Graduate students (and very recently graduated young researchers) will be presenting posters on Wednesday afternoon. Poster submissions are now closed, but you may email silvermania2015@gmail.com for late submissions.

## Poster Titles and Abstracts

Kenny Ascher, Brown University
A Fibered Power Theorem for Pairs of Log General Type
A theorem of Caporaso-Harris-Mazur states that given a family of curves with general fiber a smooth curve of genus greater than or equal to 2 (i.e. a curve of general type), some high fibered power of the family admits a dominant rational map to a variety of general type. This was generalized to families of surfaces of general type by Hassett, and to families of arbitrary dimensional varieties of general type by Abramovich. Assuming various versions of Lang's Conjecture, these results lead to uniform bounds on the number of rational points of varieties of general type. Using the moduli space of stable pairs, and recent work of Kovács-Patakfalvi, we prove an analogous fibered power theorem in the case of pairs of log general type.

Maistret Celine, University of Warwick
Local arithmetic for hyperelliptic curves of genus 2
In the context of proving the 2-parity conjecture for jacobians of hyperelliptic curves of genus 2 admitting a Richelot isogeny, we present the computation of local arithmetic invariants of both the curve and its jacobian. In particular, computation of invariants such as the Tamagawa numbers, the number of real connected components and the deficiency will be detailed.

Sneha Chaubey, University of Illinois, Urbana-Champaign
Arithmetic of Farey-Ford Packings
The Farey sequence is a natural exhaustion of the set of rational numbers between 0 and 1 by finite lists. Ford Circles are a natural family of mutually tangent circles associated to Farey fractions: they are an important object of study in the geometry of numbers and hyperbolic geometry. We compute the distributions and moments of certain statistics of geometric quantities such as (but not limited to) areas, length, slopes, and angles associated to Ford circles in a Farey-Ford Packing.

Anastassia Etropolski, Emory University
Local to global principles for Galois representations
Let $$K$$ be a number field and let $$E/K$$ be an elliptic curve whose mod $$\ell$$ Galois representation locally has image contained in a group $$G$$, up to conjugacy. We classify the possible images for the global Galois representation in the case where $$G$$ is a Cartan subgroup or the normalizer of a Cartan subgroup. When $$K = \mathbf{Q}$$, we deduce a counterexample to the local-global principle in the case where $$G$$ is the normalizer of a split Cartan and $$\ell = 13$$. In particular, there are at least three elliptic curves (up to twist) over $$\mathbf{Q}$$ whose mod $$13$$ image of Galois is locally contained in the normalizer of a split Cartan, but whose global image is not.

Rosu Eugenia, UC Berkeley
Integers that can be written as the sum of two cubes
The Birch and Swinnerton-Dyer conjecture predicts that we have non-torsion rational points on an elliptic curve iff the $$L$$-function corresponding to the elliptic curve vanishes at 1. Thus BSD predicts that a positive integer $$N$$ is the sum of two cubes if $$L(E_N, 1)=0$$, where $$L(E_N, s)$$ is the $$L$$-function corresponding to the elliptic curve $$E_N: x^3+y^3=N$$. We have computed several formulas that relate $$L(E_N, 1)$$ to the trace of a modular function at a CM point. This offers a criterion for when the integer $$N$$ is the sum of two cubes. Furthermore, when $$L(E_N, 1)$$ is nonzero we get a formula for the number of elements in the Tate-Shafarevich group.

Claire Frechette, Brown University
Combinatorial Properties of Rogers-Ramanujan-Type Identities Arising from Hall-Littlewood Polynomials
Here we consider the $$q$$-series coming from the Hall-Littlewood polynomials, defined by Griffin, Ono, and Warnaar in their work on the framework of the Rogers-Ramanujan identities. We devise a recursive method for computing the coefficients of these series when they arise within the Rogers-Ramanujan framework. Furthermore, we study the congruence properties of certain quotients and products of these series, generalizing the famous Ramanujan congruences.

