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Seminars and Colloquia

Algebra Seminar

Friday, May 03, 2013

02:00 pm - 03:00 pm

Algebra Seminar, Andrew Obus-Columbia University: "Abhyankar's Inertia Conjecture for PSL_2(s)

Abstract: It is an easy result that the complex numbers in the standard topology are simply connected. In algebraic geometry (under an appropriate definition of simply connected), the analog to this result is that the affine line A^1_C over the complex numbers is simply connected. But this no longer holds over an algebraically closed field k of characteristic p! In fact, its fundamental group \pi_1(A^1_k) is quite large, and not completely understood. But Abhyankar's Conjecture (now a theorem of Harbater and Raynaud) describes exactly what finite groups can be quotients of \pi_1(A^1_k). The etale covers of the affine line corresponding to finite quotients of \pi_1(A^1_k) are called "one-point covers." Abhyankar has a further conjecture, his inertia conjecture, which essentially states that any conceivable ramification pattern that could be associated to a one-point cover does, in fact, occur. This conjecture is wide open in general. In this talk, we will discuss the situation where the Galois group of the cover (analogous to the deck transformation group in topology) is equal to PSL_2(s), when PSL_2(s) has a cyclic p-Sylow group.

ESC 618

Friday, April 26, 2013

02:00 pm - 03:00 pm

Algebra Seminar, Bianca Viray-Brown University: " Failure of the local to global principle on surfaces obtained as the intersection of two quadrics"

Abstract: The Hasse-Minkowski theorem says that any quadratic equation has a solution over Q if and only if it has a solution over every completion, i.e. over the p-adics for every p and over the reals. Unfortunately, this does not hold in general, as examples of Lind and Reichardt show. We consider the case of an intersection of quadrics in 5 variables. Although, the local-global principle does not hold, we should that for a certain class of these surfaces, when the local-global principle fails, one can cover the surface with curves, each of which fail to have a point in some completion of Q. This is joint work with Anthony V'arilly-Alvarado.

ESC 618

Friday, March 29, 2013

02:00 pm - 03:00 pm

Algebra Seminar, Beth Malmskog-Colorado College: "The a-numbers of Jacobians of Suzuki Curves"

Abstract: For a natural number m, let q=22m+1, and Sm be the Suzuki curve defined over the field with q elements. Because of the large number of rational points relative to their genus, the Suzuki curves provide good examples of Goppa codes. The automorphism group of Sm is the Suzuki group Sz(q). However, the Jacobian of Sm has not been determined. It is fairly easy to see that the 2-rank of Jac(Sm) is 0, but the full 2-torsion group scheme of its Jacobian is not known. The a-number is a finer invariant of the isomorphism class of the 2-torsion group scheme. The a-number also places constraints on the decomposition of the Jacobian into indecomposable varieties. In this talk, I will discuss joint work with Holley Friedlander, Derek Garton, Rachel Pries, and Colin Weir in which which we computed a closed formula for the a- number of Sm using the action of the Cartier operator on H0.

ESC 618

Friday, February 22, 2013

02:00 pm - 03:00 pm

Algebra Seminar, Adam Topaz-University of Pennsylvania: "On the connection between valuation theory and Galois theory."

Abstract: In this talk I will describe the deep connection between valuation theory and the structure of Galois groups. On the one hand, a tamely-branching valuation yields a plethora of commuting elements in Galois groups via the usual structure theorems in decomposition theory. I will describe, in a purely elementary way, how similar commuting elements in Galois groups can arise from arbitrary valuations (even non-tamely-branching ones). Conversely, I will describe that such phenomena in Galois theory can, essentially, only arise from valuation theory.

ESC 618

Friday, February 15, 2013

02:00 pm - 03:00 pm

Algebra Seminar: Adam Towsley, Graduate Center CUNY "Newton's Method in Global Fields"

Abstract: Classically Newton's method is used to approximate roots of complex valued functions f by creating a sequence of points that converges to a root of f in the usual topology. For any global field K we completely describe the conditions under which Newton's method applied to a squarefree polynomial f with K-coefficients will succeed in finding a root of f in the v-adic topology for infinitely many places v of K. Furthermore, we show that Newton approximation sequence fails to converge v-adically for a positive density of places v.

