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Raymond van Bommel
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Update tcc course page
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‎index.htm‎

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</div>
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<div id="mainbar">
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<h1><a name="news" id="news"></a>News</h1>
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The web page for my TCC course is now online, see <a href="#teaching">below</a>.
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<!--<h1><a name="news" id="news"></a>News</h1>-->
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<h1><a name="contact" id="contact"></a>Contact details</h1>
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<p>
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<h1><a name="teaching" id="teaching"></a>Teaching</h1>
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<ul>
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<li>Spring 2025: <a href="tcc.htm">The Birch and Swinnerton-Dyer conjecture explained</a> (<a href="https://www.maths.ox.ac.uk/groups/tcc">Taught Course Centre</a> course, University of Bristol)
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<li>Spring 2025: <a href="teaching/tcc2025.htm">The Birch and Swinnerton-Dyer conjecture explained</a> (<a href="https://www.maths.ox.ac.uk/groups/tcc">Taught Course Centre</a> course, University of Bristol)
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<li>Autumn 2021: <a href="https://math.mit.edu/classes/18.721/Fall2021/">Introduction To Algebraic Geometry (18.721)</a> (Massachusetts Institute of Technology)</li>
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<li>Summer 2019: <a href="https://www.mathematik.uni-mainz.de/lehrveranstaltungen/?event_id=366564671866814">Computeralgebra</a> (Johannes Gutenberg-Universit&auml;t Mainz)</li>
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<li>Winter 2018/2019: <a href="https://ilias.uni-mainz.de/goto.php?client_id=JOGU&target=crs_634588">Zahlentheorie</a> (Johannes Gutenberg-Universit&auml;t Mainz)</li>

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<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1" />
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<title>TCC - The Birch and Swinnerton-Dyer conjecture explained</title>
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<script type='text/javascript' src='./jquery/jquery-1.12.0.min.js'></script>
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<script type='text/javascript' src='./jquery/jquery.rot13.min.js'></script>
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<!-- E-mailadres verbergen, met dank aan Stefan van der Lugt -->
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<script type='text/javascript'>
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$(document).ready(function(){
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</head>
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<title>The Birch and Swinnerton-Dyer conjecture explained</title>
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<link href="../css/1.css" rel="stylesheet" type="text/css" />
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<link rel="shortcut icon" type="image/ico" href="../favicon.ico" />
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<body>
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<a name="top" id="top"></a>
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<div id="mainbar">
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<h1><a name="practical" id="practical"></a>Practical information</h1>
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This course takes place on Thursday between 14:00 and 16:00 in room G.11 of the Fry Building in Bristol, and online on Teams (registration information <a href="https://www.maths.ox.ac.uk/groups/tcc">here</a>) from 1 May up to 26 June 2025, with no course on 12 June.<br /><br />
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This course took place in Bristol, and online on Teams (as part of <a href="https://www.maths.ox.ac.uk/groups/tcc">TCC</a>) in May and June 2025.<br /><br />
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Questions, comments, and remarks are encouraged.
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My contact details are on my <a href="index.htm">main page</a>.
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In case you are in need of an assessment for this course, please contact me immediately.
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My contact details are on my <a href="../index.htm">main page</a>.
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<h1><a name="Abstract" id="abstract"></a>Abstract</h1>
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The Birch and Swinnerton-Dyer conjecture is a millennium problem originally stated for elliptic curves over the rationals and later generalised to abelian varieties over number fields. In this course, we will learn about the different geometric and arithmetic invariants that occur in the conjecture and their significance in the broader context of arithmetic geometry.
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<h1><a name="notes" id="notes"></a>Notes</h1>
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You can find some notes <a href="https://www.dropbox.com/scl/fi/hy36ql9rf16ggd96ehrkv/TCC_course.pdf?rlkey=n2ipdtry6dbnqzumm52ftwrof&st=r954ouck">here</a>. This is work in progress and any errors in it are mine. Please feel encouraged to report such errors to me. Other suggestions are also welcome.
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You can find some notes <a href="tcc2025.pdf">here</a> (<a href="tcc2025.tex">TeX</a>). Any errors in it are mine. Please feel encouraged to report such errors to me. Other suggestions are also welcome.
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<h1><a name="prerequisites" id="prerequisites"></a>Prerequisites</h1>
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Certainly you don't need to know all of these topics to join the course, but if you do, it might help you connect it to other things you know about.
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<h1><a name="topics" id="topics"></a>Possible topics</h1>
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There is a good chance that I won't be able to cover all these topics in detail, but below you can find a list of topics I am considering. (Updated after lecture 4.)
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Below you can find a list of topics covered.
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<ul><li>Introduction to the conjecture and abelian varieties (lecture 1)</li>
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<li>Models of curves and abelian varieties (lecture 2)</li>
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<li>Periods and N&eacute;ron differential (lecture 3/4)</li>
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<li>Regulators, heights, and Arakelov theory (lecuture 4/5)</li>
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<li>Tate-Shafarevich group, Selmer groups, descent, local-global principle, torsors and H1 (lecture 5/6)</li>
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<li>Tamagawa numbers</li>
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<li>L-functions, Tate module, and conductor</li>
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<li>Known results: Coates-Wiles, Gross-Zagier, Kolyvagin, Heegner points</li>
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<li>Tamagawa numbers (lecture 7)</li>
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<li>L-functions, Tate module, and conductor (lecture 7)</li>
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<li>Known results: modularity, Heegner points, Gross-Zagier, Euler systems, invariance under isogeny and Weil restriction, parity (lecture 8)</li>
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</ul>
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</div>
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‎teaching/tcc2025.pdf‎

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