Abstracts
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Jared Weinstein (UCLA)
Non-abelian local class field theory and the Lubin-Tate tower
What are the algebraic extensions of a nonarchimedean local field (eg the p-adic numbers)? All of the abelian extensions of a nonarchimedean local field can be constructed by adjoining the torsion points of a one-dimensional formal module of height one: This is the crux of classical Lubin-Tate theory. The question of constructing the nonabelian extensions leads one to the study of the Lubin-Tate tower, which is a moduli space for deformations of formal modules of greater height. By results of Harris-Taylor and Boyer, the cohomology of the Lubin-Tate tower encodes precise information about non-abelian extensions of the local field (namely, it realizes the local Langlands correspondence). The Lubin-Tate tower has a horribly singular special fiber, which hinders any direct study of its cohomology, but we will show that after base extension there is a model for the tower whose reduction contains a very curious nonsingular hypersurface defined over a finite field. We will write down the equation for this hypersurface and then beg the audience for help in computing its zeta function.
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Ekin Ozman (Wisconsin)
Local Points on Quadratic Twists of the Classical Modular Curve
Let X^d(N) be the modular curve described as quadratic twist of the classical modular curve, X_0(N) by a quadratic field K=Q(sqrt{d})$ and Atkin-Lehner involution w_N. Rational points on this twist are K-rational points of X_0(N) that are fixed by sigma composed with w_N where sigma is the generator of Gal(K/Q). Unlike X_0(N), it's not immediate to say that there are points (global or local) on X^d(N). Given (N,d,p) we give necessary and sufficient conditions for existence of a Qp-rational point on X^d(N), answering the following question of Ellenberg:
For which d and N there exists points on X^d(N) for every completion of Q?
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Liang Xiao (Chicago)
Ramification theory for complete discrete valuation field with imperfect residue field
The classical ramification theory for local fields fails when the residue field is not perfect. Abbes and Saito introduced a new definition of a ramification filtration on the Galois group of a complete discrete valuation field that applies to the imperfect residue field case. We discuss their approach and study basic properties of the ramification filtration. If time permits, we will discuss some global applications.
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Guillermo Mantilla-Soler (Wisconsin)
Mordell-Weil ranks in towers of modular Jacobians
In this talk we describe a technique to bound the growth of Mordell-Weil ranks in towers of Jacobians of modular curves. In more detail, we will show our progress towards the following result. Let p > 2 be a prime, and let J_n be the Jacobian of the principal modular curve X(p^n). Let F be a number field with mu-invariant mu, and such that Q(J_1[p]) is contained in F. We show that there exists a constant C, depending on F and p, such that rank J_n(F) is at most (2[F:Q] + 4 mu) dim J_n + C p^{2n} for all n.
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Maria Stadnik (Northwestern)
Ray class fields of conductor (p)
Let K be Galois extension of the rational numbers, let H be the Hilbert class field of K, and let zeta denote a primitive p-th root of unity. We conjecture that under certain conditions on K, there are infinitely many p completely split in K for which the ray class field of conductor (p) = pO_K equals H(zeta + zeta^{-1}). We give motivation for why a conjectural density should exist and explain how to reformulate this question into a set of simpler questions about certain modules over Gal(K/Q). From there we can adapt methods from Hooley's proof of Artin's conjecture on primitive roots (which assumes the generalized Riemann hypothesis) to help solve the problem. We give results for multiquadratic fields assuming the GRH.
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Mirela Çiperiani (Austin)
Divisibility of Heegner points on Z_p extensions
Let E be an elliptic curve over Q, of analytic rank greater than 1, and p a prime of good ordinary reduction. Consider the
Heegner points which lie on E over the different layers of the anticyclotomic Z_p extension of an imaginary quadratic
extension K. Since the analytic rank of E/Q is greater than 1 the trace of a Heegner point down to K always equals zero. We
will discuss what this implies about the divisibility of the Heegner points by elements of the relevant Iwasawa algebra.
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