ESC 618

Friday, February 08, 2013

02:00 pm - 03:00 pm

Algebra Seminar, Andrew Obus - Columbia University: " Abhyankar's Inertia Conjecture for PSL_2(s)"

ESC 618

Friday, February 01, 2013

02:00 pm - 03:00 pm

Algebra Seminar, John Voight - University of Vermont: "Lattice methods for algebraic modular forms on classical groups"

Abstract: The theta series of a positive definite quadratic form is also a classical modular form: consequently, the representation numbers of quadratic forms can be used to understand spaces of modular forms in the guise of Brandt matrices. The connection between such arithmetically-defined counting functions and modular forms is one piece of the Langlands philosophy, which predicts deep connections between automorphic forms in different guises via their Galois representations. In this talk, we consider algorithms for computing systems of Hecke eigenvalues in the more general setting of algebraic modular forms, as introduced by Gross. We use Kneser's neighbor method and isometry testing for lattices due to Plesken and Souveigner to compute systems of Hecke eigenvalues associated to deﬁnite forms of classical reductive algebraic groups. This is joint work with Matthew Greenberg.

ESC 618

Friday, November 30, 2012

02:00 pm - 03:00 pm

Algebra Seminar, Brian Wynne - Wesleyan: "The 2-color Rado number of x+y+kz = 3w"

Abstract: It follows from a theorem of R. Rado that for any positive integer k there exists a smallest positive integer N, depending on k, such that every 2-coloring of 1, ...., N yields a monochromatic solution of the equation x+y+kz = 3w. Based on computer experiments, Robertson and Myers conjectured values for N depending on the congruence class of k (mod 9). Dan Saracino and I established the values of N and found that in some cases they depend on the congruence class of k (mod 27). I will discuss these results and mention some open problems.

ESC 618

Friday, November 16, 2012

02:00 pm - 03:00 pm

Algebra Seminar, Harris Daniels - UCONN: "Siegel Functions, modular curves and Serre's uniformity problem"

Abstract: Serre's uniformity problem asks whether there exists a uniform bound k such that for every prime p > k and every elliptic curve E/Q without complex multiplication, the Galois representation associated to the p-torsion of E is surjective. Serre showed that if a p-torsion representation is not surjective, then it has to be contained in either a Borel subgroup, the normalizer of a split Cartan subgroup, the normalizer of a non-split Cartan subgroup, or one of a finite list of "exceptional'' subgroups. We will focus on the case when the image of the representation is contained in the normalizer of a split Cartan subgroup. In particular, we will show that any elliptic curve whose Galois representation at 11 is contained in the normalizer of a split Cartan has complex multiplication. To prove this we compute a model for the curve X_s^+ (11) and then use ideas of Poonen and Schaefer, and Chabauty and Coleman, to show that all the rational points on this curve correspond to elliptic curves with complex multiplication.

ESC 618

Friday, November 09, 2012

02:00 pm - 03:00 pm

Algebra Seminar, Alvaro Lozano-Robledo - UCONN: " Formal Groups of Elliptic Curves with Potential Good Supersingular Reduction "

Abstract: Let L be a number field and let E / L be an elliptic curve with potential supersingular reduction at a prime ideal ℘ of L above a rational prime p. In this talk we describe a formula for the slopes of the Newton polygon associated to the multiplication-by-p map in the formal group of E , that only depends on the congruence class of p mod 12, the ℘-adic valuation of the discriminant of a model for E over L , and the valuation of the j-invariant of E. The formula is applied to prove a divisibility formula for the ramification indices in the field of definition of a p-torsion point. We will introduce all the terms as we go along.

ESC 618

Friday, November 02, 2012

02:00 pm - 03:00 pm

Algebra Seminar, Abbey Bourdon-Wesleyan: " Uniform bound on Certain CM Cases of the Rasmussen/Tamagawa Conjecture

Abstract: Let A be an abelian variety of dimension g defined over a number field F. For a prime ℓ we can consider the field F ( A[ℓ∞]), which is an extension of F generated by coordinates of ℓ-powered torsion points of A. While the structure of this field is of interest in its own right, such fields also may help illuminate part of the structure of the absolute Galois group of F. Of particular interest are cases when F ( A[ℓ∞]) is a pro-ℓ extension of F(μℓ) and A has good reduction away from ℓ. Unfortunately, abelian varieties satisfying both requirements are relatively rare. In 2008, Rasmussen and Tamagawa conjectured that for a fixed F and g such examples will exist for only a finite number of primes ℓ. While the finiteness has since been resolved, if we restrict to the case where A is an elliptic curve with complex multiplication we can in fact show that the bound is uniform over all the number fields of degree n.

ESC 618

Friday, October 26, 2012

02:00 pm - 03:00 pm

Algebra Seminar, James Stankewicz - Wesleyan: "Twists of Shimura curves 2: Embeddings and Abelian varieties

Abstract: In an earlier talk we motivated the study of Shimura Curves and their twists. In this installment, we give several theorems on the arithmetic of quaternion orders and quadratic subrings. We will then show how these theorems apply to the theory of abelian varieties, and in turn to Twists of Shimura Curves.

