https://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTFall2019&feed=atom&action=historyNTS ABSTRACTFall2019 - Revision history2024-03-29T12:09:02ZRevision history for this page on the wikiMediaWiki 1.39.5https://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTFall2019&diff=18492&oldid=prevShusterman: /* Dec 05 */2019-11-25T22:09:06Z<p><span dir="auto"><span class="autocomment">Dec 05</span></span></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| bgcolor="#BCD2EE" | What is the probability that the ring of integers in a number field contains a unit of mixed signature? In this talk, we present Cohen-Lenstra style heuristics for unit signatures and narrow class groups of odd abelian number fields. In addition, we analyze the equation $x^3 - ax^2 + bx - 1 = 0$ to prove that there are infinitely many cyclic cubic number fields with no units of mixed signature. This is joint work with Noam Elkies, Ila Varma, and John Voight.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| bgcolor="#BCD2EE" | What is the probability that the ring of integers in a number field contains a unit of mixed signature? In this talk, we present Cohen-Lenstra style heuristics for unit signatures and narrow class groups of odd abelian number fields. In addition, we analyze the equation $x^3 - ax^2 + bx - 1 = 0$ to prove that there are infinitely many cyclic cubic number fields with no units of mixed signature. This is joint work with Noam Elkies, Ila Varma, and John Voight.</div></td></tr>
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</table>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTFall2019&diff=18491&oldid=prevShusterman at 22:08, 25 November 20192019-11-25T22:08:58Z<p></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| bgcolor="#BCD2EE" | Fix a number field $K$ and a finite transitive subgroup $G \le S_n$. Malle's conjecture proposes asymptotics for counting the number of $G$-extensions of number fields $F/K$ with discriminant bounded above by $X$. A recent and fruitful approach to this problem introduced by Lemke Oliver, Wang, and Wood is to count inductively. Fix a normal subgroup $T \triangleleft G$. Step one: for each $G/T$-extension $L/K$, first count the number of towers of fields $F/L/K$ with ${\mathrm Gal}(F/L) \cong T$ and ${\mathrm Gal}(F/K)\cong G$ with discriminant bounded above by $X$. Step two: sum over all choices for the $G/T$-extension $L/K$. In this talk we discuss the close relationship between step one of this method and the first Galois cohomology group. This approach suggests a refinement of Malle's conjecture which gives new insight into the problem. We give the solution to step one when $T$ is an abelian normal subgroup of $G$, and convert this into nontrivial lower bounds for Malle's conjecture whenever $G$ has an abelian normal subgroup.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| bgcolor="#BCD2EE" | Fix a number field $K$ and a finite transitive subgroup $G \le S_n$. Malle's conjecture proposes asymptotics for counting the number of $G$-extensions of number fields $F/K$ with discriminant bounded above by $X$. A recent and fruitful approach to this problem introduced by Lemke Oliver, Wang, and Wood is to count inductively. Fix a normal subgroup $T \triangleleft G$. Step one: for each $G/T$-extension $L/K$, first count the number of towers of fields $F/L/K$ with ${\mathrm Gal}(F/L) \cong T$ and ${\mathrm Gal}(F/K)\cong G$ with discriminant bounded above by $X$. Step two: sum over all choices for the $G/T$-extension $L/K$. In this talk we discuss the close relationship between step one of this method and the first Galois cohomology group. This approach suggests a refinement of Malle's conjecture which gives new insight into the problem. We give the solution to step one when $T$ is an abelian normal subgroup of $G$, and convert this into nontrivial lower bounds for Malle's conjecture whenever $G$ has an abelian normal subgroup.</div></td></tr>
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<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">| bgcolor="#BCD2EE" align="center" | On unit signatures and narrow class groups of odd abelian number fields: structure and heuristics</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">|-</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">| bgcolor="#BCD2EE" | What is the probability that the ring of integers in a number field contains a unit of mixed signature? In this talk, we present Cohen-Lenstra style heuristics for unit signatures and narrow class groups of odd abelian number fields. In addition, we analyze the equation $x^3 - ax^2 + bx - 1 = 0$ to prove that there are infinitely many cyclic cubic number fields with no units of mixed signature. This is joint work with Noam Elkies, Ila Varma, and John Voight.</ins></div></td></tr>
