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April 6 |
Nha Truong (Hawaii)
The slopes of modular forms are the $p$-adic valuations of the eigenvalues of the Hecke operators $T_p$. The study of slopes plays an important role in understanding the geometry of the eigencurves, introduced by Coleman and Mazur. The study of the slope began in the 1990s when Gouvea and Mazur computed many numerical data and made several interesting conjectures. Later, Buzzard, Calegari, and other people made more precise conjectures by studying the space of overconvergent modular forms. Recently, Bergdall and Pollack introduced the ghost conjecture that unifies the previous conjectures in most cases. The ghost conjecture states that the slope can be predicted by an explicitly defined power series. We prove the ghost conjecture under a certain mild technical condition. In the pre-talk, I will explain an example in the quaternionic setting which was used as a testing ground for the proof. This is joint work with Ruochuan Liu, Liang Xiao, and Bin Zhao. |
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April 13 |
Alina Bucur (UC San Diego)
A guiding question in number theory, specifically in arithmetic statistics, is counting number fields of fixed degree and Galois group as their discriminants grow to infinity. We will discuss the history of this question and take a closer look at the story in the case of quartic fields. In joint work with Florea, Serrano Lopez, and Varma, we extend and make explicit the counts of extensions of an arbitrary number field that was done over the rationals by Cohen, Diaz y Diaz, and Olivier. |
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April 20 |
Keegan Ryan (UC San Diego) (paper)
We introduce a new lattice basis reduction algorithm with approximation guarantees analogous to the LLL algorithm and practical performance that far exceeds the current state of the art. We achieve these results by iteratively applying precision management techniques within a recursive algorithm structure and show the stability of this approach. We analyze the asymptotic behavior of our algorithm, and show that the heuristic running time is $O(n^{\omega}(C+n)^{1+\varepsilon})$ for lattices of dimension $n$, $\omega\in (2,3]$ bounding the cost of size reduction, matrix multiplication, and QR factorization, and $C$ bounding the log of the condition number of the input basis $B$. This yields a running time of $O\left(n^\omega (p + n)^{1 + \varepsilon}\right)$ for precision $p = O(\log \|B\|_{max})$ in common applications. Our algorithm is fully practical, and we have published our implementation. We experimentally validate our heuristic, give extensive benchmarks against numerous classes of cryptographic lattices, and show that our algorithm significantly outperforms existing implementations. |
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April 27 |
Somnath Jha (IIT Kanpur)
The classical Diophantine problem of determining which integers can be written as a sum of two rational cubes has a long history; it includes works of Sylvester, Selmer, Satgé, Leiman and the recent work of Alpöge-Bhargava-Shnidman-Burungale-Skinner. In this talk, we will use Selmer groups of elliptic curves and integral binary cubic forms to study some cases of the rational cube sum problem. This talk is based on joint works with D. Majumdar, P. Shingavekar and B. Sury. |
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May 4 |
Hanlin Cai (Utah) (paper)
In this talk I'll talk about a (perfectoid) mixed characteristic version of F-signature and Hilbert-Kunz multiplicity by utilizing the perfectoidization functor of Bhatt-Scholze and Faltings' normalized length. These definitions coincide with the classical theory in equal characteristic. Moreover, perfectoid signature detects BCM regularity and transforms similarly to F-signature or normalized volume under quasi-étale maps. As a consequence, we can prove that BCM-regular rings have finite local étale fundamental group and torsion part of their divisor class groups. This is joint work with Seungsu Lee, Linquan Ma, Karl Schwede and Kevin Tucker. |
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May 11 |
Samit Dasgupta (Duke)
In 1976, Ken Ribet used modular techniques to prove an important relationship between class groups of cyclotomic fields and special values of the zeta function. Ribet’s method was generalized to prove the Iwasawa Main Conjecture for odd primes p by Mazur-Wiles over Q and by Wiles over arbitrary totally real fields. Central to Ribet’s technique is the construction of a nontrivial extension of one Galois character by another, given a Galois representation satisfying certain properties. Throughout the literature, when working integrally at p, one finds the assumption that the two characters are not congruent mod p. For instance, in Wiles’ proof of the Main Conjecture, it is assumed that p is odd precisely because the relevant characters might be congruent modulo 2, though they are necessarily distinct modulo any odd prime. In this talk I will present a proof of Ribet’s Lemma in the case that the characters are residually indistinguishable. As arithmetic applications, one obtains a proof of the Iwasawa Main Conjecture for totally real fields at p=2. Moreover, we complete the proof of the Brumer-Stark conjecture by handling the localization at p=2, building on joint work with Mahesh Kakde for odd p. Our results yield the full Equivariant Tamagawa Number conjecture for the minus part of the Tate motive associated to a CM abelian extension of a totally real field, which has many important corollaries. This is joint work with Mahesh Kakde, Jesse Silliman, and Jiuya Wang. |
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May 18 |
Andrew Kobin (Emory)
Zeta and L-functions are ubiquitous in modern number theory. While some work in the past has brought homotopical methods into the theory of zeta functions, there is in fact a lesser-known zeta function that is native to homotopy theory. Namely, every suitably finite decomposition space (aka 2-Segal space) admits an abstract zeta function as an element of its incidence algebra. In this talk, I will show how many 'classical' zeta functions in number theory and algebraic geometry can be realized in this homotopical framework. I will also discuss work in progress towards a categorification of motivic zeta and L-functions. |
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June 1 |
Catherine Hsu (Swarthmore College)
In his seminal work on modular curves and the Eisenstein ideal, Mazur studied the existence of congruences between certain Eisenstein series and newforms, proving that Eisenstein ideals associated to weight 2 cusp forms of prime level are locally principal. In this talk, we'll explore generalizations of Mazur's result to squarefree level, focusing on recent work, joint with P. Wake and C. Wang-Erickson, about a non-optimal level N that is the product of two distinct primes and where the Galois deformation ring is not expected to be Gorenstein. First, we will outline a Galois-theoretic criterion for the deformation ring to be as small as possible, and when this criterion is satisfied, deduce an R=T theorem. Then we'll discuss some of the techniques required to computationally verify the criterion. |