Difference between revisions of "NTS ABSTRACT"

From Math
Jump to: navigation, search
(Sep 22)
(Dec 14)
 
(58 intermediate revisions by 5 users not shown)
Line 1: Line 1:
Return to [https://www.math.wisc.edu/wiki/index.php/NTS NTS Spring 2016]
+
Return to [https://www.math.wisc.edu/wiki/index.php/NTS NTS Spring 2017]
  
== Sep 8 ==
+
 
 +
== Sept 7 ==
  
 
<center>
 
<center>
 
{| 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"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Arunabha Biswas'''
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''David Zureick-Brown'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Limiting values of higher Mahler Measure and cyclotomic polynomials.
+
| bgcolor="#BCD2EE"  align="center" | Progress on Mazur’s program B
 
|-
 
|-
| bgcolor="#BCD2EE"  |   We consider the k-higher Mahler measure m_k(P) of a Laurent polynomial P as the integral of log^k |P| over the complex unit circle. In number theory, Lehmer's conjecture and the appearance of higher Mahler measures in L-functions are the main sources of motivation for studying various properties of m_k(P). Beyond number theory, Mahler measure has connections with topological entropies of dynamical systems and polynomial knot invariants. In this talk I shall present (1) an explicit formula for the value of |m_k(P)|/k! as k approaches infinity, (2) some asymptotic results regarding m_k(P) and (3) a scheme to approximate special values of a class of L-functions.
+
| 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.
  
 
|}                                                                         
 
|}                                                                         
Line 17: Line 18:
 
<br>
 
<br>
  
== Sep 15 ==
+
 
 +
 
 +
== Sept 14 ==
  
 
<center>
 
<center>
 
{| 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"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" |'''Naser T. Sardari'''
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Solly Parenti'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Discrete Log problem for the algebraic group PGL_2.
+
| bgcolor="#BCD2EE"  align="center" | Unitary CM Fields and the Colmez Conjecture
 
|-
 
|-
| bgcolor="#BCD2EE"  | We consider the problem of finding the shortest path between a given pair of vertices in the LPS Ramanujan graphs $X_{p,q}$ where $p$ is a fixed prime number and $q$ is an integer. We give a polynomial time algorithm in $\log(q)$ which returns the shortest path between  two diagonal vertices under a standard conjecture on the distribution of integers representable as sum of two squares and assuming one can factor quickly. Numerically, for a typical pair of vertices corresponded to diagonal elements the minimal path has a length about $3\log(q)+ O(1)$ while provably, there are pairs of points with distance at least $4\log(q)+ O(1)$ .  For a general pair of vertices, we write it as a product of three Euler angels and as a result for a typical pair we find a path with distance $9 \log(q)+ O(1)$.  
+
| 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.
 
+
 
+
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
Line 34: Line 35:
 
<br>
 
<br>
  
== Sep 22 ==
+
== Sept 21 ==
  
 
<center>
 
<center>
 
{| 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"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Alex Smith'''
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Chao Li '''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Statistics for 8-class groups and 4-Selmer groups
+
| bgcolor="#BCD2EE"  align="center" | Goldfeld's conjecture and congruences between Heegner points
 
|-
 
|-
| bgcolor="#BCD2EE"  | Assuming the grand Riemann hypothesis, we verify that the set of quadratic imaginary fields has the distribution of 8-class groups predicted by the Cohen-Lenstra heuristic. To do this we prove that, in families of quadratic fields parameterized by a single prime p, the 8-class rank is determined by the Artin symbol of p in a certain extension of the rationals. Using Chebotarev's density theorem, we find that the distribution of 8-class ranks in most of these small families is given by the Cohen-Lenstra heuristic. We can bundle these small families together to get the full result, with GRH necessary to control error bounds in this process. By analogous means, we also find the distribution of 4-Selmer groups in the quadratic twist family of an elliptic curve with full 2-torsion.
+
| 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.
  
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
 
+
== Sept 28 ==
<br>
+
 
+
== Sep 29 ==
+
  
 
<center>
 
<center>
 
{| 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"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" |'''Steve Lester'''
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Daniel Hast '''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | title coming soon
+
| bgcolor="#BCD2EE"  align="center" | Rational points on solvable curves over Q via non-abelian Chabauty
 
|-
 
|-
| bgcolor="#BCD2EE"  | abstract coming soon
+
| 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).
  