ESC 618

Friday, October 19, 2012

02:00 pm - 03:00 pm

Algebra Seminar, Steven J. Miller, Williams University: "Cookie Monster Meets the Fibonacci Numbers. Mmmmmm -- Theorems!"

Abstract: A beautiful theorem of Zeckendorf states that every positive integer can be written uniquely as a sum of non-consecutive Fibonacci numbers. Once this has been shown, it's natural to ask how many Fibonacci numbers are needed. Lekkerkerker proved that the average number of such summands needed for integers in [F_n, F_{n+1}) is n / (phi^2 + 1), where phi is the golden mean. We present a combinatorial proof of this through the cookie problem and differentiating identities, and further prove that the fluctuations about the mean are normally distributed and the distribution of gaps between summands is exponentially decreasing. These techniques apply to numerous generalizations, which we'll discuss as time permits. This is joint work with several students from Williams' SMALL summer REU program. The only background required is elementary probability, combinatorics, and a love of cookies.

ESC 618

Friday, September 28, 2012

02:00 pm - 05:00 pm

Algebra Seminar, Zhiren Wang, Yale: "Dynamical Studies of Euclidean Minima"

Abstract: The Euclidean minimum is a numerical indicator that detects whether there is an Euclidean algorithm in a number field with respect to its algebraic norm. In this talk, we will briefly survey the history of its studies, in particular Cerri's work and his algorithm for the computation of Euclidean minima. Then we will discuss how facts from dynamical systems can be applied to show computability in finite time for all fields of degree 7 or higher and to produce computational complexity bounds for most fields. The talk will be based on recent joint works with Uri Shapira, as well as on previous works with Elon Lindenstrauss.

ESC 618

Friday, September 21, 2012

02:00 pm - 05:00 pm

Algebra Seminar: Sungmun Cho, Northeastern University "Group schemes and local densities of quadratic lattices in residue characteristic 2"

Abstract: The celebrated Smith-Minkowski-Siegel mass formula expresses the mass of quadratic lattice (L,Q) as a product of local factors, called the local densities of (L,Q). This mass formula is an essential tool for the classification of integral quadratic lattices. In this talk, I will explain the local density formula by observing the existence of a smooth affine group scheme G over ℤ2 with the generic fiber Autℚ〖_2〗 (L,Q), which satisfies G(ℤ2) = Autℤ〖_2〗(L,Q). This method works for any unramified finite extensions of ℚ2. Consequently, this provides the long awaited proof for the local density formula of Conway and Sloane and its generalization to unramified finite extensions of ℚ2. As an example, I will give the mass formula for the integral quadratic form Qn(x1,∙∙∙,xn) = x_(1 )^2 + ∙∙∙ +x_n^2 associated to a number filed k which is totally real and such that the ideal (2) is unramified over k.

ESC 618

Friday, September 14, 2012

02:00 pm - 05:00 pm

Algebra Seminar: James Stankewicz, Wes, "Twists of Shimura curves and applications"

Abstract: The classical modular curves of level N are important objects in the study of elliptic curves and modular forms. Shimura curves are a natural generalization of classical modular curves which may be formed using quaternion algebras. In this talk, we will give a short introduction to Shimura curves and their twists. We will then describe some techniques for determining if Atkin-Lehner twists of Shimura curves have rational points. Finally, we will give applications to the study of rational points on varieties, CM and supersingular elliptic curves, the inverse Galois problem and others.

ESC 618

Friday, September 07, 2012

02:10 pm - 05:00 pm

Algebra Seminar, Adam Gamzon, Hebrew University: Local Torsion on Abelian Surfaces With Real Mulitplication by Q (√5 )

Abstract: Fix an integer d ≥ 1. In 2008, David and Weston showed that, on average, an elliptic curve over Q picks up a nontrivial p-torsion point defined over a finite extension K of the p-adics of degree at most d for only finitely many primes p. This talk will discuss the frequency with which a principally polarized abelian surface A over Q with real multiplication by Q (√5 ) has a nontrivial p-torsion point defined over K. Averaging by height, the main result shows that A picks up a nontrivial p-torsion point over K for only finitely many p. Furthermore, we will indicate how this result connects with the deformation theory of modular Galois representations.

