Difference between revisions of "NTS ABSTRACT"

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Return to [https://www.math.wisc.edu/wiki/index.php/NTS NTS Fall 2015]
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Return to [https://www.math.wisc.edu/wiki/index.php/NTS NTS Spring 2017]
  
== Sep 03 ==
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== Sept 7 ==
  
 
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Kiran Kedlaya'''
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''David Zureick-Brown'''
 
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| bgcolor="#BCD2EE"  align="center" | ''On the algebraicity of (generalized) power series''
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| bgcolor="#BCD2EE"  align="center" | Progress on Mazur’s program B
 
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| bgcolor="#BCD2EE"  | I’ll discuss recent progress on Mazur’s ”Program B”, including my own recent work with Jeremy Rouse which completely classifies the possibilities for the 2-adic image of Galois associated to an elliptic curve over the rationals. I will also discuss a large number of other very recent results by many authors.
A remarkable theorem of Christol from 1979 gives a criterion for detecting whether a power series over a finite field of characteristic p represents an algebraic function: this happens if and only if the coefficient of the n-th power of the series variable can be extracted
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from the base-p expansion of n using a finite automaton. We will describe a result that extends this result in two directions: we allow
 
an arbitrary field of characteristic p, and we allow "generalized power series" in the sense of Hahn-Mal'cev-Neumann. In particular, this gives
 
a concrete description of an algebraic closure of a rational function field in characteristic p (and corrects a mistake in my previous attempt
 
to give this description some 15 years ago).
 
 
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== Sep 10 ==
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== Sept 14 ==
  
 
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Sean Rostami'''
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Solly Parenti'''
 
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| bgcolor="#BCD2EE"  align="center" | ''Fixers of Stable Functionals''
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| bgcolor="#BCD2EE"  align="center" | Unitary CM Fields and the Colmez Conjecture
 
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| bgcolor="#BCD2EE"  | Pierre Colmez conjectured a formula for the Faltings height of a CM abelian variety in terms of log derivatives of Artin L-functions arising from the CM type. We will study the relevant class functions in the case where our CM field contains an imaginary quadratic field and use this to extend the known cases of the conjecture.
The epipelagic representations of Reeder-Yu, a generalization of the "simple supercuspidals" of Gross-Reeder, are certain low-depth supercuspidal representations of reductive algebraic groups G. Given a "stable functional" f, which is a suitably 'generic' linear functional on a vector space coming from a Moy-Prasad filtration for G, one can create such a representation. It is known that the representations created in this way are compactly induced from the fixer in G of f and it is important to identify explicitly all the elements that belong to this fixer. This work is in-progress.
 
 
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== Sep 17 ==
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== Sept 21 ==
  
 
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''David Zureick-Brown'''
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Chao Li '''
 
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| bgcolor="#BCD2EE"  align="center" | ''Tropical geometry and uniformity of rational points''
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| bgcolor="#BCD2EE"  align="center" | Goldfeld's conjecture and congruences between Heegner points
 
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| bgcolor="#BCD2EE"  | Given an elliptic curve E over Q, a celebrated conjecture of Goldfeld asserts that a positive proportion of its quadratic twists should have analytic rank 0 (resp. 1). We show this conjecture holds whenever E has a rational 3-isogeny. We also prove the analogous result for the sextic twists of j-invariant 0 curves. For a more general elliptic curve E, we show that the number of quadratic twists of E up to twisting discriminant X of analytic rank 0 (resp. 1) is  >> X/log^{5/6}X, improving the current best general bound towards Goldfeld's conjecture due to Ono--Skinner (resp. Perelli--Pomykala). We prove these results by establishing a congruence formula between p-adic logarithms of Heegner points based on Coleman's integration. This is joint work with Daniel Kriz.
Let X be a curve of genus g over a number field F of degree d = [F:Q]. The conjectural existence of a uniform bound N(g,d) on the number #X(F) of F-rational points of X is an outstanding open problem in arithmetic geometry, known to follow from the Bomberi-Lang conjecture. We prove a special case of this conjecture - we give an explicit uniform bound when X has Mordell-Weil rank r &le; g-3. This generalizes recent work of Stoll on uniform bounds on hyperelliptic curves. Using the same techniques, we give an explicit, unconditional uniform bound on the number of F-rational torsion points of J lying on the image of X under an Abel-Jacobi map. We also give an explicit uniform bound on the number of geometric torsion points of J lying on X when the reduction type of X is highly degenerate. Our methods combine Chabauty-Coleman's p-adic integration, non-Archimedean potential theory on Berkovich curves, and the theory of linear systems and divisors on metric graphs. This is joint work with Joe Rabinoff and Eric Katz.
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== Sept 28 ==
  