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</table>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTFall2019&diff=18481&oldid=prevShusterman: /* Nov 26 */2019-11-22T23:26:19Z<p><span dir="auto"><span class="autocomment">Nov 26</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 23:26, 22 November 2019</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| bgcolor="#BCD2EE" | Fix a number field $K$ and a finite transitive subgroup $G \le S_n$. Malle's conjecture proposes asymptotics for counting the number of $G$-extensions of number fields $F/K$ with discriminant bounded above by $X$. A recent and fruitful approach to this problem introduced by Lemke Oliver, Wang, and Wood is to count inductively. Fix a normal subgroup $T\<del style="font-weight: bold; text-decoration: none;">trianglelefteq </del>G$. Step one: for each $G/T$-extension $L/K$, first count the number of towers of fields $F/L/K$ with ${\mathrm Gal}(F/L) \cong T$ and ${\mathrm Gal}(F/K)\cong G$ with discriminant bounded above by $X$. Step two: sum over all choices for the $G/T$-extension $L/K$. In this talk we discuss the close relationship between step one of this method and the first Galois cohomology group. This approach suggests a refinement of Malle's conjecture which gives new insight into the problem. We give the solution to step one when $T$ is an abelian normal subgroup of $G$, and convert this into nontrivial lower bounds for Malle's conjecture whenever $G$ has an abelian normal subgroup.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| bgcolor="#BCD2EE" | Fix a number field $K$ and a finite transitive subgroup $G \le S_n$. Malle's conjecture proposes asymptotics for counting the number of $G$-extensions of number fields $F/K$ with discriminant bounded above by $X$. A recent and fruitful approach to this problem introduced by Lemke Oliver, Wang, and Wood is to count inductively. Fix a normal subgroup $T \<ins style="font-weight: bold; text-decoration: none;">triangleleft </ins>G$. Step one: for each $G/T$-extension $L/K$, first count the number of towers of fields $F/L/K$ with ${\mathrm Gal}(F/L) \cong T$ and ${\mathrm Gal}(F/K)\cong G$ with discriminant bounded above by $X$. Step two: sum over all choices for the $G/T$-extension $L/K$. In this talk we discuss the close relationship between step one of this method and the first Galois cohomology group. This approach suggests a refinement of Malle's conjecture which gives new insight into the problem. We give the solution to step one when $T$ is an abelian normal subgroup of $G$, and convert this into nontrivial lower bounds for Malle's conjecture whenever $G$ has an abelian normal subgroup.</div></td></tr>
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</table>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTFall2019&diff=18480&oldid=prevShusterman: /* Nov 26 */2019-11-22T23:26:05Z<p><span dir="auto"><span class="autocomment">Nov 26</span></span></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| bgcolor="#BCD2EE" | Fix a number field $K$ and a finite transitive subgroup $G \le S_n$. Malle's conjecture proposes asymptotics for counting the number of G-extensions of number fields $F/K$ with discriminant bounded above by X. A recent and fruitful approach to this problem introduced by Lemke Oliver, Wang, and Wood is to count inductively. Fix a normal subgroup $T\trianglelefteq G$. Step one: for each $G/T$-extension $L/K$, first count the number of towers of fields $F/L/K$ with ${\mathrm Gal}(F/L) \cong T$ and ${\mathrm Gal}(F/K)\cong G$ with discriminant bounded above by $X$. Step two: sum over all choices for the $G/T$-extension $L/K$. In this talk we discuss the close relationship between step one of this method and the first Galois cohomology group. This approach suggests a refinement of Malle's conjecture which gives new insight into the problem. We give the solution to step one when $T$ is an abelian normal subgroup of $G$, and convert this into nontrivial lower bounds for Malle's conjecture whenever $G$ has an abelian normal subgroup.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| bgcolor="#BCD2EE" | Fix a number field $K$ and a finite transitive subgroup $G \le S_n$. Malle's conjecture proposes asymptotics for counting the number of <ins style="font-weight: bold; text-decoration: none;">$</ins>G<ins style="font-weight: bold; text-decoration: none;">$</ins>-extensions of number fields $F/K$ with discriminant bounded above by <ins style="font-weight: bold; text-decoration: none;">$</ins>X<ins style="font-weight: bold; text-decoration: none;">$</ins>. A recent and fruitful approach to this problem introduced by Lemke Oliver, Wang, and Wood is to count inductively. Fix a normal subgroup $T\trianglelefteq G$. Step one: for each $G/T$-extension $L/K$, first count the number of towers of fields $F/L/K$ with ${\mathrm Gal}(F/L) \cong T$ and ${\mathrm Gal}(F/K)\cong G$ with discriminant bounded above by $X$. Step two: sum over all choices for the $G/T$-extension $L/K$. In this talk we discuss the close relationship between step one of this method and the first Galois cohomology group. This approach suggests a refinement of Malle's conjecture which gives new insight into the problem. We give the solution to step one when $T$ is an abelian normal subgroup of $G$, and convert this into nontrivial lower bounds for Malle's conjecture whenever $G$ has an abelian normal subgroup.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|} </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|} </div></td></tr>
