NTS ABSTRACTFall2021
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Sep 9
Anwesh Ray |
Arithmetic statistics and diophantine stability for elliptic curves |
In 2017, B. Mazur and K. Rubin introduced the notion of diophantine stability for a variety defined over a number field. Given an elliptic curve E defined over the rationals and a prime number p, E is said to be diophantine stable at p if there are abundantly many p-cyclic extensions $L/\mathbb{Q}$ such that $E(L)=E(\mathbb{Q})$. In particular, this means that given any integer $n>0$, there are infinitely many cyclic extensions with Galois group $\mathbb{Z}/p^n\mathbb{Z}$, such that $E(L)=E(\mathbb{Q})$. It follows from more general results of Mazur-Rubin that $E$ is diophantine stable at a positive density set of primes p. In this talk, I will discuss diophantine stability of average for pairs $(E,p)$, where $E$ is a non-CM elliptic curve and $p\geq 11$ is a prime number at which $E$ has good ordinary reduction. First, I will fix the elliptic curve and vary the prime. In this context, diophantine stability is a consequence of certain properties of Selmer groups studied in Iwasawa theory. Statistics for Iwasawa invariants were studied recently in joint work with Debanjana Kundu. As an application, one shows that if the Mordell Weil rank of E is zero, then, $E$ is diophantine stable at $100\%$ of primes $p$. One also shows that standard conjectures (like rank distribution) imply that for any prime $p\geq 11$, a positive density set of elliptic curves (ordered by height) is diophantine stable at $p$. I will also talk about related results for stability and growth of the p-primary part of the Tate-Shafarevich group in cyclic p-extensions. |
Sep 16
Qiao He |
Kudla-Rapoport conjecture at a ramified prime |
Kudla-Rapoport conjecture predicts that there is an identity between the intersection number of special cycles on unitary Rapoport-Zink space and the derivative of local density of certain Hermitian form. However, the original conjecture was only formulated at an unramified prime. In this talk, I will motivate the original conjecture and discuss how to modify it at a ramified prime. Finally, I will sketch how to verify the modified conjecture for n=3. This is a joint work with Yousheng Shi and Tonghai Yang. |