# Abstracts

**Alex Bartel** (Warwick)

*Elliptic curves whose Selmer groups grow in all "small" extensions*

For most elliptic curves over Q it is known (and for all others it is conjectured) that there exists a quadratic extension of Q over which the size of the 2-Selmer group of the curve is the same as over Q. I will show that this is no longer true if "quadratic" is replaced by "bi-quadratic". Indeed, it turns out that there are lots of elliptic curves over Q whose 2-Selmer group grows in size in all bi-quadratic extensions, and, for all odd primes p, whose p-Selmer group grows in size in all dihedral extensions of degree 2p and all elementary abelian extensions of degree p^2. This is not a parity phenomenon, as I shall also demonstrate.

**Manjul Bhargava **(Princeton)

*Explicit 2-descent and the average size of the 2-Selmer group of Jacobians of odd hyperelliptic curves*

We show how to construct explicit models for elements in the 2-Selmer groups of Jacobians of odd hyperelliptic curves, through a study of the rational (and integral) orbits of a certain natural representation of the odd split orthogonal group. As a consequence of this study, we show that the average size of the 2-Selmer group of the Jacobians of odd hyperelliptic curves over Q (of any given genus) is 3. This implies that the average rank of the Jacobians of such hyperelliptic curves is bounded (by 3/2). Via Chabauty methods, the result also then implies a uniform bound on the number of rational points on the majority of these curves. This is joint work with Dick Gross.

**Werner Bley **(LMU München)

*Equivariant BSD conjecture in cyclic p-power extensions*

We report on a joint project with D.M.Castillo. Let $E$ be an elliptic curve defined over $\Q$ and let $K / \Q$ be a finite cyclic extension of $p$-power order with group $G$, $p$ an odd prime. Assuming a variety of conditions, we describe a very explicit reformulation of the relevant case of the ETNC. This is essentially based on the fact, that under our hypothesis the $p$-completion of the Mordell-Weil group $E(K)$ is a $\Z_p[G]$-permutation module, and the explicit computation of the equivariant regulator. If time permits, we present some numerical examples.

**Peter Bruin **(Zurich)

*Optimal bounds for the difference between the Néron-Tate height and the Weil height on elliptic curves over \Qbar*

It is known that for any elliptic curve $E$ over $\Qbar$ in Weierstraß form, the difference $h-\hat h$ between the Weil height and the

Néron–Tate height is a bounded function on $E(\Qbar)$. I will describe an algorithm that, given $E$ as above and a real number $\epsilon>0$, computes the supremum and the infimum of $h-\hat h$ on $E(\Qbar)$ with accuracy $\epsilon$.

**Nils Bruin **(Simon Fraser)

*Classgroups and congruent primes*

For a given prime p we consider two classical properties. The first is the class number of the imaginary quadratic field Q(sqrt(-p)), the second whether p can be the area of a right angled triangle with all sides rational.

For either statement one can partially answer these questions based on congruence conditions on p. We push this classification one step further and find, quite surprisingly, that the newly obtained criteria are not completely aligned anymore between the two properties.

In the process we see quite explicitly the way in which Tate-Shafarevich groups and class groups are analogous objects.

**Brian Conrey** (American Institute of Mathematics and Bristol)

*Rank two in the family of quadratic twists of an elliptic curve*

We combine techniques from Random Matrix theory and analytic number theory to give conjectural statistics about rank 2 elliptic curves in a family of twists.

**Vladimir Dokchitser **(Cambridge)

*Growth of Sha in towers of number fields*

**Victor Flynn** (Oxford)

*Descent via (3,3)-isogeny on Jacobians of genus 2 curves*

We give a parametrisation of curves of genus 2 for which the Mordell-Weil group of the Jacobian contains two independent

points of order 3, and develop the theory required to perform descent via (3,3)-isogeny. We apply this to several examples, where it can shown that non-reducible Jacobians have nontrivial 3-part of the Tate-Shafarevich group.

**Wojciech Gajda **(Poznan)

*Abelian varieties over function fields and independence of l-adic representations*

I will discuss some recent calculations of monodromies for abelian varieties defined over finitely generated fields of arbitrary

characteristic. In the second part of my talk I am going to report on the independence of \ell-adic Galois representations (in the sense of

Serre) attached to abelian varieties and (more generally) to etale cohomology of separated schemes over function fields. This is a joint work with S.Arias-de-Reyna, G.Boeckle and S.Petersen.

**Wei Ho **(Columbia)

*Average sizes of Selmer groups in families of elliptic curve
*

We discuss results on determining the average size of Selmer groups-and thereby bounding the average Mordell-Weil rank-in certain

natural families of elliptic curves over the rational numbers. The main ideas involve finding explicit descriptions of the relevant moduli spaces, typically as orbits of representations, and then counting integral points on them using the geometry of numbers. This is joint work with Manjul Bhargava.