</table>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTFall2019&diff=18479&oldid=prevShusterman: /* Nov 26 */2019-11-22T23:25:32Z<p><span dir="auto"><span class="autocomment">Nov 26</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 23:25, 22 November 2019</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| bgcolor="#BCD2EE" align="center" | Counting Towers of Number Fields</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| bgcolor="#BCD2EE" align="center" | Counting Towers of Number Fields</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| bgcolor="#BCD2EE" | Fix a number field $K$ and a finite transitive subgroup $G \le S_n$. Malle's conjecture proposes asymptotics for counting the number of G-extensions of number fields $F/K$ with discriminant bounded above by X. A recent and fruitful approach to this problem introduced by Lemke Oliver, Wang, and Wood is to count inductively. Fix a normal subgroup $T\trianglelefteq G$. Step one: for each $G/T$-extension $L/K$, first count the number of towers of fields $F/L/K$ with ${\mathrm Gal}(F/L) \cong T$ and ${\<del style="font-weight: bold; text-decoration: none;">rm </del>Gal}(F/K)\cong G$ with discriminant bounded above by $X$. Step two: sum over all choices for the $G/T$-extension $L/K$. In this talk we discuss the close relationship between step one of this method and the first Galois cohomology group. This approach suggests a refinement of Malle's conjecture which gives new insight into the problem. We give the solution to step one when $T$ is an abelian normal subgroup of $G$, and convert this into nontrivial lower bounds for Malle's conjecture whenever $G$ has an abelian normal subgroup.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| bgcolor="#BCD2EE" | Fix a number field $K$ and a finite transitive subgroup $G \le S_n$. Malle's conjecture proposes asymptotics for counting the number of G-extensions of number fields $F/K$ with discriminant bounded above by X. A recent and fruitful approach to this problem introduced by Lemke Oliver, Wang, and Wood is to count inductively. Fix a normal subgroup $T\trianglelefteq G$. Step one: for each $G/T$-extension $L/K$, first count the number of towers of fields $F/L/K$ with ${\mathrm Gal}(F/L) \cong T$ and ${\<ins style="font-weight: bold; text-decoration: none;">mathrm </ins>Gal}(F/K)\cong G$ with discriminant bounded above by $X$. Step two: sum over all choices for the $G/T$-extension $L/K$. In this talk we discuss the close relationship between step one of this method and the first Galois cohomology group. This approach suggests a refinement of Malle's conjecture which gives new insight into the problem. We give the solution to step one when $T$ is an abelian normal subgroup of $G$, and convert this into nontrivial lower bounds for Malle's conjecture whenever $G$ has an abelian normal subgroup.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|} </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|} </div></td></tr>
</table>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTFall2019&diff=18478&oldid=prevShusterman: /* Nov 26 */2019-11-22T23:25:02Z<p><span dir="auto"><span class="autocomment">Nov 26</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 23:25, 22 November 2019</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| bgcolor="#BCD2EE" align="center" | Counting Towers of Number Fields</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| bgcolor="#BCD2EE" align="center" | Counting Towers of Number Fields</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| bgcolor="#BCD2EE" | Fix a number field $K$ and a finite transitive subgroup $G \le S_n$. Malle's conjecture proposes asymptotics for counting the number of G-extensions of number fields $F/K$ with discriminant bounded above by X. A recent and fruitful approach to this problem introduced by Lemke Oliver, Wang, and Wood is to count inductively. Fix a normal subgroup $T\trianglelefteq G$. Step one: for each $G/T$-extension $L/K$, first count the number of towers of fields $F/L/K$ with ${\<del style="font-weight: bold; text-decoration: none;">rm </del>Gal}(F/L) \cong T$ and ${\rm Gal}(F/K)\cong G$ with discriminant bounded above by $X$. Step two: sum over all choices for the $G/T$-extension $L/K$. In this talk we discuss the close relationship between step one of this method and the first Galois cohomology group. This approach suggests a refinement of Malle's conjecture which gives new insight into the problem. We give the solution to step one when $T$ is an abelian normal subgroup of $G$, and convert this into nontrivial lower bounds for Malle's conjecture whenever $G$ has an abelian normal subgroup.