Mathematics & Computer Science

Seminars and Colloquia

Algebra Seminar

Friday, April 04, 2014

01:10 pm - 02:00 pm

Algebra Seminar and PhD Defense, Bonita Graham (Wesleyan): "A construction of rigid analytic cohomology classes for split reductive linear algebraic groups.

Abstract: Ash and Stevens showed that it is possible to lift ordinary classical Hecke Eigensymbols on connected reductive groups to overconvergent Hecke Eigensymbols. Pollack and Pollack showed explicitly how to compute these lifts in the case of GL_3. I extend this constructive proof to any split connected reductive algebraic group G.<br/>The key step is constructing a suitable filtration on a vector space related to G. An explicit formula for the filtration is given, allowing the computation of approximations of the overconvergent eigenclasses.<br/>

ESC 618

Friday, March 28, 2014

01:10 pm - 02:00 pm

Algebra Seminar, Keith conrad (UConn): "The $p$-adic generalized arithmetic-geometric mean."

Abstract: The arithmetic-geometric mean $M(x,y)$ of two positive real numbers $x$ and $y$ is the common limit of the two sequences $\{x_n\}$ and $\{y_n\}$ where $x_1 = x$, $y_1 = y$, $x_{n+1} = (x_n + y_n)/2$ and $y_{n+1} = \sqrt{x_ny_n}$. Convergence is fast. In 1989, Henniart and Mestre showed in $p$-adic fields that essentially the same construction leads to a common limit, where positivity is replaced by $y_1/x_1$ being sufficiently close to 1.<br/><br/>The construction of the limit $M(x,y)$ can be generalized from two positive numbers to $n$ positive numbers, using elementary symmetric polynomials in place of $x+y$ and $xy$. We will discuss a $p$-adic analogue.<br/>

ESC 618

Friday, February 21, 2014

01:10 pm - 02:00 pm

Algebra Seminar, Han Li (Yale): "Indefinite Integral Quadratic Forms Beyond Reduction Theory"

Abstract: The classical reduction theory of integral quadratic forms was developed by Hermite, Minkowski, Siegel and many others. It is known that a non-degenerate integral quadratic form in n-variables is integrally equivalent to a form whose height (the maximum value of the coefficients) is less than its determinant (up to a multiple constant), and whose value at (1, 0,...0) is less than the n-th root of its determinant. However, for indefinite forms in at least 3 variables it turns out that neither of the estimates is optimal. In this talk we will discuss some classical results and recent effort in improving these estimates. This is a joint work with Prof. Margulis.

ESC 618

Friday, February 14, 2014

01:10 pm - 02:00 pm

Algebra Seminar, Matthew Gelvin (Wes): "Fusion systems, orbit counting, and broken chains"

Abstract: If we're interested in studying a finite group $G$, we will often consider the action of $G$ on finite sets. There are two ways of understanding a $G$-set $X$: Either break it up into minimal pieces, each of which is itself a $G$-set, or record the number of $H$-fixed points of $X$ for all subgroups $H\leq G$. Both approaches contain the same data, with the \emph{table of marks} serving as the dictionary connecting the two. In the first part of this talk we will describe how to understand this table of marks in terms of the poset of subgroups of $G$ and a modified M\"obius inversion process.<br/><br/>Now let $S$ be a finite $p$-group and $\mathcal{F}$ a fusion system on $S$; for our purposes, $\mathcal{F}$ is simply the data of which subgroups of $S$ are \emph{fused} together, generalizing the notion of conjugate subgroups of a finite group. The analogue of a $G$-set is an $S$-set that respects this new fusion data. Earlier work with Reeh has shown that the additive monoid of these $\mathcal{F}$-sets has a basis of minimal elements. Unfortunately, the construction went via an inductive algorithm, so it is in general difficult to describe what exactly these basis elements should look like. The last part of this talk will detail joint work with Reeh and Yal\cc in, where we give a new description of these basis elements in terms of the combinatorics of the fusion system and the group theory of $S$.<br/>

ESC 618

Friday, December 06, 2013

01:10 pm - 02:00 pm

Algebra Seminar, Justin Lynd (Rutgers): "Fusion systems and punctured groups"

Abstract: Transporter systems, linking systems, and fusion systems, which are categories having objects some collection of subgroups of a fixed finite p-group S, give progressively coarser encodings of the p-local structure (i.e. roughly the amalgam of normalizers of<br/>p-subgroups) of a hypothetical finite group having S as Sylow p-subgroup. Chermak recently showed that to every fusion system F there is a unique corresponding linking system L. I'll give an an introduction to these objects and Chermak's technique of "descent" for constructing new transporter systems from old. Then I'll present some preliminary work (joint with Ellen Henke) examining in some small special cases a question arising out of Chermak's work: need there exist a "punctured group" associated to an (exotic) fusion system F over S, that is, a transporter system over the collection of nonidentity subgroups of S? For an exotic fusion system (i.e. one not arising from a finite group), such an associated punctured group could be viewed as the most group-like object approximating the nonexistent finite group at the given prime.<br/>