ESC 618

Friday, May 04, 2012

04:10 pm - 06:00 pm

Algebra Seminar and PhD Defense, Glenn Henshaw: Search Bounds for Points in Linear and Quadratic Spaces

Abstract: Let F be a quadratic form in N variables over K where K is a global field or Q ̅. Suppose F has a non-trivial zero outside of a finite union of K -varieties. We show that there exists such a point of relatively small height. This generalizes results by Cassels and Masser in several directions. We also investigate analogous questions over positive definite quaternion algebras.

ESC 618

Friday, April 27, 2012

04:10 pm - 06:00 pm

Algebra Seminar, Peter Loth, Sacred Heart University: Infinitary equivalence of Warfield groups

Abstract: Warfield groups are direct summands of simply presented abelian groups or, alternatively, are abelian groups possessing a nice decomposition basis with simply presented cokernel. They have been classified up to isomorphism by their Ulm-Kaplansky and Warfield invariants. The concept of decomposition basis was generalized to the notion of partial decomposition basis by Jacoby. In this talk, we discuss a model-theoretic classification of abelian groups with partial decomposition bases in L_∞^δ𝛚. This is joint work with Jacoby, Leistner and Str|ngmann and generalizes results by Barwise and Eklof.

ESC 618

Friday, April 20, 2012

04:10 pm - 06:00 pm

Algebra Seminar, Aaron Levin Michigan State: A generalization of the Nagell-Lutz theorem

Abstract: The classical Nagell-Lutz theorem states that a rational torsion point on an elliptic curve in Weierstrass form (with integral coefficients) has integral coordinates. We will discuss a generalization of this theorem that arose from studying a certain question coming from Diophantine approximation. In particular, our method seemingly gives a new proof of the classical Nagell-Lutz theorem that notably does not appear to explicitly use the group structure of the elliptic curve (e.g., formal groups, division polynomials, etc.). This is joint work with Umberto Zannier.

ESC 618

Friday, March 30, 2012

04:10 pm - 06:00 pm

Algebra Seminar Benjamin Linowitz, Dartmouth College: Quaternion orders and arithmetic hyperbolic geometry

Abstract: A well-known construction associates to an order in a quaternion algebra (defined over a totally real number field) a hyperbolic surface. In 1980 Vigneras used this construction in order to prove the existence of hyperbolic surfaces which were isospectral (have the same spectrum with respect to the Laplace-Beltrami operator) but not isometric. Key to Vigneras' method was a characterization of the values contained in the spectrum of an arithmetic manifold as embedding numbers of certain rank two commutative orders into quaternion orders. After discussing a few recent developments in the embedding theory of quaternion orders we will report on recent work with John Voight and Peter Doyle. This work deals with explicit examples of isospectral but not isometric hyperbolic surfaces. The example appearing in Vigneras' original paper was a pair of manifolds of genus 100801. We will use the arithmetic of quaternion orders in to exhibit substantially simpler examples: a pair of genus 6 manifolds and a pair of orbifolds whose underlying surface has genus 0. These examples have minimal volume amongst all isospectral surfaces arising from maximal arithmetic Fuchsian groups.

ESC 618

Friday, February 24, 2012

04:10 pm - 06:00 pm

(Algebra Seminar) Tom Tucker, University of Rochester: "Primitive divisors in dynamical sequences of integers"

Abstract: Let f(x) in Z[x] be a polynomial of degree d > 1 that is not conjugate to xd and let a be an integer that is not preperiodic under the action of f. A result of Silverman shows that there are infinitely many primes that appear as divisors of f n(a). Various authors have shown that in many cases, one has the following: for all but finitely many n, f n(a) has a "primitive divisor", that is, a prime divisor that is not a divisor of any f m(a) for m < n (that is, the divisor did not appear earlier in the dynamical sequence). We will show this in full generality for number fields, assuming the abc-conjecture, and prove it unconditionally in the analogous situation of function fields of characteristic 0.

ESC 618

Friday, February 10, 2012

04:10 pm - 06:00 pm

Algebra Seminar, Ken Ono, Emory University: Mock Modular periods and L-functions

Abstract: Spaces of cusp forms are images of spaces of harmonic Maass forms under a certain differential operator. Here we investigate the implications of this fact for spaces of integer weight cusp forms, and spaces of half-integral weight cusp forms. We shall obtain new theorems about period polynomials, and we obtain an extended Eichler-Shimura theory. In the case of half-integral weight forms, we show how to extend classical works of Waldspurger and Shimura on central values of quadratic twists of modular L-functions to include central derivatives of these L-functions.