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|-
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Daniel Hast '''
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| bgcolor="#BCD2EE"  align="center" | Rational points on solvable curves over Q via non-abelian Chabauty
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| bgcolor="#BCD2EE"  | By Faltings' theorem, any curve over Q of genus at least two has only finitely many rational points—but the bounds coming from known proofs of Faltings' theorem are often far from optimal. Chabauty's method gives much sharper bounds for curves whose Jacobian has low rank, and can even be refined to give uniform bounds on the number of rational points. This talk is concerned with Minhyong Kim's non-abelian analogue of Chabauty's method, which uses the unipotent fundamental group of the curve to remove the restriction on the rank. Kim's method relies on a "dimension hypothesis" that has only been proven unconditionally for certain classes of curves; I will give an overview of this method and discuss my recent work with Jordan Ellenberg where we prove this dimension hypothesis for any Galois cover of the projective line with solvable Galois group (which includes, for example, any hyperelliptic curve).
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== Sep 22 ==
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== Oct 12 ==
  
 
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Joseph Gunther'''
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Matija Kazalicki '''
 
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| bgcolor="#BCD2EE"  align="center" | ''Embedding Curves in Surfaces and Stabilization of Hypersurface Singularity Counts''
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| bgcolor="#BCD2EE"  align="center" | Supersingular zeros of divisor polynomials of elliptic curves of prime conductor and Watkins' conjecture
 
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| bgcolor="#BCD2EE"  | For a prime number p, we study the mod p zeros of divisor polynomials of elliptic curves E/Q of conductor p. Ono made the observation that these zeros of are often j-invariants of supersingular elliptic curves over F_p. We relate these supersingular zeros to the zeros of the quaternionic modular form associated to E, and using the later partially explain Ono's findingsWe notice the curious connection between the number of zeros and the rank of elliptic curve.
We'll present two new applications of Poonen's closed point sieve over finite fieldsThe first is that the obvious local obstruction to embedding a curve in a smooth surface is the only global obstruction.  The second is a proof of a recent conjecture of Vakil and Wood on the asymptotic probability of hypersurface sections having a prescribed number of singularities.
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In the second part of the talk, we briefly explain how a special case of Watkins' conjecture on the parity of modular degrees of elliptic curves follows from the methods previously introduced.  This is a joint work with Daniel Kohen.
  
''Note: The day of this seminar is '''Tuesday'''.''
 
 
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== Oct 19 ==
 
 
== Sep 24 ==
 
  
 
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Brandon Alberts'''
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Andrew Bridy'''
 
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| bgcolor="#BCD2EE"  align="center" | ''The Moments Version of Cohen-Lenstra Heuristics for Nonabelian Groups''
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| bgcolor="#BCD2EE"  align="center" | Arboreal finite index for cubic polynomials
 
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| bgcolor="#BCD2EE"  | Let K be a global field of characteristic 0. Let f \in K[x] and b \in K, and set K_n = K(f^{-n}(b)). The projective limit of the groups Gal(K_n/K) embeds into the automorphism group of an infinite rooted tree. A major problem in arithmetic dynamics is to find conditions that guarantee the index is finite; a complete answer would give a dynamical analogue of Serre's celebrated open image theorem. I solve the finite index problem for cubic polynomials over function fields by proving a complete list of necessary and sufficient conditions. For number fields, the proof of sufficiency is conditional on both the abc conjecture and a form of Vojta's conjecture. This is joint work with Tom Tucker.
Cohen-Lenstra heuristics posit the distribution of unramified abelian extensions of quadratic fields. A natural question to ask would be how to get an analogous heuristic for nonabelian groups. In this talk I take and extend on recent work in the area of unramified extensions of imaginary quadratic fields and bring it all together under one Cohen-Lenstra style heuristic.
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== Oct 19 ==
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== Oct 08 ==
 
