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| bgcolor="#BCD2EE" align="center" | Bounds on the 2-torsion in the class groups of number fields | | bgcolor="#BCD2EE" align="center" | Bounds on the 2-torsion in the class groups of number fields | ||
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| bgcolor="#BCD2EE" | | | bgcolor="#BCD2EE" | (Joint with M. Bhargava, T. Taniguchi, F. Thorne, J. Tsimerman, Y. Zhao) | ||
Given a number field K of fixed degree n over Q, a classical theorem of Brauer--Siegel asserts that the size of the class group of K is bounded by O_\epsilon(|Disc(K)|^(1/2+\epsilon). For any prime p, it is conjectured that the p-torsion | |||
subgroup of the class group of K is bounded by O_\epsilon(|Disc(K)|^\epsilon. Only the case n=p=2 of this conjecture in known. In fact, for most pairs (n,p), the best known bounds come from the "convex" Brauer--Siegel bound. | |||
In this talk, we will discuss a proof of a subconvex bound on the size of the 2-torsion in the class groups of number fields, for all degrees n. We will also discuss an application of this result towards improved bounds on the number of integral points on elliptic curves. | |||
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Revision as of 22:05, 18 January 2017
Return to NTS Spring 2017
Jan 19
| Bianca Viray |
| On the dependence of the Brauer-Manin obstruction on the degree of a variety |
| Let X be a smooth projective variety of degree d over a number field k. In 1970 Manin observed that elements of the Brauer group of X can obstruct the existence of a k-point, even when X is everywhere locally soluble. In joint work with Brendan Creutz, we prove that if X is geometrically abelian, Kummer, or bielliptic then this Brauer-Manin obstruction to the existence of a k-point can be detected from only the d-primary torsion Brauer classes. |
Jan 26
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Feb 2
| Arul Shankar |
| Bounds on the 2-torsion in the class groups of number fields |
| (Joint with M. Bhargava, T. Taniguchi, F. Thorne, J. Tsimerman, Y. Zhao)
Given a number field K of fixed degree n over Q, a classical theorem of Brauer--Siegel asserts that the size of the class group of K is bounded by O_\epsilon(|Disc(K)|^(1/2+\epsilon). For any prime p, it is conjectured that the p-torsion subgroup of the class group of K is bounded by O_\epsilon(|Disc(K)|^\epsilon. Only the case n=p=2 of this conjecture in known. In fact, for most pairs (n,p), the best known bounds come from the "convex" Brauer--Siegel bound. In this talk, we will discuss a proof of a subconvex bound on the size of the 2-torsion in the class groups of number fields, for all degrees n. We will also discuss an application of this result towards improved bounds on the number of integral points on elliptic curves. |
Feb 9
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Feb 16
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Feb 23
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Mar 2
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Mar 9
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Mar 16
Mar 30
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Apr 6
| Celine Maistret |
Apr 13
Apr 20
| Yueke Hu |
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Apr 27
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May 4
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