Morningside Seminar on Number Theory


This seminar is chaired by professor Ye Tian, Song Wang, Yichao Tian, Weizhe Zheng, Wenwei Li in Morningside Center of Mathematics.

Lectures & Seminars in June 2015














































Place: 817 of South Building


Series Lectures

¨ John Coates, June 10 – July 3, the first lecture will start at 10:45 on June 10.

Title: Quadratic twists of X_0(49)

Abstract: The quadratic twists of X_0(49) are the only elliptic curves defined over Q for which the prime 2 is potentially ordinary. For reasons which we still do not understand fully, this makes their Iwasawa theory more accessible, especially at the prime 2. In the course, I hope to prove the conjecture of Birch and Swinnerton-Dyer for the quadratic twists of X_0(49), whose complex L-series does not vanish at s=1, concentrating in particular on the difficult 2-primary part of the exact Birch-Swinnerton-Dyer formula for the order of the Tate-Shafarevich group of such a twist. In the discussion of the formula for the 2-primary part, I will be following work of Gonzalez-Aviles.

¨ Ralph Greenberg, June 15-July 13, the first lecture will start at 9:30 on June 15.

Title: Introduction to Iwasawa Theory

Abstract: This course will be a broad introduction to the ideas of Iwasawa Theory as they have developed over its long history. We will start with studying the behavior of ideal class groups in a $\ZZ_p$-extension of a number field. The main objective will be to formulate Iwasawa's Main Conjecture. Then we will study the analogous questions for elliptic curves. Finally, we will describe how these special cases fit into a vastly more general framework.


Other talks

¨ Yunqing Tang, 3:30pm-4:30pm, June 16

Title: Cycles in the de Rham cohomology of abelian varieties over number fields

Abstract: In his 1982 paper, Ogus defined a class of cycles in the de Rham cohomology of smooth proper varieties over number fields. In the case of abelian varieties, this class includes all the Hodge cycles by the work of Deligne, Ogus and Blasius, and Ogus predicted that all such cycles are Hodge. I confirm Ogus' prediction for some families of abelian varieties under the assumption that the cycles lie in the Betti cohomology with real coefficients. These families include abelian varieties that have prime dimension and nontrivial endomorphism ring. The proof is based on a theorem of Bost and the known cases of the Mumford--Tate conjecture.

¨ Kestutis Cesnavicius, 4:45pm-5:45pm, June 16

Title: The p-parity conjecture for elliptic curves with a p-isogeny

Abstract: For an elliptic curve E over a number field K, one consequence of the Birch and Swinnerton-Dyer conjecture is the parity conjecture: the global root number should match the parity of the Mordell-Weil rank. Its weaker but more approachable version is the p-parity conjecture for a fixed prime p: the global root number should match the parity of the Z_p-corank of the p-infinity Selmer group. After surveying what is known on the p-parity conjecture, we will discuss its proof in the case when E has a K-rational p-isogeny.

¨ Xin Wan, 3:30pm -6pm, June 22

Title: Iwasawa theory and BSD conjecture

Abstract: In the first talk I'll introduce some backgrounds on Iwasawa theory and BSD conjecture, and review some recent progresses. In the second talk I'll discuss some techniques involved in the proof in details.

¨ Otmar Venjakob, 4:00pm – 5:00pm, June 23

Title: Iwasawa cohomolgy and (\phi,\Gamma)-modules over Lubin Tate extensions

Abstract: We generalize work of Fontaine and Herr about expressing local Galois cohomology and Iwasawa cohomology in terms of (\phi,\Gamma)-modules from the cyclotomic case to the case of Lubin-Tate extensions. In particular we show a kind of reciprocity law which admits to calculate the image of an adequate Kummer map in the firrst Iwasawa cohomology group. This is joint work with Peter Schneider (Münster).

¨ Mingmin Shen, 3:30pm-5:00pm, June 30 (then on July 1, 7, 8, 14, 15)

Title: The rationality problem in algebraic geometry

Abstract: One fundamental problem in algebraic geometry is to determine whether a variety is rational or not. Here being rational can be understood as being isomorphic to the projective space modulo lower dimensional subvarieties on both sides. In the lectures, I will explain how the problem gets more complicated as dimension increases. After reviewing the classical solution in low dimensional case, I will explain the work of Clemens—Griffiths on cubic threefold. Then a major part will be devoted to the recent results obtained by Voisin and Totaro via cycle-theoretical approach. I will also discuss the case of cubic fourfolds where the question is still open.

¨ Liang Xiao, (TBA)

Title: Galois representations attached to torsion coherent cohomology of Hilbert modular varieties

Abstract: We prove that for each eigenclass in the mod p^n coherent cohomology of automorphic line bundles on Hilbert modular varieties, one can attach a pseudo-Galois representation.  In the case of Hilbert modular surface and of parallel weight one, we show that the representation we obtain is unramified at p. This is a joint work with Matthew Emerton and Davide Reduzzi. If time permits, we discuss a potential approach to generalize this to general Hilbert modular varieties.

¨ Jieyu Zhang, 9:00-10:00, July 14

Title: An effective criterion for Eulerian multiple zeta values in characteristic p

Abstract: A classical multiple zeta value (MZV) is said to be Eulerian if it is a rational multiple of a power of 2$\pi$i. Typical examples (from Euler's formula) are the Riemann zeta values (depth one MZV's) at even positive integers. It is a natural question whether one has a criterion for Eulerian MZV's, but it is still open. In this talk, we will consider the analogous question for those positive characteristic MZV's introduced by Thakur. We will present an effective criterion for these Eulerian MZV's, and show how this criterion can be implemented by Magma to determine whether a given MZV is Eulerian. This is a joint work with M. Papanikolas and J. Yu.

¨ Ziyang Gao, (TBA)

Title: Recent developments on the André-Oort conjecture
Abstract: The André-Oort conjecture predicts that any subvariety of a mixed Shimura variety containing a Zariski dense subset of special points is again a moduli space of some mixed Hodge structures with some Hodge tensors. An interesting example is when the ambient mixed Shimura variety is the universal abelian variety, in which case special points are precisely the points corresponding to torsion points on CM abelian varieties. This conjecture was reduced to a lower bound for the size of Galois orbits of special points by a series of work (Klingler-Ullmo-Yafaev, Pila-Tsimerman, Gao) and hence proved for mixed Shimura varieties of abelian type by the recent work of Tsimerman and Yuan-Zhang/Andreatta-Goren-Howard-Pera. In the proof, a transcendental and a distribution theorem (Ax-Lindemann and its corollary) of independent interest were proved. In my talk I will explain this conjecture and sketch its proof. In particular I will explain the very recent result of Tsimerman about how to prove the lower bound using the Colmez conjecture in average.


A book of Yau - A History in Sum
Local indecomposability of Galois representations