  
 
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Ana Caraiani'''
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | ''Jiuya Wang''
 
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| bgcolor="#BCD2EE"  align="center" | ''On vanishing of torsion in the cohomology of Shimura varieties''
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| bgcolor="#BCD2EE"  align="center" | Malle's conjecture for compositum of number fields
 
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| bgcolor="#BCD2EE"  | Abstract: Malle's conjecture is a conjecture on the asymptotic distribution of number fields with bounded discriminant. We propose a general framework to prove Malle's conjecture for compositum of number fields based on known examples of Malle's conjecture and good uniformity estimates. By this method, we prove Malle's conjecture for $S_n\times A$ number fields for $n = 3,4,5$ and $A$ in an infinite family of abelian groups. As a corollary, we show that Malle's conjecture is true for $C_3\wr C_2$ in its $S_9$ representation, whereas its $S_6$ representation is the first counter example of Malle's conjecture given by Kl\"uners. By a sieve method, we further prove the secondary term for $S_3\times A$ extensions for infinitely many odd abelian groups $A$ over $\mathbb{Q}$.
I will discuss joint work in progress with Peter Scholze showing that torsion in the cohomology of certain compact unitary Shimura varieties occurs in the middle degree, under a genericity assumption on the corresponding Galois representation.
 
 
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== Nov 2 ==
 
 
== Oct 15 ==
 
  
 
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Valentin Blomer'''
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '' Carl Wang-Erickson''
 
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| bgcolor="#BCD2EE"  align="center" | ''Arithmetic, geometry and analysis of a senary cubic form''
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| bgcolor="#BCD2EE"  align="center" | The rank of the Eisenstein ideal
 
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| bgcolor="#BCD2EE"  | Abstract: In his landmark 1976 paper "Modular curves and the Eisenstein ideal", Mazur studied congruences modulo p between cusp forms and an Eisenstein series of weight 2 and prime level N. We use deformation theory of pseudorepresentations to study the corresponding Hecke algebra. We will discuss how this method can be used to refine Mazur's results, quantifying the number of Eisenstein congruences. Time permitting, we'll also discuss some partial results in the composite-level case. This is joint work with Preston Wake.  
We establish an asymptotic formula (with power saving error term) for the number of rational points of bounded height for a certain cubic fourfold, thereby proving a strong form of Manin's conjecture for this algebraic variety by techniques of analytic number theory.
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== Oct 22 ==
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== Nov 9 ==
  
 
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Brian Cook'''
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | ''Masahiro Nakahara''
 
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| bgcolor="#BCD2EE"  align="center" | ''Configurations in dense subsets of Euclidean spaces''
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| bgcolor="#BCD2EE"  align="center" | Index of fibrations and Brauer-Manin obstruction
 
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| bgcolor="#BCD2EE"  | Abstract: Let X be a smooth projective variety with a fibration into varieties that either satisfy a condition on representability of zero-cycles or that are torsors under an abelian variety. We study the classes in the Brauer group that never obstruct the Hasse principle for X. We prove that if the generic fiber has a zero-cycle of degree d over the generic point, then the Brauer classes whose orders are prime to d do not play a role in the Brauer--Manin obstruction. As a result we show that the odd torsion Brauer classes never obstruct the Hasse principle for del Pezzo surfaces of degree 2, certain K3 surfaces, and Kummer varieties.
A result of Katznelson and Weiss states that given a suitably dense (measurable) subset of the Euclidean plane realizes every sufficiently large distance, that is, for every prescribed (sufficiently large) real number the set contains two elements whose distance is this number. The analogue of this statement for finding three equally spaced points on a line, i.e. for finding three term arithmetic progressions, in a given set is false, and in fact false in every dimension. In this talk we revisit the case of three term progressions when the standard Euclidean metric is replaced by other metrics.
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== Oct 29 ==
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== Nov 16 ==
  
 
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Aaron Levin'''
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | ''Joseph Gunther''
 
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| bgcolor="#BCD2EE"  align="center" | ''Integral points and orbits in the projective plane''
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| bgcolor="#BCD2EE"  align="center" | Irrational points on random hyperelliptic curves
 
