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January 9 |
ORGANIZATIONAL MEETING |
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January 16 |
NO MEETING |
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January 23 |
Adam Logan (Carleton University/TIMC) Using the Torelli theorem for K3 surfaces of Pyatetskii-Shapiro and Shafarevich one can describe the automorphism group of a K3 surface over {\mathbb C} up to finite error as the quotient of the orthogonal group of its Picard lattice by the subgroup generated by reflections in classes of square -2. We will give a similar description valid over an arbitrary field in which the reflection group is replaced by a certain subgroup. We will then illustrate this description by giving several examples of interesting behaviour of the automorphism group, and by showing that the automorphism groups of two families of K3 surfaces that arise from Diophantine problems are finite. This is joint work with Martin Bright and Ronald van Luijk (University of Leiden). |
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January 23: colloquium (4pm, APM 6402) |
Aaron Pollack (Duke) By a "modular form" for a reductive group G we mean an automorphic form that has some sort of very nice Fourier expansion. The classic example are the holomorphic Siegel modular forms, which are special automorphic functions for the group Sp_{2g}. Following work of Gan, Gross, Savin, and Wallach, it turns out that there is a notion of modular forms on certain real forms of the exceptional groups. I will define these objects and explain what is known about them. |
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January 30 |
Robert Hough (Stony Brook University) In his thesis, M. Bhargava proved parameterizations and identified local conditions which he used to give asymptotic counts for $S_4$ quartic and quintic number fields, ordered by discriminant. This talk will discuss results in an ongoing project to add detail to Bhargava's work by considering in addition to the field discriminant, the lattice shape of the ring of integers in the canonical embedding, and by giving strong rates with lower order terms in the asymptotics. These results build on earlier work of Taniguchi-Thorne, Bhargava-Shankar-Tsimerman and Bhargava-Harron. |
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February 6 |
James Upton (UC Irvine) Let X be a smooth affine variety over a finite field of characteristic p. The Dwork-Monsky trace formula is a fundamental tool in understanding the L-functions of p-adic representations of 1(X). We extend this result to the study of representations valued in a higher-dimensional local ring R. The special case R=Zp[[T]] arises naturally in the study of etale Zp-towers over X. Time permitting, we discuss some spectral-halo type results and conjectures describing the p-adic variation of slopes in certain Zp-towers. |
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February 13 |
Everett Howe Sutherland and Voloch have investigated the idea of distinguishing two curves over a finite field by comparing the zeta functions of the curves and of certain covers of the curves. To test the limits of this method, it is instructive to find examples of nonisomorphic curves that have covers whose Jacobians are unexpectedly isogenous. I will describe a construction that, heuristically, should produce infinitely many examples of pairs of genus-2 curves over finite fields such that the Jacobians of the curves are isogenous and the Jacobians of the (genus-17) maximal unramified abelian 2-covers of the curves are also isogenous. Implementing this construction led to the discovery of a (very!) unlikely intersection in characteristic zero, whose existence proves that indeed there exist infinitely many examples of genus-2 curves as above. |
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February 20 |
Edgar Costa (MIT) In this talk, we will focus on how one can deduce some geometric invariants of an abelian variety or a K3 surface by studying their Frobenius polynomials. In the case of an abelian variety, we show how to obtain the decomposition of the endomorphism algebra, the corresponding dimensions, and centers. Similarly, by studying the variation of the geometric Picard rank, we obtain a sufficient criterion for the existence of infinitely many rational curves on a K3 surface of even geometric Picard rank. |
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February 20: colloquium (4pm, APM 6402) |
Chao Li (Columbia) Can an integer $n$ be represented as a sum of two squares $n=x^2+y^2$? If so, how many different representations are there? We begin with the answers to these classical questions due to Fermat and Jacobi. We then illustrate Hurwitz's class number formula for binary quadratic forms, and put all these classical formulas under the modern perspective of the Siegel-Weil formula. We explain how the latter perspective led Gross-Keating to discover a new type of identity between arithmetic intersection numbers on modular surfaces andderivatives of certain Eisenstein series. After outlining the influential program of Kudla and Rapoport for generalization to higher dimensions, we report a recent proof (joint with W. Zhang) of the Kudla-Rapoport conjecture and hint at the usage of the uncertainty principle in the proof. |
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February 22 (Saturday) |
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February 27 |
Nathan Green (UCSD) Classically, the transcendence (and even the irrationality) of odd zeta values is widely conjectured, but yet unproven. However, for zeta values defined over the rational function field, Jing Yu succeeded in proving their transcendence in 1991, and many other transcendence results (including algebraic independence) followed in the intervening years. In this work (joint with T. Ngo Dac), we prove the algebraic independence of zeta values defined over the function field of an elliptic curve. The main technique we use is to construct a Tannakian category of t-motives whose associated periods contain these zeta values - thus we may exploit the existence of a motivic Galois group to study the transcendence degree. We also discuss the difficulties and pathway to proving algebraic independence for zeta values of function fields of arbitrary curves. |
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March 5 |
Jeff Manning (UCLA) In his work on modularity theorems, Wiles proved a numerical criterion for a map of rings R->T to be an isomorphism of complete intersections. He used this to show that certain deformation rings and Hecke algebras associated to a mod p Galois representation at non-minimal level were isomorphic and complete intersections, provided the same was true at minimal level. In addition to proving modularity theorems, this numerical criterion also implies a connection between the order of a certain Selmer group and a special value of an L-function. In this talk I will consider the case of a Hecke algebra acting on the cohomology a Shimura curve associated to a quaternion algebra. In this case, one has an analogous map of ring R->T which is known to be an isomorphism, but in many cases the rings R and T fail to be complete intersections. This means that Wiles' numerical criterion will fail to hold. I will describe a method for precisely computing the extent to which the numerical criterion fails (i.e. the 'Wiles defect"), which will turn out to be determined entirely by local information at the primes dividing the discriminant of the quaternion algebra. This is joint work with Gebhard Bockle and Chandrashekhar Khare. |
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March 12 |
Luciena Xiao (Caltech) Central leaves in the special fiber of Shimura varieties are the loci where the isomorphism class of the universal $p$-divisible group remains constant. The Hecke orbit conjecture asserts that every prime-to-$p$ Hecke orbit in a PEL type Shimura variety is dense in the central leaf containing it. This conjecture is proved for Hilbert modular varieties by C.-F. Yu, and for Siegel modular varieties by Chai and Oort. In this talk I give an overview of the conjecture and present my work that generalizes Chai and Oort's strategy to irreducible components of certain Newton strata on Shimura varieties of PEL type. |