ESC 618

Friday, December 02, 2011

01:00 pm - 05:00 pm

Christina Ballantine, College of the Holy Cross: Ramanujan Bigraphs (Algebra Seminar)

Abstract: Expander graphs are well-connected yet sparse graphs. The expansion property of a regular graph is governed by the second largest eigenvalue of the adjacency matrix. One can consider quotients of the Bruhat-Tits building of GL(n), n=2,3, over a p-adic field and view them as graphs. In this context the relationship between regular expander graphs and the Ramanujan Conjecture is well understood and has led to the definition and construction of asymptotically optimal regular expanders called Ramanujan graphs. The notion o fRamanujan graph can be extended to bigraphs (i.e., biregular, bipartitegraphs). In this talk I will use the representation theory of SU(3) over a p-adic field to investigate whether certain quotients of the associated Bruhat-Tits tree are Ramanujan bigraphs. I will show that a quotient of the Bruhat-Tits tree associated with a quasi-split form G of SU(3) is Ramanujan if and only if G satisfies a Ramanujan type conjecture. (This is joint work with Dan Ciubotaru).

ESC 618

Friday, November 11, 2011

04:00 pm - 05:00 pm

Algebra Seminar: Weyl group multiple Dirichlet series of type C

Speaker: Jennifer Beineke, Western New England University Abstract: One of the main achievements in the recent study of multiple Dirichlet series is the construction of Weyl group multiple Dirichlet series. These are Dirichlet series in several complex variables associated to a root system, and in this talk, we will investigate the case where the root system is of type C. We will discuss properties of these series, and will detail their construction, using tools and patterns from algebraic combinatorics.

ESC 618

Friday, November 04, 2011

01:00 pm - 05:00 pm

Algebra Seminar: Automorphisms of Certain Maximal Curves

Speaker: Beth Malmskog Abstract: The Hasse-Weil bound restricts the number of points of a curve which are defined over a finite field; if the number of points meets this bound, the curve is called maximal. Giulietti and Korchmaros introduced a curve 𝐶₃ which is maximal over 𝔽q6 and determined its automorphism group. Garcia, Guneri, and Stichtenoth generalized this construction to a family of curves 𝐶n, indexed by an odd integer n ≥ 3, such that 𝐶n is maximal over 𝔽q2n. Rachel Pries, Robert Guralnick, and I determined the automorphism group Aut(𝐶n ) when n > 3; in contrast with the case n = 3, it fixes the point at infinity on 𝐶n. The proof uses ramification groups and results from group theory. I will discuss maximal curves, automorphism groups, and outline our proof, including new results since my last talk at Wesleyan in April.

ESC 618

Friday, October 28, 2011

01:00 pm - 05:00 pm

Algebra Seminar: Euclidean quadratic forms and ADC forms

Speaker: Peter Clark, University of Georgia Abstract: Let R be an integral domain with fraction field K. Consider the following pleasant property of a quadratic form q(x) over R: for every d in R, if there exists x in K^n such that q(x) = d, then there exists y in R^n such that q(y) = d. I call such forms "ADC forms", after a theorem of Aubry and Davenport-Cassels which gives a Euclidean-like sufficient condition for a positive definite integral quadratic form to be an ADC form. In turn, when an integral domain R is equipped with a multiplicative norm function, one can consider "Euclidean quadratic forms" and the result that Euclidean implies ADC carries over to this context. Among other things, this raises the prospect of pursuing the "geometry of numbers" in a fairly general class of integral domains. However, classical cases are still quite interesting, in particular consider the case of a "Hasse domain" (i.e., an S-integer ring in a global field). About a year ago I proved that a quadratic form over a Hasse domain R is ADC iff it is regular (i.e., represents every element represented by its genus) and locally ADC: i.e., is ADC when viewed as a quadratic form over each non-Archimedean completion of R. More recently I have classified ADC forms over certain complete discrete valuation rings. I am quite close to a classification of positive definite ADC forms over Z, and I will tell you the latest word on that. Some of this work is joint with W.C. Jagy and is closely related to past and present work of his.

ESC 618

Friday, October 14, 2011

01:00 pm - 05:00 pm

Algebra Seminar: Hilbert's Tenth Problem and Mazur's conjectures in large subrings of number fields

Speaker: Kirsten Eisentraeger, Penn State Abstract: Hilbert's Tenth Problem in its original form was to find an algorithm to decide, given a multivariate polynomial equation with integer coefficients, whether it has a solution over the integers. In 1970 Matijasevich, building on work by Davis, Putnam and Robinson, proved that no such algorithm exists, i.e. Hilbert's Tenth Problem is undecidable. In this talk we will consider generalizations of Hilbert's Tenth Problem and Mazur's conjectures for large subrings of number fields. We will show that Hilbert's Tenth Problem is undecidable for large complementary subrings of number fields and that the analogues of Mazur's conjectures do not hold in these rings.