ESC 618

Friday, November 22, 2013

01:10 pm - 02:00 pm

Algebra Seminar, Kate Thompson (UGA): "Towards Analytic Proofs of Universality of Quadratic Forms"

Abstract: The 290-Theorem of Bhargava and Hanke says that a positive-definite quadratic form in four variables over the integers is universal iff the form represents 1 through 290 and classifies up to equivalence all such forms. in this talk I will describe some of the issues arising when one tries to generalize this result to rings of integers in other number fields. <br/><br/>The methods used to prove universality for a fixed form Q are very analytic in nature, drawing from the theory of (classical) modular forms and Siegel local densities. These tools generalize and can be used to provide universality theorems for positive-definite quaternary quadratic forms over O_K (for K a totally real number field). I will give an overview of how these analytic tools work, pinpointing what changes significantly when generalizing to a number field, and highlighting theoretical versus computational difficulties.<br/>

ESC 618

Friday, November 08, 2013

01:10 pm - 02:00 pm

Algebra Seminar, David Pollack (Wes): "Explicit eigencurves for GL_3"

Abstract: A theorem of Hida tells us that ordinary modular forms on $\operatorname{GL}_2$ deform in families of varying p-adic weight. These families are defined over a p-adic neighborhood in weight space. Certain homology classes on $\GL_3$ are also known to deform in p-adic families. In this case the weight space (up to twists) is two-dimensional, and the family may only be defined over (an open subset of) a one-dimensional curve in weight space. In this talk we will give an introduction to these p-adic families and their relation to deformations of Galois representations, and report on ongoing work with Avner Ash to compute approximations to the curves over which these deformations exist.

ESC 618

Friday, November 01, 2013

01:10 pm - 02:00 pm

Algebra Seminar, Holly Krieger (MIT): "The arithmetic of dynamical sequences"

Abstract: Understanding the arithmetic of sequences that are dynamically defined, such as the Fibonacci and Mersenne numbers, is of classical interest but generally results are difficult; for example, the infinitude of primes in either of these sequences is still open.<br/> However, results on the existence of primitive prime divisors and perfect powers in these sequences have been achieved, including the notable theorem of Bugeaud, Mignotte, and Siksek listing the Fibonacci powers. These questions and methods generalize to sequences which are forward orbits under iteration of certain dynamical systems, and I will discuss results on the arithmetic of such sequences, which rely on techniques from Diophantine approximation, arithmetic dynamics, and complex dynamics.<br/>

ESC 618

Friday, October 25, 2013

01:10 pm - 02:00 pm

Algebra Seminar: Three Wesleyan Graduate Student Talks by Abbey Bourdon, Bonita Graham, and James Ricci

Abstract: This seminar will consist of three separate talks by graduate students on the material in their theses:<br/><br/>Abbey Bourdon will present "A Uniform Version of a Finiteness Conjecture for CM Elliptic Curves." <br/><br/>Bonita Graham will present "A Construction of Rigid Analytic Cohomology Classes for Split Reductive Linear Algebraic Groups."<br/><br/>James Ricci will present "Finiteness results for regular ternary quadratic polynomials."<br/>

ESC 618

Friday, October 11, 2013

01:10 pm - 02:00 pm

Algebra Seminar, Asher Auel (Yale): A game on the exceptional group G_2 and a conjecture concerning flag varieties

Abstract: I will introduce a simple combinatorial game that can be played on any lattice in euclidean space. When played on the weight lattice of a linear algebraic group, this game is related to a longstanding conjecture on the derived category of flag varieties. Without any use of flag varieties, I will describe the solution to this game for all weight lattices in two dimensions. The most interesting case is the exceptional group G_2, which is related to octonion algebras. At the end, I will explain the conjecture via the case of quadrics, where the classical Clifford algebras play a role. This is joint work with Alexey Ananyevskiy, Skip Garibaldi, and Kirill Zainoulline.<br/>

ESC 618

Friday, September 13, 2013

01:10 pm - 02:30 pm

Algebra Seminar, Christopher Rasmussen (Wes): Trigonal curves good away from 3

Abstract: In the 1990s, Nigel Smart determined all genus two curves defined over Q with good reduction away from 2. Following Smart's general approach, we give a preliminary report on joint work with Beth Malmskog (Colorado College), to classify trigonal Picard curves with good reduction away from 3 defined over Q. Such curves have applications to cryptography, and also provide explicit of examples of varieties (via their Jacobians) whose (3-adic) Galois representations may be unusually constrained.<br/>

ESC 618

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

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.<br/>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.<br/>

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)"

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

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

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.<br/>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.<br/>

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.<br/><br/>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.<br/>

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 <br/>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. <br/>

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.<br/><br/>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.<br/>

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<br /><br/><br/>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<br/><br/>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<br/>maximal over 𝔽q2n. Rachel Pries, Robert Guralnick, and I determined<br/>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.<br/>

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<br/><br/>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. <br/><br/>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.<br/>

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<br/><br/>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<br/><br/>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.<br/>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.<br/>

ESC 618