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| bgcolor="#BCD2EE"  | Abstract:Let d and g be positive integers with 1 < d < g.  If d is odd, we prove there exists B_d such that a positive proportion of odd genus g hyperelliptic curves over Q have at most B_d points of degree dIf d is even, we similarly bound the degree d points not lazily pulled back from degree d/2 points of the projective line.  The proofs use tropical geometry work of Park, as well as results of Bhargava and Gross on average ranks of hyperelliptic Jacobians. This is joint work with Jackson Morrow.
We will discuss the problem of classifying the behavior of integral points on affine subsets of the projective planeAs an application, we will examine the problem of classifying endomorphisms of the projective plane with an orbit containing a Zariski dense set of integral points (with respect to some plane curve). This is joint work with Yu Yasufuku.
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Time willing, we'll discuss rich, delicious interactions with work of next week's speaker.
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== Nov 12 ==
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== Nov 30 ==
  
 
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Vlad Matei'''
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | ''Reed Gordon-Sarney''
 
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| bgcolor="#BCD2EE"  align="center" | ''A geometric perspective on Landau's theorem over function fields''
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| bgcolor="#BCD2EE"  align="center" |Zero-Cycles on Torsors under Linear Algebraic Groups
 
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| bgcolor="#BCD2EE"  | Abstract:In this talk, the speaker will discuss his thesis on the following question of Totaro from 2004: if a torsor under a connected linear algebraic group has index d, does it have a close etale point of degree d? The d = 1 case is an open question of Serre from the `60s. The d > 1 case, surprisingly, has a negative answer.
We revisit the recent [http://arxiv.org/abs/1504.06809 result] of Lior-Bary-Soroker. It deals with a function field analogue of Landau's classical result about the asymptotic density of numbers which are sums of two integer squares. The results obtained are just in the large characteristic and large degree regime. We obtain a characterization as q<sup>n</sup> goes to infinity, which is the desired analogue of the result over the integers. We take a geometric perspective in computing the number of polynomials of degree n which are split in the extension <b>F</b><sub>q</sub>[T<sup>1/2</sup>] / <b>F</b><sub>q</sub>[T] and we obtain a geometric explanation for the "mysterious" binomial coefficients appearing in his asymptotic.
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== Dec 17 ==
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== Dec 7 ==
  
 
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Tonghai Yang'''
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | ''Rafe Jones''
 
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| bgcolor="#BCD2EE"  align="center" | ''A generating function of arithmetic divisors in a unitary Shimura variety: modularity and application''
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| bgcolor="#BCD2EE"  align="center" |How do you (easily) find the genus of a plane curve?
 
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| bgcolor="#BCD2EE"  | Abstract: If you’ve ever wanted to show a plane curve has only finitely many rational points, you’ve probably wished you could invoke Faltings’ theorem, which requires the genus of the curve to be at least two. At that point, you probably asked yourself the question in the title of this talk. While the genus is computable for any given irreducible curve, it depends in a delicate way on the singular points. I’ll talk about a much nicer formula that applies to irreducible “variables separated” curves, that is, those given by A(x) = B(y) where A and B are rational functions.  
In the original Gross-Zagier formula and Zhang's extension to Shimura curves, the modularity of the generating function [[File:Tonghai1.jpg]]
 
  
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Then I’ll discuss how to use this to resolve the question that motivated me originally: given an integer m > 1 and a rational function f defined over a number field K, does f possess a K-orbit containing infinitely many mth powers of elements of K?  The answer turns out to be no unless f has a very special form: for m > 4 the map f must essentially be the mth power of some rational function, while for smaller m other exceptions arise, including maps closely related to multiplication on elliptic curves. If time permits I’ll discuss a connection to an arithmetic dynamical analogue of the Mordell-Lang conjecture.
  
  
is a very important step, where Z(m) are the Heeger divisors and [\xi] is the rational canonical divisor of degree 1 (associated to Hodge bundle).  In the proof, they actually use arithmetic version in calculation: [\hat{\xi}] + \sum \hat{Z}(m) q^m \in \widehat{CH}^1(X) \otimes \C which is also modular. In this talk, we define  a generalization of this arithmetic generating function to unitary Shimura variety of type (n, 1) and prove that it is modular.  It has application to Colmez conjecture and Gross-Zagier type formula.
 
