\/pdf\/zeta.pdf\"><img decoding=\"async\" src=\"wp-content\/uploads\/pdf.png\" alt=\"pdf\" \/><\/a><br \/>first published: 2001-07-15<br \/>MSC Primary 11M06, Secondary 11B68, 30B50<\/li>\n<\/ul>\n<p>This note gathers the salient features of Dirichlet-series and their convergence, in particular the Riemann \u03b6-function, some special values, including Bernoulli numbers. We follow Riemann for the proof of its <em>functional equation<\/em>, which was added in version 1.1 in 2011.<\/p>\n<p>The appendix treats the <em>analytical continuation<\/em> of the \u03b6-function in an elementary way, without using the functional equation.<\/p>\n<h3><a name=\"sl2\"><\/a>My thesis: Kohomologie S-arithmetischer Untergruppen der SL(2) \u00fcber einem Funktionenk\u00f6rper und arithmetische Eigenschaften automorpher Formen<\/h3>\n<ul>\n<li>version 1.5, 2015-03-04 : <a title=\"sl2.pdf\" href=\"wp-content\/uploads\/pdf\/sl2.pdf\"><img decoding=\"async\" src=\"wp-content\/uploads\/pdf.png\" alt=\"pdf\" \/><\/a> (deutsch)<br \/>first published: 1999-07-09<br \/> MSC Primary 11F75, Secondary 11F70, 11F67<\/li>\n<\/ul>\n<p>My thesis was written under the guidance of <a title=\"G\u00fcnter Harder\" href=\"https:\/\/de.wikipedia.org\/wiki\/G%C3%BCnter_Harder\" target=\"_blank\" rel=\"noopener noreferrer\">G\u00fcnter Harder<\/a>. The results in part one confirmed a conjecture of his: the vanishing of the rational cohomology H<sup>i<\/sup>(\u0393,<strong>Q<\/strong>) = 0 of an S-arithmetic group \u0393 \u2282 SL(2,F) for a function field F\/<strong>F<\/strong><sub>q<\/sub> in degree i \u2260 0, card(S). This was generalized by Harder to other algebraic groups (published in Inventiones in 1977). The other parts deal with the theory of Eisenstein series and its application to the construction of automorphic cohomology classes.<\/p>\n<p>The new version 1.5 loads the package hyperref with option colorlinks, instead of german.sty the package babel is used. The source is now under git control (git repository). <\/p>\n","protected":false},"excerpt":{"rendered":"<p>Modular Forms and Dirichlet series by Andrew OGG version 1.0, 2018-08-29 : MSC Primary 11F11, Secondary 11F25, 11F66 This is a re-issue of Ogg\u2019s book published in 1969, typeset with TeX. Class field theory version 1.0, 2015-03-04 : first published: 2012-01-10 MSC Primary 11R37, Secondary 11S31 These old notes summarize class field theory. As such [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":57,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"ngg_post_thumbnail":0,"footnotes":""},"class_list":["post-87","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/berndt-schwerdtfeger.de\/index.php?rest_route=\/wp\/v2\/pages\/87","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/berndt-schwerdtfeger.de\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/berndt-schwerdtfeger.de\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/berndt-schwerdtfeger.de\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/berndt-schwerdtfeger.de\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=87"}],"version-history":[{"count":26,"href":"https:\/\/berndt-schwerdtfeger.de\/index.php?rest_route=\/wp\/v2\/pages\/87\/revisions"}],"predecessor-version":[{"id":1028,"href":"https:\/\/berndt-schwerdtfeger.de\/index.php?rest_route=\/wp\/v2\/pages\/87\/revisions\/1028"}],"up":[{"embeddable":true,"href":"https:\/\/berndt-schwerdtfeger.de\/index.php?rest_route=\/wp\/v2\/pages\/57"}],"wp:attachment":[{"href":"https:\/\/berndt-schwerdtfeger.de\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=87"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}