ESC 618

Friday, October 07, 2011

01:00 pm - 05:00 pm

Algebra Seminar: Some applications of the trace formula

Speaker: Andrew Knightly, University of Maine Abstract: I will give an introduction to the trace formula, and describe some of the ways it can be used in the study of modular L-functions. Using a test function attached to the simple supercuspidal representations constructed by Gross and Reeder, we deduce spectral information pertaining to cuspidal newforms (Maass or holomorphic) of cubic level. This is joint work with Charles Li.

ESC 618

Friday, September 16, 2011

04:00 pm - 05:00 pm

Combinatorics of the Gindikin-Karpelevich formula

Speaker: Ben Salisbury, UCONN Abstract: A crystal is a combinatorial framework to be used to discuss certain properties of Lie algebras and their representations. In particular, the crystal of the negative part of the universal enveloping algebra of a Lie algebra may be used to expand important products arising from integrals over p-adic groups as sums. In this talk, we explain how the realization of this crystal in terms of Young tableaux yields a statistic defining a coefficient making this expansion possible.

ESC 618

Friday, April 22, 2011

03:00 pm - 04:00 pm

Automorphisms of Generalized GK Curves

Speaker: Beth Malmskog, Colorado State University Abstract: Curves with as many points as possible over a finite field $\mathbb{F}_q$ under the Hasse-Weil bound are called maximal curves. Besides being interesting as extremal objects, maximal curves have applications in coding theory. Maximal curves may also have a great deal of symmetry, i.e. have an automorphism group which is large compared to the curve's genus. I will discuss two families of maximal curves and find a large subgroup of each curve's automorphism group. We also give an upper bound for the size of the automorphism group.

ESC 618

Friday, February 25, 2011

03:00 pm - 04:00 pm

Cyclic Extensions and the Local Lifting Problem

Speaker: Andrew Obus, Columbia Abstract: The lifting problem we will consider roughly asks: given a smooth projective curve X over an algebraically closed field of characteristic p and a finite group G of automorphisms of X, does there exist a smooth, projective curve X' in characteristic zero and a finite group of automorphisms G' of X' such that (X', G') lifts (X, G)? It turns out that solving this lifting problem reduces to solving a local lifting problem in a formal neighborhood of each point of X where G acts with non-trivial inertia. The Oort conjecture states that this local lifting problem should be solvable whenever the inertia group is cyclic. A new result of Stefan Wewers and the speaker shows that the local lifting problem is solvable whenever the inertia group is cyclic of order not divisible by p^4, and in many cases even when the inertia group is cyclic and arbitrarily large. We will discuss this result, after giving a good amount of background on the local lifting problem in general.

ESC 618

Friday, February 18, 2011

03:00 pm - 04:00 pm

Computing Hilbert Modular forms, or Nonsolvable number fields ramified at one prime

Speaker: John Voight Abstract: We use explicit methods for computing with quaternion algebras and the cohomology of Shimura curves to compute spaces of Hilbert modular forms. As a consequence, we construct nonsolvable finite extensions of the rational numbers ramified only at 3 and 5, respectively. This is joint work with Matthew Greenberg and Lassina Dembele.

ESC 618

Friday, February 11, 2011

03:00 pm - 04:00 pm

2 Nilpotent Real Section Conjecture

Speaker: Kirsten Wickelgren, Harvard Abstract: Grothendieck's anabelian conjectures say that hyperbolic curves over certain fields should be K(pi,1)'s in algebraic geometry. It follows that points on such a curve are conjecturally the sections of etale pi_1 of the structure map. These conjectures are analogous to equivalences between fixed points and homotopy fixed points of Galois actions on related topological spaces. This talk will start with an introduction to Grothendieck's anabelian conjectures, and then present a 2 nilpotent real section conjecture: for a smooth curve X over R with negative Euler characteristic, pi_0(X(R)) is determined by the maximal 2-nilpotent quotient of the fundamental group with its Galois action, as the kernel of an obstruction of Jordan Ellenberg. This implies that the set of real points equipped with a real tangent direction of the smooth compactification of X is determined by the maximal 2-nilpotent quotient of Gal(C(X)) with its Gal(R) action, showing a 2-nilpotent birational real section conjecture.

ESC 618

Friday, January 28, 2011

03:00 pm - 04:00 pm

Computing Isogeny Volcanoes

Speaker: Dustin Moody, NIST Abstract: Isogenies are maps between elliptic curves. Isogeny volcanoes are an interesting structure that have had several recent applications in cryptography. An isogeny volcano is a connected component of a larger graph. We further explore properties of and how to compute volcanoes given that we have already computed one of a different degree. This allows us to compute volcanoes of composite degree more efficiently than a direct construction using modular polynomials.

