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Series: Algebra Seminar

Let X be a curve over a p-adic field K with semi-stable reduction and let $\omega$ be a meromorphic differential on X. There are two p-adic integrals one may associated to this data. One is the Vologodsky (abelian, Zarhin, Colmez) integral, which is a global function on the K-points of X defined up to a constant. The other is the collection of Coleman integrals on the subdomains reducing to the various components of the smooth locus. In this talk I will prove the following Theorem, joint with Sarah Zerbes: The Vologodsky integral is given on each subdomain by a Coleman integrals, and these integrals are related by the condition that their differences on the connecting annuli form a harmonic 1-cocyle on the edges of the dual graph of the special fiber.I will further explain the implications to the behavior of the Vologodsky integral on the connecting annuli, which has been observed independently and used, by Stoll, Katz-Rabinoff-Zureick-Brown, in works on global bounds on the number of rational points on curves, and an interesting product on 1-forms used in the proof of the Theorem as well as in work on p-adic height pairings. Time permitting I will explain the motivation for this result, which is relevant for the interesting question of generalizing the result to iterated integrals.

Series: Algebra Seminar

Real-valued smooth differential forms on Berkovich analytic spaces were introduced by Chambert-Loir and Ducros. They show many fundamental properties analogous to smooth real differential forms on complex manifolds, which are used for example in Arakelov geometry. In particular, these forms define a real valued bigraded cohomology theory for Berkovich analytic space, called tropical Dolbeault cohomology. I will explain the definition and properties of these forms and their link to tropical geometry. I will then talk about results regarding the tropical Dolbeault cohomology of varietes and in particular curves. In particular, I will look at finite dimensionality and Poincar\'e duality.

Series: Algebra Seminar

Given a Galois cover of curves X to Y with Galois group G which is totally ramified at a point x and unramified elsewhere,
restriction to the punctured formal neighborhood of x induces a Galois extension of Laurent series rings k((u))/k((t)). If
we fix a base curve Y , we can ask when a Galois extension of Laurent series rings comes from a global cover of Y in this
way. Harbater proved that over a separably closed field, this local-to-global principle holds for any base curve if G is a
p-group, and gave a condition for the uniqueness of such an extension. Using a generalization of Artin-Schreier theory to
non-abelian p-groups, we characterize the curves Y for which this lifting property holds and when it is unique, but over
a more general ground field.

Series: Algebra Seminar

In this talk we will discuss the following question: When does there exist a curve of degree d and genus g passing through
n general points in P^r? We will focus primarily on what is known in the case of space curves (r=3).

Series: Algebra Seminar

We determine the average size of the $\phi$-Selmer group in any quadratic twist family of abelian varieties having an isogeny $\phi$ of degree 3 over any number field. This has several applications towards the rank statistics in such families of quadratic twists. For example, it yields the first known quadratic twist families of absolutely simple abelian varieties over $\mathbb{Q}$, of dimension greater than one, for which the average rank is bounded; in fact, we obtain such twist families in arbitrarily large dimension. In the case that $E/F$ is an elliptic curve admitting a 3-isogeny, we prove that the average rank of its quadratic twists is bounded; if $F$ is totally real, we moreover show that a positive proportion of these twists have rank 0 and a positive proportion have $3$-Selmer rank 1. We also obtain consequences for Tate-Shafarevich groups of quadratic twists of a given elliptic curve. This is joint work with Manjul Bhargava, Zev Klagsbrun, and Ari Shnidman.

Series: Algebra Seminar

In this talk, I will summarize forthcoming work with Griffin, Ono, and Zagier. In 1927 Pólya proved that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for Riemann's Xi-function. This hyperbolicity has been proved for degrees $d\leq 3$. We obtain an arbitrary precision asymptotic formula for the derivatives $\Xi^{(2n)}(0)$, which allows us to prove thehyperbolicity of 100% of the Jensen polynomials of each degree. We obtain a general theorem which models such polynomials by Hermite polynomials. This general condition also confirms a conjecture of Chen, Jia, and Wang.

Series: Algebra Seminar

A classical theorem in modern homotopy theory states that functors from finite pointed sets to spaces satisfying certain conditions model infinite loop spaces (Segal 1974). This theorem offers a recognition principle for infinite loop spaces. An analogous theorem for Morel-Voevodsky's motivic homotopy theory has been sought for since its inception. In joint work with Marc Hoyois, Adeel Khan, Vladimir Sosnilo and Maria Yakerson, we provide such a theorem. The category of finite pointed sets is replaced by a category where the objects are smooth schemes and the maps are spans whose "left legs" are finite syntomic maps equipped with a K-theoretic trivialization of its contangent complex. I will explain what this means, how it is not so different from finite pointed sets and why it was a natural guess. In particular, I will explain some of the requisite algebraic geometry.Time permitting, I will also provide 1) an explicit model for the motivic sphere spectrum as a torsor over a Hilbert scheme and,2) a model for all motivic Eilenberg-Maclane spaces as simplicial ind-smooth schemes.

Series: Algebra Seminar

postponed from September 18

In this talk I first wish to review my work with Balakrishnan and Muller, giving an algorithm for finding integral points on curves under certain (strong) assumptions. The main ingredients are the theory of p-adic height pairings and the theory of p-adic metrized line bundles. I will then explain a new proof of the main result using a p-adic version of Zhang's adelic metrics, and a third proof which only uses the metric at one prime p. At the same time I will attempt to explain why I think this last proof is interesting, being an indication that there may be new p-adic methods for finding integral points.

Series: Algebra Seminar

Chai and Oort have asked
the following question: For any algebraically closed field $k$, and for
$g \geq 4$, does there exist an abelian variety over $k$ of dimension
$g$ not isogenous to a Jacobian? The answer in characteristic 0 is now
known to be yes.
We present a heuristic which suggests that for certain $g \geq 4$, the
answer in characteristic $p$ is no. We will also construct a proper
subvariety of $X(1)^n$ which intersects every isogeny class, thereby
answering a related question, also asked by Chai
and Oort. This is joint work with Jacob Tsimerman.

Series: Algebra Seminar

I will discuss the interplay between tangent lines of algebraic and tropical curves. By tropicalizing all the tangent lines
of a plane curve, we obtain the tropical dual curve, and a recipe
for computing the Newton polygon of the dual projective curve.
In the case of canonical curves, tangent lines are closely related
with various phenomena in algebraic geometry such as double covers, theta characteristics and Prym varieties. When degenerating
them in families, we discover analogous constructions in tropical
geometry, and links between quadratic forms, covers of graphs and
tropical bitangents.