T. Alden Gassert, University of Colorado
Let $$f(x) = x^2+c \in \mathbb Z[x]$$, and let $$K$$ be a number field generated by a root of $$f^n(x)$$ (assuming $$f^n(x)$$ is irreducible). The purpose of this work is to determine the multiplicities of primes dividing the discriminant of $$K$$. As a consequence of our result, we identify a sufficient condition for $$K$$ to be monogenic. Namely, $$K$$ is monogenic if $$f(0)$$, $$f^2(0)$$, $$f^3(0),\ldots, f^n(0)$$ are all square-free.

Heidi Goodson, University of Minnesota - Twin Cities
Hypergeometric Point Counts for Dwork K3-Surfaces
In 1995, Koike showed that the trace of Frobenius for elliptic curves in the Legendre family can be expressed in terms of Greene's finite field hypergeometric series. Further connections between hypergeometric series and algebraic varieties have been studied since then, though the focus has largely been on elliptic curves and Calabi-Yau threefolds. We extend this work by showing that the number of points on the family of Dwork K3-surfaces over finite fields can be expressed in terms of Greene's finite field hypergeometric series.

Richard Griffon, Institut Mathématique de Jussieu
A Brauer-Siegel theorem for elliptic curves over function fields
The Brauer-Siegel theorem bounds the product of the regulator and the class number of a number field in terms of its discriminant. It can be seen as an measure of the "arithmetic complexity" of this number field. Now consider an elliptic curve $$E$$ defined over a global field, assuming its Tate-Shafarevich group is finite, one can form the product of the order of this group and the Néron-Tate regulator of $$E$$. Heuristically, this product measures the complexity of computing the Mordell-Weil group of $$E$$. This prompts the question of bouding this quantity in terms of simpler invariants of $$E$$, e.g. its height. Unfortunately, it seems unlikely that a perfect analogue of the classical Brauer-Siegel theorem holds in this setting. This poster describes my investigations on the behaviour of this "Brauer-Siegel ratio" for a family of twists of an elliptic curve over a function field in positive characteristic.

Joseph Gunther, City University of New York (CUNY)
Embedding Curves into Surfaces Over Finite Fields
When can a curve, possibly singular, be embedded into some smooth surface, defined over the same field? This was answered elegantly by Altman and Kleiman in the 1970s for any infinite perfect field. We extend their result to curves defined over finite fields. The methods involved are quite different; the key tool used is Poonen's geometric closed-point sieve.

Sebastián Herrero, Pontificia Universidad Católica De Chile
Let $$x$$ be a point in the modular curve of level one $$X_0(1)$$ over the complex numbers such that $$x$$ is not a cusp. A well known result states that the Hecke points $$T_n(x)$$ associated to $$x$$ become equidistributed, with respect to the hyperbolic measure, as $$n$$ goes to infinity. In the case of the modular curve over $$\mathbb{C}_p$$ the situation is rather different. In most cases we have equidistribution of $$T_n(x)$$ to the canonical point of a certain Berkovich space associated to $$X_0(1)$$. The type of reduction of $$x$$ mod $$p$$ plays an important role in this setting. This is joint work with Ricardo Menares and Juan Rivera Letelier.

Hideaki Ikoma, University of Tokyo
On the arithmetic restricted volumes and arithmetic base loci
We study fundamental properties of the arithmetic restricted volumes and the arithmetic multiplicities of the adelically metrized line bundles. The arithmetic restricted volumes have the concavity property and characterize the arithmetic augmented base loci as the null loci. We also show a generalized Fujita approximation for the arithmetic restricted volumes.

Kenneth S. Jacobs, University of Georgia
A Logarithmic Equidistribution Result in Non-Archimedean Dynamics
Let $$K$$ be a complete, algebraically closed, non-Archimedean valued field, and let $$\phi$$ be a rational map of degree $$d\gt1$$ defined over $$K$$. Rumely has defined a discrete probability measure, called the crucial measure, supported on the interior of the Berkovich line over $$K$$ which carries information about the reduction of $$\phi$$. In this poster we show that the crucial measures attached to the iterates of $$\phi$$ converge weakly to the equilibrium measure $$\mu_\phi$$ when integrated against functions with logarithmic singularities.