ESC 618

Friday, December 03, 2010

01:00 pm - 02:30 pm

Reducible Galois representations and Hecke eigenclasses

Speaker: Avner Ash, Boston College Abstract: Serre's conjecture (now a theorem due to Khare-Wintenberger and Kisin) gives a tight relationship between odd 2-dimensional mod p irreducible representations of G_Q (the absolute Galois group of Q) and mod p modular forms. In 2000, W. Sinnott and I published a generalization of this conjecture in which we connect mod p n-dimensional representations rho of G_Q and mod p cohomology of arithmetic subgroups of SL(n,Z). An interesting feature of this conjecture is the complicated parity condition on rho when rho is reducible. I have recently proved this conjecture in many cases when rho is a direct sum of 1-dimensional representations. This talk will explain these things from scratch, more or less.

ESC 618

Friday, November 12, 2010

01:00 pm - 02:30 pm

A Compactification of the Space of Algebraic Maps from P^1 to a Grassmannian

Speaker: Yijun Shao Abstract: The moduli space of algebraic maps (i.e., morphisms) of degree d from P^1 to a Grassmannian is a nonsingular, noncompact, quasi-projective variety. In this talk, I will describe an explicit construction of a compactification for this space satisfying the following properties: 1. the compactification is a nonsingular projective variety; 2. the boundary is a divisor with normal crossings (i.e., a union of nonsingular codimension 1 subvarieties which intersect transversally). The construction is based on a compactification given by a Quot scheme. This Quot scheme is a nonsingular projective variety, but the boundary is in general singular and of codimension>1. The main tool of the construction is blowup. We first define a sequence of nested closed subschemes of the boundary, and then blowup the Quot scheme successively along these closed subschemes. The final outcome of the blowups is a compactification with the desired properties.

ESC 618

Friday, November 05, 2010

01:00 pm - 02:30 pm

Descent on the Congruent Number Elliptic Curves

Speaker: Nick Rogers, University of Rochester Abstract: A classical Diophantine problem asks which positive integers n occur as areas of right triangles with rational sides. Such "congruent numbers" correspond to elliptic curves y^2 = x^3 - n^2 x with positive rank. In this talk I'll describe how to compute certain Selmer groups associated to these elliptic curves. These descent calculations lead to results on non-congruent numbers and Tate-Shafarevich groups, and shed some light on the behavior of ranks in families of quadratic twists.

ESC 618

Friday, October 15, 2010

01:00 pm - 02:30 pm

The Dwork Family and Hypergeometric Functions

Speaker: Adriana Salerno Abstract: In his work studying the Zeta functions of families of hypersurfaces, Dwork came upon a one-parameter family of hypersurfaces in $\mathbb{P}^{n-1}$ (now known as the Dwork family), defined by:$$X_{\lambda}:x_1^n+\cdots +x_n^n-n\lambda x_1\cdots x_n=0.$$ These examples were not only useful to Dwork in his study of his deformation theory for computing Zeta functions of families, but they have also proven to be extremely useful to physicists working in mirror symmetry. A startling result is that these families are very closely linked to hypergeometric functions. This phenomenon was carefully studied by Dwork in the cases where $n=3,4$ and for $n=5$ by Candelas, de la Ossa, and Rodr\'{i}guez-Villegas. Dwork, Candelas, et.al. observed that, for these families, the differential equation associated to the Gauss-Manin connection is in fact hypergeometric. We have developed a computer algorithm, implemented in Pari-GP, which can check this result for larger values of $n$ by computing the Gauss-Manin connection and the parameters of the hypergeometric differential equation.

ESC 618

Friday, October 08, 2010

01:00 pm - 02:30 pm

Almost Universal Ternary Sums of Triangular Numbers

Speaker: Anna Haensch, Wesleyan University Abstract: A fundamental question in number theory is to study the representations of positive integers as sums of polygonal numbers. This problem has a rich history, beginning with Gauss who showed in 1796 that all positive integers can be expressed as a sum of three triangular numbers. This is equivalent to representing every positive integer of the form 8n+3 by a sum of three squares. Extending this idea, in 1862 Liouville determined all triples (a,b,c) of positive integers for which the ternary sums aT_x+bT_y+cT_z are universal, that is, representing all positive integers. Here T_x = x(x+1)/2 is the polynomial representing all triangular numbers. The next logical question is to determine for which triples the ternary sums would be almost universal, that is, representing all but finitely many positive integers. Recently in 2008 Kane and Sun compile a list of sufficient conditions, and formulate a conjecture for necessary conditions. Later in 2009 Chan and Oh resolve this conjecture by providing a complete classification of almost universal ternary sums of triangular numbers. In this talk I will explain the results of this paper.