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
<br>
+
== Oct 12 ==
 
+
== Oct 6 ==
+
  
 
<center>
 
<center>
 
{| 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"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" |'''Nicole Looper'''
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Matija Kazalicki '''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Arboreal Galois representations of higher degree polynomials and Odoni's Conjecture
+
| bgcolor="#BCD2EE"  align="center" | Supersingular zeros of divisor polynomials of elliptic curves of prime conductor and Watkins' conjecture
 
|-
 
|-
| bgcolor="#BCD2EE"  | Since the mid-1980s, when the study of arboreal Galois representations first began, most results have concerned the representations induced by quadratic rational maps. In the higher degree case, by contrast, very little has been knownI will discuss some recent results pertaining to higher degree polynomials over number fields. This will include a partial solution to a conjecture made by R.W.K. Odoni in 1985.
+
| 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.
 +
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.
  
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
<br>
+
== Oct 19 ==
 
+
== Oct 13 ==
+
  
 
<center>
 
<center>
 
{| 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"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" |'''Ling Long'''
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Andrew Bridy'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | title coming soon
+
| bgcolor="#BCD2EE"  align="center" | Arboreal finite index for cubic polynomials
 
|-
 
|-
| bgcolor="#BCD2EE"  | abstract coming soon
+
| 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.
  
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
 
+
== Oct 19 ==
<br>
+
 
+
== Oct 20 ==
+
  
 
<center>
 
<center>
 
{| 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"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Jack Klys'''
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | ''Jiuya Wang''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | title coming soon
+
| bgcolor="#BCD2EE"  align="center" | Malle's conjecture for compositum of number fields
 
|-
 
|-
| bgcolor="#BCD2EE"  | abstract coming soon
+
| 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 $\mathbb{Q}$.
 
+
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
<br>
+
== Nov 2 ==
 
+
== Oct 27 ==
+
  
 
<center>
 
<center>
 
{| 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"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''William Duke'''
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '' Carl Wang-Erickson''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" |
+
| bgcolor="#BCD2EE"  align="center" | The rank of the Eisenstein ideal
 
|-
 
|-
| bgcolor="#BCD2EE"  |  
+
| 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.
  
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
<br>
 
  
== Nov 3 ==
 
  
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
| bgcolor="#F0A0A0" align="center" style="font-size:125%" |
 
|-
 
| bgcolor="#BCD2EE"  align="center" |
 
|-
 
| bgcolor="#BCD2EE"  | 
 
  
|}                                                                       
 
</center>
 
  
<br>
 
  
== Nov 10 ==
+
 
 +
 
 +
 
 +
== Nov 9 ==
  
 
<center>
 
<center>
 
{| 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"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" |
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | ''Masahiro Nakahara''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" |
+
| bgcolor="#BCD2EE"  align="center" | Index of fibrations and Brauer-Manin obstruction
 
|-
 
|-
| bgcolor="#BCD2EE"  |  
+
| 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.
 +
 
  
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
<br>
 
  
== Nov 17 ==
+
 
 +
== Nov 16 ==
  
 
<center>
 
<center>
 
{| 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"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" |
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | ''Joseph Gunther''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" |
+
| bgcolor="#BCD2EE"  align="center" | Irrational points on random hyperelliptic curves
 
|-
 
|-
| bgcolor="#BCD2EE"  |  
+
| 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 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.
 +
 
 +
 
  
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
<br>
 
  
== Dec 1 ==
+
 
 +
== Nov 30 ==
  
 
<center>
 
<center>
 
{| 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"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" |
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | ''Reed Gordon-Sarney''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" |  
+
| bgcolor="#BCD2EE"  align="center" |Zero-Cycles on Torsors under Linear Algebraic Groups
 
|-
 
|-
| bgcolor="#BCD2EE"  |
+
| 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.
 +
 
  
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
<br>
 
  
== Dec 8 ==
+
== Dec 7 ==
  
 
<center>
 
<center>
 
{| 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"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" |
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | ''Rafe Jones''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" |  
+
| bgcolor="#BCD2EE"  align="center" |How do you (easily) find the genus of a plane curve?
 
|-
 
|-
| bgcolor="#BCD2EE"  |
+
| 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.
  
|} 
+
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.
</center>   
+
  
<br>
 
  
== Dec 15 ==
+
|}                                                                       
 +
</center>
 +
 
 +
== Dec 14 ==
  
 
<center>
 
<center>
 
{| 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"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Efrat Bank'''
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | ''Robert J. Lemke Oliver''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Primes in short intervals on curves over finite fields
+
| bgcolor="#BCD2EE"  align="center" |Selmer groups, Tate-Shafarevich groups, and ranks of abelian varieties in quadratic twist families
 
|-
 
|-
| bgcolor="#BCD2EE"  | abstract coming soon
+
| 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.
  
 
+
|}                                                                      
|}
+
                                                                   
+
 
</center>
 
</center>

Latest revision as of 23:44, 3 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.