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| bgcolor="#BCD2EE" | Fix a number field $K$ and a finite transitive subgroup $G \le S_n$. Malle's conjecture proposes asymptotics for counting the number of G-extensions of number fields $F/K$ with discriminant bounded above by X. A recent and fruitful approach to this problem introduced by Lemke Oliver, Wang, and Wood is to count inductively. Fix a normal subgroup $T\trianglelefteq G$. Step one: for each $G/T$-extension $L/K$, first count the number of towers of fields $F/L/K$ with ${\<ins style="font-weight: bold; text-decoration: none;">mathrm </ins>Gal}(F/L) \cong T$ and ${\rm Gal}(F/K)\cong G$ with discriminant bounded above by $X$. Step two: sum over all choices for the $G/T$-extension $L/K$. In this talk we discuss the close relationship between step one of this method and the first Galois cohomology group. This approach suggests a refinement of Malle's conjecture which gives new insight into the problem. We give the solution to step one when $T$ is an abelian normal subgroup of $G$, and convert this into nontrivial lower bounds for Malle's conjecture whenever $G$ has an abelian normal subgroup.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|} </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|} </div></td></tr>
</table>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTFall2019&diff=18477&oldid=prevShusterman at 23:24, 22 November 20192019-11-22T23:24:50Z<p></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| bgcolor="#BCD2EE" | In 1966 Artin and Tate constructed a canonical pairing on the Brauer group of a surface over a finite field, and conjectured it to be alternating. This duality has analogous incarnations across arithmetic and topology, namely the Cassels-Tate pairing for a Jacobian variety, and the linking form on a 5-manifold. I will explain a proof of the conjecture, which is based on a surprising connection to Steenrod operations. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| bgcolor="#BCD2EE" | In 1966 Artin and Tate constructed a canonical pairing on the Brauer group of a surface over a finite field, and conjectured it to be alternating. This duality has analogous incarnations across arithmetic and topology, namely the Cassels-Tate pairing for a Jacobian variety, and the linking form on a 5-manifold. I will explain a proof of the conjecture, which is based on a surprising connection to Steenrod operations. </div></td></tr>
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<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">|-</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Brandon Alberts'''</ins></div></td></tr>
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<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">| bgcolor="#BCD2EE" align="center" | Counting Towers of Number Fields</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">|-</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">| bgcolor="#BCD2EE" | Fix a number field $K$ and a finite transitive subgroup $G \le S_n$. Malle's conjecture proposes asymptotics for counting the number of G-extensions of number fields $F/K$ with discriminant bounded above by X. A recent and fruitful approach to this problem introduced by Lemke Oliver, Wang, and Wood is to count inductively. Fix a normal subgroup $T\trianglelefteq G$. Step one: for each $G/T$-extension $L/K$, first count the number of towers of fields $F/L/K$ with ${\rm Gal}(F/L) \cong T$ and ${\rm Gal}(F/K)\cong G$ with discriminant bounded above by $X$. Step two: sum over all choices for the $G/T$-extension $L/K$. In this talk we discuss the close relationship between step one of this method and the first Galois cohomology group. This approach suggests a refinement of Malle's conjecture which gives new insight into the problem. We give the solution to step one when $T$ is an abelian normal subgroup of $G$, and convert this into nontrivial lower bounds for Malle's conjecture whenever $G$ has an abelian normal subgroup.</ins></div></td></tr>
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</table>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTFall2019&diff=18440&oldid=prevShusterman: /* Sep 19 */2019-11-17T18:28:36Z<p><span dir="auto"><span class="autocomment">Sep 19</span></span></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| bgcolor="#BCD2EE" | </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| bgcolor="#BCD2EE" | </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>An abelian variety in characteristic <math>p</math> is said to be ordinary if its <math>p</math> torsion is as large as possible. In 2012, Cais, Ellenberg and Zureick-Brown made some conjectures about the distribution of the size of the <math>p</math> -torsion of an abelian variety. I will talk about some families which do not obey these heuristics, namely Jacobians of Artin-Schreier and superelliptic curves, and discuss the structure of the respective moduli spaces that make it so. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>An abelian variety in characteristic <math>p</math> is said to be ordinary if its <math>p</math> torsion is as large as possible. In 2012, Cais, Ellenberg and Zureick-Brown made some conjectures about the distribution of the size of the <math>p</math> -torsion of an abelian variety. I will talk about some families which do not obey these heuristics, namely Jacobians of Artin-Schreier and superelliptic curves, and discuss the structure of the respective moduli spaces that make it so. </div></td></tr>