 
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== Dec 14 ==
 
 
== Dec 17 ==
 
  
 
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Nathan Kaplan'''
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | ''Robert J. Lemke Oliver''
 
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| bgcolor="#BCD2EE"  align="center" | Coming soon...
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| bgcolor="#BCD2EE"  align="center" |Selmer groups, Tate-Shafarevich groups, and ranks of abelian varieties in quadratic twist families
 
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| bgcolor="#BCD2EE"  | Abstract: 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 towardsthe 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.
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Latest revision as of 00:44, 4 December 2017

Return to NTS Spring 2017


Sept 7

David Zureick-Brown
Progress on Mazur’s program B
I’ll discuss recent progress on Mazur’s ”Program B”, including my own recent work with Jeremy Rouse which completely classifies the possibilities for the 2-adic image of Galois associated to an elliptic curve over the rationals. I will also discuss a large number of other very recent results by many authors.



Sept 14

Solly Parenti
Unitary CM Fields and the Colmez Conjecture
Pierre Colmez conjectured a formula for the Faltings height of a CM abelian variety in terms of log derivatives of Artin L-functions arising from the CM type. We will study the relevant class functions in the case where our CM field contains an imaginary quadratic field and use this to extend the known cases of the conjecture.


Sept 21

Chao Li
Goldfeld's conjecture and congruences between Heegner points
Given an elliptic curve E over Q, a celebrated conjecture of Goldfeld asserts that a positive proportion of its quadratic twists should have analytic rank 0 (resp. 1). We show this conjecture holds whenever E has a rational 3-isogeny. We also prove the analogous result for the sextic twists of j-invariant 0 curves. For a more general elliptic curve E, we show that the number of quadratic twists of E up to twisting discriminant X of analytic rank 0 (resp. 1) is >> X/log^{5/6}X, improving the current best general bound towards Goldfeld's conjecture due to Ono--Skinner (resp. Perelli--Pomykala). We prove these results by establishing a congruence formula between p-adic logarithms of Heegner points based on Coleman's integration. This is joint work with Daniel Kriz.

Sept 28

Daniel Hast
Rational points on solvable curves over Q via non-abelian Chabauty
By Faltings' theorem, any curve over Q of genus at least two has only finitely many rational points—but the bounds coming from known proofs of Faltings' theorem are often far from optimal. Chabauty's method gives much sharper bounds for curves whose Jacobian has low rank, and can even be refined to give uniform bounds on the number of rational points. This talk is concerned with Minhyong Kim's non-abelian analogue of Chabauty's method, which uses the unipotent fundamental group of the curve to remove the restriction on the rank. Kim's method relies on a "dimension hypothesis" that has only been proven unconditionally for certain classes of curves; I will give an overview of this method and discuss my recent work with Jordan Ellenberg where we prove this dimension hypothesis for any Galois cover of the projective line with solvable Galois group (which includes, for example, any hyperelliptic curve).

Oct 12

Matija Kazalicki
Supersingular zeros of divisor polynomials of elliptic curves of prime conductor and Watkins' conjecture
For a prime number p, we study the mod p zeros of divisor polynomials of elliptic curves E/Q of conductor p. Ono made the observation that these zeros of are often j-invariants of supersingular elliptic curves over F_p. We relate these supersingular zeros to the zeros of the quaternionic modular form associated to E, and using the later partially explain Ono's findings. We notice the curious connection between the number of zeros and the rank of elliptic curve.

In the second part of the talk, we briefly explain how a special case of Watkins' conjecture on the parity of modular degrees of elliptic curves follows from the methods previously introduced. This is a joint work with Daniel Kohen.

Oct 19

Andrew Bridy
Arboreal finite index for cubic polynomials
Let K be a global field of characteristic 0. Let f \in K[x] and b \in K, and set K_n = K(f^{-n}(b)). The projective limit of the groups Gal(K_n/K) embeds into the automorphism group of an infinite rooted tree. A major problem in arithmetic dynamics is to find conditions that guarantee the index is finite; a complete answer would give a dynamical analogue of Serre's celebrated open image theorem. I solve the finite index problem for cubic polynomials over function fields by proving a complete list of necessary and sufficient conditions. For number fields, the proof of sufficiency is conditional on both the abc conjecture and a form of Vojta's conjecture. This is joint work with Tom Tucker.