ESC 618

Friday, October 01, 2010

01:00 pm - 02:30 pm

Special Values of Goss L-Functions in Positive Characteristics

Speaker: Matt Papanikolas, Texas A&M Abstract: Values of Dirichlet L-functions at positive integers are expressible in terms of powers of pi and values of polylogarithms at algebraic numbers. In this talk we will focus on finding analogies of these results over function fields of positive characteristic. In particular, we will consider special values of Goss L-functions for Dirichlet characters, which take values in the completion of the rational function field in one variable over a finite field. Building on work of Anderson for the case of L(1,chi), we deduce various power series identities on tensor powers of the Carlitz module that are "log-algebraic" and in turn use these formulas to determine exact values of L(n,chi) for arbitrary n > 0. Moreover, we relate these L-series values to powers of the Carlitz period and values of Carlitz polylogarithms at algebraic points.

ESC 618

Friday, September 24, 2010

01:00 pm - 02:30 pm

Embedding Orders in Central Simple Algebras

Speaker: Ben Linowitz, Dartmouth College Abstract: Let $K$ be a number field and $B$ be a central simple algebra defined over $K$. In 1932, Albert, Brauer, Hasse and Noether explicitly characterized the finite extensions $L$ of $K$ which may be embedded into $B$. An integral refinement immediately suggests itself: given an order $\Omega\subset \mathcal{O}_L$ and a maximal order $\mathcal M \subset B$, when does there exist an embedding of $\Omega$ into $\mathcal M$? This question turns out to be more subtle. In the case that $B$ is a quaternion algebra satisfying the Eichler condition, Chinburg and Friedman have shown that the proportion of isomorphism classes of maximal orders of $B$ admitting an embedding of $\Omega$ is either $0$, $\frac{1}{2}$ or $1$. Arenas-Carmona considered a broad class of higher rank central simple algebras and determined the maximal orders admitting an embedding of $\mathcal{O}_L$. We consider central simple algebras of dimension $p^2$ for $p$ an odd prime and show that arbitrary commutative orders $\Omega$ in a degree $p$ extension of $K$ embed into none, all or exactly one out of every $p$ isomorphism classes of maximal orders. The maximal orders admitting an embedding of $\Omega$ are explicitly characterized using the Bruhat-Tits building for $SL_p$.

ESC 618

Friday, September 17, 2010

01:00 pm - 02:30 pm

The Restriction Map in the Galois Cohomology of Linear Algebraic Groups

Speaker: Jodi Black, Emory University Abstract: Let k be a field and let G be a connected linear algebraic group over k. We denote the first Galois cohomology set of k with values in G by H^1(k, G). Let L_1, ....L_m be a set of finite field extensions of k of coprime degree. The following is an open question of Jean-Pierre Serre: "Does the product of the restriction maps from H^1(k,G) to H^1(L_1, G) x ...xH^1(L_m, G) have trivial kernel?" We discuss the significance of a positive answer to this question for important open problems in the area and show that it has positive answer for certain families of groups G.

ESC 618

Friday, September 10, 2010

01:00 pm - 05:00 pm

On a Certain PSL(2, Z) 1-Cocycle

Speaker: Sheldon Joyner Abstract: Classically, if some manifold M is equipped with an action of a subgroup G of PSL(2, Z) under which a certain space F of functions on M transforms via a 1-cocycle, the latter is referred to as an automorphy factor, and the functions F are said to be G-modular. In this talk, I will produce an injective 1-cocycle of PSL(2, Z) into a certain group of formal power series which extends the well-known identification of the fundamental group of P^1\{0,1,infty} with associated Chen series. This cocycle may be regarded as a quasi-automorphy factor for sections of the universal prounipotent bundle with connection on PSL(2, Z) - in particular for the polylogarithm generating series Li(z). I will go on to show that the quasi-modularity of Li(z) may be used to give a family of proofs of the analytic continuation and functional equation for the Riemann zeta function. Moreover, under this cocycle, the involutive generator of PSL(2, Z) maps to the Drinfeld associator, while the infinite cyclic generator maps to an R-matrix, in Drinfel'd's formal model of the quasi-triangular quasi-Hopf algebras, thereby producing a representation of PSL(2, Z) into tensor products of certain qtqH algebras. Underlying the whole story is a path space realization of PSL(2, Z) using Deligne's idea of tangential basepoint.

ESC 618