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</table>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTFall2019&diff=18439&oldid=prevShusterman: /* Oct 3 */2019-11-17T18:28:24Z<p><span dir="auto"><span class="autocomment">Oct 3</span></span></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Wiles's proof of the modularity of semistable elliptic curves over the rationals uses the Langlands-Tunnell theorem as a starting point. In order to feed this into a modularity lifting theorem, one needs to use congruences between modular forms of weight one and modular forms of higher weight. Similar congruences are not known over imaginary quadratic fields and Wiles's strategy runs into problems right from the start. We circumvent this congruence problem and show that mod 3 Galois representations over imaginary quadratic fields arise from automorphic forms that are the analog of higher weight modular forms. Our argument relies on a 2-adic automorphy lifting theorem over CM fields together with a "2-3 switch." As an application, we deduce that a positive proportion of elliptic curves over imaginary quadratic fields are modular. This is joint work in progress with Chandrashekhar Khare and Jack Thorne.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Wiles's proof of the modularity of semistable elliptic curves over the rationals uses the Langlands-Tunnell theorem as a starting point. In order to feed this into a modularity lifting theorem, one needs to use congruences between modular forms of weight one and modular forms of higher weight. Similar congruences are not known over imaginary quadratic fields and Wiles's strategy runs into problems right from the start. We circumvent this congruence problem and show that mod 3 Galois representations over imaginary quadratic fields arise from automorphic forms that are the analog of higher weight modular forms. Our argument relies on a 2-adic automorphy lifting theorem over CM fields together with a "2-3 switch." As an application, we deduce that a positive proportion of elliptic curves over imaginary quadratic fields are modular. This is joint work in progress with Chandrashekhar Khare and Jack Thorne.</div></td></tr>
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</table>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTFall2019&diff=18438&oldid=prevShusterman: /* Oct 10 */2019-11-17T18:28:09Z<p><span dir="auto"><span class="autocomment">Oct 10</span></span></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| bgcolor="#BCD2EE" | Let $K$ be a field. Fix a projective curve $X \subset \mathbb{P}^r_K$ of degree $d$. A general hyperplane $H \in \mathbb{P}^{r*}$ intersects $X$ in $d$ points; the monodromy of $X \bigcap H$ as $H$ varies is a subgroup $G_X$ of $S_d$ known as the sectional monodromy group of $X$. When $K=\mathbb{C}$ (or in fact for $\mathrm{char} K = 0$), the equality $G_X=S_d$ was shown by Castelnuovo; this large monodromy fact is important in studying the degree-genus problem for projective curves. I will talk about the behaviour of sectional monodromy groups in positive characteristic. I will show that for a large class of curves the inclusion $G_X \supset A_d$ holds. On the other hand, for a seemingly simple family of curves $X_{m,n}$ given by the equation $x^n=y^mz^{n-m}$ in $\mathbb{P}^2$ I will completely characterize the possibilities for $G_{X_{n,m}}$; the list of possibilities includes linear groups $\mathrm{AGL}_n(q)$, $\mathrm{PGL}_2(q)$ as well as some sporadic simple groups.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| bgcolor="#BCD2EE" | Let $K$ be a field. Fix a projective curve $X \subset \mathbb{P}^r_K$ of degree $d$. A general hyperplane $H \in \mathbb{P}^{r*}$ intersects $X$ in $d$ points; the monodromy of $X \bigcap H$ as $H$ varies is a subgroup $G_X$ of $S_d$ known as the sectional monodromy group of $X$. When $K=\mathbb{C}$ (or in fact for $\mathrm{char} K = 0$), the equality $G_X=S_d$ was shown by Castelnuovo; this large monodromy fact is important in studying the degree-genus problem for projective curves. I will talk about the behaviour of sectional monodromy groups in positive characteristic. I will show that for a large class of curves the inclusion $G_X \supset A_d$ holds. On the other hand, for a seemingly simple family of curves $X_{m,n}$ given by the equation $x^n=y^mz^{n-m}$ in $\mathbb{P}^2$ I will completely characterize the possibilities for $G_{X_{n,m}}$; the list of possibilities includes linear groups $\mathrm{AGL}_n(q)$, $\mathrm{PGL}_2(q)$ as well as some sporadic simple groups.</div></td></tr>
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