[Liste-CICMA] SÉM. QUÉBEC-VERMONT NUMBER THEORY (03/04/2014, R. Lemke Olivier, J. Walcher)

Guillermo Martinez-Zalce martinez at crm.umontreal.ca
Tue Apr 1 11:06:37 EDT 2014


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SÉMINAIRE QUÉBEC-VERMONT NUMBER THEORY

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DATE :
Le jeudi 3 avril 2014 / Thursday, April 3, 2014

HEURE / TIME :
10 h 30 - 12 h / 10:30 a.m. - 12:00 p.m.

CONFERENCIER(S) / SPEAKER(S) :
Robert Lemke Olivier (Stanford University)

TITRE / TITLE :
2-Selmer ranks of elliptic curves and Erdos-Kac theorems

LIEU / PLACE :
McGill University, Burnside Hall 920 

RESUME / ABSTRACT :
The problem of determining the distribution of the 2-Selmer ranks of quadratic twists of an elliptic curve has received a great deal of recent attention, both in works conjecturing distributions and in those providing solutions; in both cases, the nature of the two-torsion of the elliptic curve plays a cruical role. In particular, if E/Q has full two-torsion, the distribution is known, due to work of Heath-Brown, Swinnerton-Dyer, and Kane, and if E possesses no two-torsion, then, again, the distribution is known, due to work of Klagsbrun, Mazur, and Rubin, though with the caveat that one arranges discriminants in a non-standard way. In stark contrast to these two cases, we show that if K is a number field and E/K is an elliptic curve with partial two-torsion, then no limiting distribution on 2-Selmer ranks exists. We do so by showing that, for any fixed integer r, at least half of the twists of E have 2-Selmer rank greater than r, and we establish an analogous result for simultaneous twists, either for multiple elliptic curves twisted by the same discriminant or for a single elliptic curve twisted by a tuple of discriminants. These results depend upon connecting the 2-Selmer rank of twists to the values of an additive function and then establishing results analogous to the classical Erdos-Kac theorem. This work is joint with Zev Klagsbrun.


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DATE :
Le jeudi 3 avril 2014 / Thursday, April 3, 2014

HEURE / TIME :
14 h - 15 h 30 / 2:00 p.m. - 3:30 p.m.

CONFERENCIER(S) / SPEAKER(S) :
Johannes Walcher (McGill University)

TITRE / TITLE :
Algebraic cycles on Calabi-Yau threefolds and special values of L-functions

LIEU / PLACE :
Concordia University, 9th floor of the Library Building

RESUME / ABSTRACT :
Curves on Calabi-Yau 3-folds provide some of the first examples of non-classical behaviour of Abel-Jacobi maps on algebraic cycles. The situation is arithmetically rich and also plays a role in theoretical physics in the context of mirror symmetry. In this talk, I will explain, as an example, the complete calculation of the normal function associated with the van Geemen family of lines on the Dwork pencil of quintic 3-folds. The limiting value of the normal function under degeneration of the underlying family of varieties is identified (numerically) with the special value of an L-function of the residue field. I also want to give an idea of the motivation for these calculations from mirror symmetry and the interest of the non-constant part of the expansion of the normal function. Some of the calculations that I'll present were done with G. Laporte and R. Jefferson.


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Responsable(s) :
Henri Darmon (darmon at math.mcgill.ca)
Andrew Granville (andrew at dms.umontreal.ca)
Dimitris Koukoulopoulos (koukoulo at CRM.UMontreal.CA)
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http://www.math.mcgill.ca/darmon/qvnts/qvnts.html


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