Elliptic Curves
and their friends and family
Elliptic Curves
Elliptic Curves

Workshop at the University of Nottingham

September 12 and 13, 2019

Elliptic Curves and their friends and family

A workshop following the LMS Midlands regional meeting.

## Speakers

• Werner Bley (München)
• Kazim Büyükboduk (Dublin)
• Nirvana Coppola (Bristol)
• Vladimir Dokchitser (University College London)
• Giada Grossi (University College London)
• Henri Johnston (Exeter)
• Yukako Kezuka (Regensburg)
• Jaclyn Lang (Paris)

## Programme

See here for the schedule for the LMS Meeting on Wednesday September 11.
Unlike that first day, all talks on Thursday and Friday are in Physics C05.

Click on the title to get the abstract.

### Thursday, September 12

The Birch-Swinnerton-Dyer conjecture famously predicts that the rank of an elliptic curve or of an abelian variety agrees with the order of the zero at $$s=1$$ of its $$L$$-function. I will explain that the conjecture correctly gives the parity of the rank for semistable abelian surfaces, assuming finiteness of the Tate-Shafarevich group and the analytic continuation and functional equation of the $$L$$-function. This is joint work with Céline Maistret.

• 10:00 : Coffee break (C04)
• 10:30 : Yukako Kezuka

In this talk, we will look at elliptic curves $$E$$ of the form $$x^3+y^3=2p$$ or $$2p^2$$, where $$p$$ is an odd prime number congruent to 2 or 5 modulo 9. They are cubic twists of the Fermat curve $$x^3+y^3=1$$. We will first study the 3-adic valuation of the algebraic part of the value of the complex $$L$$-series at $$s=1$$. We will then discuss a relation between the 2-rank of the ideal class group of $$\mathbb{Q}(\sqrt[3]{p})$$ and the 2-part of the Tate-Shafarevich group of $$E$$. In particular, we obtain some evidence that there are many elliptic curves with rank 1 and non-trivial 2-part of the Tate-Shafarevich group. This is joint work in progress with Yongxiong Li.

• 11:40 : Jaclyn Lang

There is a general philosophy that the image of a Galois representation should be as large as possible, subject to its symmetries. This can be seen in Serre's open image theorem for non-CM elliptic curves, Ribet and Momose's work on Galois representations attached to modular forms, and recent work of the speaker and Conti-Iovita-Tilouine on Galois representations attached to $$p$$-adic families of modular forms. Recently, Bellaïche developed a way to measure the image of an arbitrary pseudorepresentations taking values in a local ring $$A$$. Under the assumptions that $$A$$ is a domain and the residual representation is not too degenerate, we explain how the symmetries of such a pseudorepresentation are reflected in its image. This is joint work with Andrea Conti and Anna Medvedovsky.

• 12:40 : Lunch break (on campus)

Let $$E$$ be an elliptic curve over the rationals and $$p$$ a prime of good reduction such that $$E$$ admits a $$p$$-isogeny over $$\mathbb{Q}$$ satisfying some assumptions. In particular the Galois representation of its $$p$$-torsion is reducible. In a joint work with C. Skinner and J. Lee, for suitable quadratic imaginary fields $$K$$, we can prove the anticyclotomic Iwasawa main conjecture for $$E/K$$. I will explain our strategy and how this, combined with complex and $$p$$-adic Gross-Zagier formulae, is used to prove the $$p$$-part of the Birch—Swinnerton-Dyer formula when $$E$$ has analytic rank one, extending the result of Jetchev-Skinner-Wan, who considered the residually irreducible case. If time permits, I will also discuss about how to use the main conjecture to prove a Gross—Zagier-Kolyvagin $$p$$-converse theorem, which is due to Skinner in the residually irreducible case and to Burungale-Skinner-Tian in the CM case.

• 4:00 : Kazim Büyükboduk

I will report on joint work with R. Pollack and S. Sasaki, where we prove a $$p$$-adic Gross-Zagier formula for critical slope $$p$$-adic $$L$$-functions. Besides the strategy for our proof, which involves interpolation of Heegner cycles in Coleman families, I will illustrate a number of applications. The first is the proof of a conjecture of Perrin-Riou, which predicts an explicit construction of a generator of the Mordell-Weil group of an elliptic curve of analytic rank one, in terms of corresponding $$p$$-adic $$L$$-values. The second is a BSD formula for elliptic curves of analytic rank one which are not necessarily semistable.

### Friday, September 13

• 9:00 : Nirvana Coppola

An important invariant associated to an elliptic curve is its $$\ell$$-adic Galois representation. In this talk I will consider elliptic curves over a local field with potentially good reduction and describe how to determine the Galois representation under the assumption that the image of inertia is non-abelian.

• 10:00 : Coffee break (C04)
• 10:30 : Henri Johnston

The equivariant Tamagawa number conjecture (ETNC) at $$s=0$$ for a finite Galois extension of number fields can be viewed as an equivariant refinement of the analytic class number formula. This is known to imply the strong Stark conjecture and the Rubin-Stark conjecture. We shall discuss a partial converse that holds under certain ramification hypotheses. This leads to an unconditional proof of the ETNC at $$s=0$$ for certain finite abelian extensions of imaginary quadratic fields. This is joint work in progress with Daniel Macias Castillo.

• 11:40 : Werner Bley

Let $$E$$ be an elliptic curve defined over a number field $$k$$ and $$F$$ a finite cyclic extension of $$k$$ of $$p$$-power degree for an odd prime $$p$$. Under certain technical hypotheses, we describe a reinterpretation of the equivariant Tamagawa Number Conjecture (`eTNC') for $$E$$, $$F/k$$ and $$p$$ as an explicit family of $$p$$-adic congruences involving values of derivatives of the Hasse-Weil $$L$$-functions of twists of $$E$$, normalised by completely explicit twisted regulators.
An important ingredient in the normalization of the equivariant regulators are certain determinants of matrices with coefficients defined in terms of the Mazur-Tate height pairing. Our reinterpretation makes the eTNC amenable to numerical verification and furthermore leads to explicit predictions which refine well-known conjectures of Mazur and Tate.
This is a report on joint work with Daniel Macias Castillo

• 12:40 : Lunch (in the city?)

## Organisation

The local organiser:

Explanations of how to reach the University of Nottingham, information about accommodation and other helpful things can be found here.