**Zev Klagsbrun** (Wisconsin)

New results concerning the distribution of 2-Selmer ranks within the quadratic twist family of an elliptic curve

Given an elliptic curve E defined over a number field K, we can ask what proportion of quadratic twists of E have 2-Selmer rank r for any non-negative integer r. I will present new results obtained by Mazur, Rubin, and myself about this distribution, including some surprising results relating to parity that have implications regarding Goldfeld's conjecture over number fields as well as some of my own results in the special case when E(Q)[2] = Z/2 that conflict with the conjectured distribution arising from the Delaunay heuristics on the Tate-Shafaravich group.

**Filip Najman** (Zagreb)

Ranks* of elliptic curves with prescribed torsion over number fields*

We investigate how the rank of elliptic curves depends on the torsion over number fields and show that, unlike over Q, the torsion subgroup of an elliptic curve can in many instances give some information on the rank. We introduce a phenomenon that we call *false complex multiplication*, and show that it explains why any elliptic curve over any quadratic field with a point of order 13 or 18 and any elliptic curve over any quartic field with a point of order 22 has even rank.

This is joint work with J. Bosman, P. Bruin and A. Dujella.

**Karl Rubin **(UC Irvine)

*Higher rank Kolyvagin systems*

A rank-one Euler system and a rank-one Kolyvagin system both consist of families of cohomology classes with appropriate properties and interrelationships. Given a rank-one Euler system, Kolyvagin's derivative construction produces a rank-one Kolyvagin system, and a rank-one Kolyvagin system gives a bound on the size of a Selmer group. In some situations an Euler system is more naturally a collection of elements in the $r$-th exterior powers of cohomology groups. In this situation, Barry Mazur and I define a rank-$r$ Kolyvagin system, and we show how a rank-$r$ Kolyvagin system bounds the size of the corresponding Selmer group.

**Alice Silverberg **(UC Irvine)

*Explicit points on a family of Jacobians of superelliptic curves over global function fields*

In a project initiated at an AIM workshop, Lisa Berger, Chris Hall, René Pannekoek, Jennifer Park, Rachel Pries, Shahed Sharif, Alice Silverberg, and Douglas Ulmer are studying arithmetic questions about the Jacobian variety of the curve $y^r=x^{r-1}(x+1)(x+t)$ over fields of the form $K_d= F_p(\mu_d,t^{1/d})$ where $d=1+p^f$ and $r$ divides $d$. In particular, we find explicit rational points and "high" rank, generalizing earlier work of Ulmer on the Legendre elliptic curve. This talk will give a report on our progress.

**Michael Stoll** (Bayreuth)

*Many curves with few rational points*

We combine recent results by Bhargava and Gross on the average size of 2-Selmer groups of Jacobians of odd degree hyperelliptic curves with Chabauty's method to show that "many" odd degree hyperelliptic curves of given genus only have "few" rational points, in a precise sense.

**Peter Swinnerton-Dyer **(Cambridge)

*Diagonal hypersurfaces and the Bloch-Kato conjecture*

**Damiano Testa **(Warwick)

*The Büchi K3 surface and its rational points*

In order to extend Matiyasevich's resolution of Hilbert's Tenth Problem, Büchi introduced a sequence of affine algebraic surfaces: he showed that if the surfaces in this sequence eventually only have trivial integral solutions, then the proof of undecidability can be extended to the case of systems of diagonal quadratic equations. Vojta later showed that a weak form of the Lang's Conjectures implies that, with finitely many exceptions, the "Büchi surfaces" do indeed only have trivial integral solutions.

In my talk I will report on joint work in progress with M. Artebani and A. Laface on the rational (not necessarily integral!) points of the first non-rational surface in Büchi's sequence. I will mention some of the geometric properties of this surface and show that it is a moduli space of vector bundles. The modular interpretation of this problem naturally leads to a question on integral structures on a moduli spaces of vector bundles, to which we do not know the answer.

**Mark Watkins **(University of Sydney)

*Large ranks of quadratic twists of the congruent number curve*

Rogers has found many high rank curves $dy^2=x^3-x$, the largest being of rank 7 with $d=797507543735$. His computations were later partly extended by Dujella, Janfada, and Salami. Today about 15 rank 7 twists are known, but no rank 8 twist. More than 10000 times as much processor time as with the first rank 7 twist has now been thrown at the problem, but if $2^{49}$ and $2^{64}$ are suspected ranges, this might not be unexpected!

We review the above situation with rank 7 and (lack of) rank 8 twists, describing some of the computational methods used. We also describe a fortuitous heuristic of Granville on the matter, and hope to say something about the growth of the number of rank 6 twists in the range of computation.

**Christian Wuthrich** (Nottingham)

*On the class group pairing on elliptic curves*

There is a pairing on the Mordell-Weil group of an elliptic curve over a number field with values in the class group of the field. It sits

somewhere between the monodromy pairing and the Neron-Tate height pairing. I would like to discuss how this pairing shows up in

questions on Galois module structures, how one can compute it effectively and how it links to p-descent.

**Shun'ichi Yokoyama** (Kyushu)

*A database project of elliptic curves having everywhere good reduction*

Determination problem of elliptic curves having everywhere good reduction over number fields is very fascinating from the viewpoint of “modularity conjecture” and related problems/applications that lead important relationships on algebraic and analytic number theory. In this talk, we report our database project of such curves with lower degree, especially over real quadratic fields and pure cubic fields.