Oct 19

Jiuya Wang
Malle's conjecture for compositum of number fields
Abstract: Malle's conjecture is a conjecture on the asymptotic distribution of number fields with bounded discriminant. We propose a general framework to prove Malle's conjecture for compositum of number fields based on known examples of Malle's conjecture and good uniformity estimates. By this method, we prove Malle's conjecture for $S_n\times A$ number fields for $n = 3,4,5$ and $A$ in an infinite family of abelian groups. As a corollary, we show that Malle's conjecture is true for $C_3\wr C_2$ in its $S_9$ representation, whereas its $S_6$ representation is the first counter example of Malle's conjecture given by Kl?\"uners. By a sieve method, we further prove the secondary term for $S_3\times A$ extensions for infinitely many odd abelian groups $A$ over $\mathbb{Q}$.

Nov 2

Carl Wang-Erickson
The rank of the Eisenstein ideal
Abstract: In his landmark 1976 paper "Modular curves and the Eisenstein ideal", Mazur studied congruences modulo p between cusp forms and an Eisenstein series of weight 2 and prime level N. We use deformation theory of pseudorepresentations to study the corresponding Hecke algebra. We will discuss how this method can be used to refine Mazur's results, quantifying the number of Eisenstein congruences. Time permitting, we'll also discuss some partial results in the composite-level case. This is joint work with Preston Wake.





Nov 9

Masahiro Nakahara
Index of fibrations and Brauer-Manin obstruction
Abstract: Let X be a smooth projective variety with a fibration into varieties that either satisfy a condition on representability of zero-cycles or that are torsors under an abelian variety. We study the classes in the Brauer group that never obstruct the Hasse principle for X. We prove that if the generic fiber has a zero-cycle of degree d over the generic point, then the Brauer classes whose orders are prime to d do not play a role in the Brauer--Manin obstruction. As a result we show that the odd torsion Brauer classes never obstruct the Hasse principle for del Pezzo surfaces of degree 2, certain K3 surfaces, and Kummer varieties.



Nov 16

Joseph Gunther
Irrational points on random hyperelliptic curves
Abstract:Let d and g be positive integers with 1 < d < g. If d is odd, we prove there exists B_d such that a positive proportion of odd genus g hyperelliptic curves over Q have at most B_d points of degree d. If d is even, we similarly bound the degree d points not lazily pulled back from degree d/2 points of the projective line. The proofs use tropical geometry work of Park, as well as results of Bhargava and Gross on average ranks of hyperelliptic Jacobians. This is joint work with Jackson Morrow.

Time willing, we'll discuss rich, delicious interactions with work of next week's speaker.



Nov 30

Reed Gordon-Sarney
Zero-Cycles on Torsors under Linear Algebraic Groups
Abstract:In this talk, the speaker will discuss his thesis on the following question of Totaro from 2004: if a torsor under a connected linear algebraic group has index d, does it have a close etale point of degree d? The d = 1 case is an open question of Serre from the `60s. The d > 1 case, surprisingly, has a negative answer.



Dec 7

Rafe Jones
How do you (easily) find the genus of a plane curve?
Abstract: If you’ve ever wanted to show a plane curve has only finitely many rational points, you’ve probably wished you could invoke Faltings’ theorem, which requires the genus of the curve to be at least two. At that point, you probably asked yourself the question in the title of this talk. While the genus is computable for any given irreducible curve, it depends in a delicate way on the singular points. I’ll talk about a much nicer formula that applies to irreducible “variables separated” curves, that is, those given by A(x) = B(y) where A and B are rational functions.

Then I’ll discuss how to use this to resolve the question that motivated me originally: given an integer m > 1 and a rational function f defined over a number field K, does f possess a K-orbit containing infinitely many mth powers of elements of K? The answer turns out to be no unless f has a very special form: for m > 4 the map f must essentially be the mth power of some rational function, while for smaller m other exceptions arise, including maps closely related to multiplication on elliptic curves. If time permits I’ll discuss a connection to an arithmetic dynamical analogue of the Mordell-Lang conjecture.


Dec 14

Robert J. Lemke Oliver
Selmer groups, Tate-Shafarevich groups, and ranks of abelian varieties in quadratic twist families
Abstract: 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 towardsthe 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.