Shanta Laishram, Indian Statistical Institute, New Delhi, India
Perfect Powers in products of Elliptic Divisibility Sequences
In a joint work with L. Hajdu and M. Szikszai, we show that there are only finitely many explicitly computable solutions for the equation given by the products of terms of Elliptic Divisibility Sequences with indices in arithmetic progression being a perfect power. Our method also gives a way to solve such equations completely and we show it by some examples.

Junghun Lee, Nagoya University
J-Stability in non-Archimedean dynamics
Roughly speaking, $$J$$-stability means that the dynamical systems on the Julia sets of two given rational maps are dynamically equivalent if those two rational maps are close enough. This notion was introduced by R. Mane, P. Sad, and D. Sullivan in complex dynamics. They also proved a $$J$$-stability theorem, which states that a rational map is $$J$$-stable if it has a neighborhood in the set of rational maps on which the number of attracting cycles is constant. In this talk, we will see an analogue of R. Mane, P. Sad, and D. Sullivan's $$J$$-stability theorem of immediately expanding rational maps in non-Archimedean dynamics.

Francesca Malagoli, Università di Pisa
Polynomial analogues of Zaremba's and McMullen's conjectures
Let $$L$$ be the field of Laurent series over a field $$K$$; as shown by Artin, an analogue of the theory of continued fractions can be developed over $$L$$. We will consider the analogues of Zaremba’s and McMullen’s conjectures. The former states that for any polynomial $$f$$ there exists a polynomial $$g$$, relatively prime to $$f$$, such that all the partial quotients of $$f/g$$ have degree 1. The second one states that for any polynomial $$D$$ whose square root is well defined there exists $$r$$ in $$K(T,\sqrt D)$$ such that all the partial quotients of $$r$$ have degree 1. We have that Zaremba’s conjecture holds if $$K$$ is infinite and that McMullen’s holds if $$K$$ is uncountable and if $$K$$ is the algebraic closure of $$\mathbb{Q}$$. Moreover, if Zaremba’s conjecture holds over finite fields, then McMullen’s conjecture holds over finite fields and over $$\mathbb{Q}$$.

Travis Morrison, Penn State University
Diophantine Sets in Global Fields
We give a new proof that the set of non-squares in a global field are diophantine, which was originally shown by Poonen. We also show that the set of elements which are not norms from a quadratic extension are diophantine. This is based on work by Koenigsmann and Park.

Cara Mullen, University of Illinois, Chicago
In complex dynamics, Hubbard trees offer a combinatorial description of the dynamics of post-critically finite (PCF) polynomials. What are the analogous objects in a non-Archimedean setting: what is a p-adic Hubbard tree? We begin to explore this question by studying the critical orbit trees associated to quadratic maps $$f_c(z)=z^2+c$$, with $$c\in\mathbb{Z}_p$$ (for $$p>2$$).

Bharathwaj Palvannan, University of Washington
The classical main conjecture in Iwasawa theory (formulated over a regular local ring) predicts that the characteristic ideal of a Selmer group is generated by a $$p$$-adic $$L$$-function. The main conjecture of Iwasawa theory formulated by Ralph Greenberg predicts a precise relationship between Selmer groups and $$p$$-adic $$L$$-functions but generalized to arbitrary normal domains that appear as deformation rings. We wish to exhibit algebraic analog of a phenomenon involving factorization of $$p$$-adic $$L$$-functions that occurs on the analytic side. As an example, results of Dasgupta involve factoring a certain three-variable $$p$$-adic $$L$$-function (constructed by Hida) into a product of two $$p$$-adic $$L$$-functions (constructed by Coates-Schmidt and Kubota-Leopoldt respectively) along a certain plane. We wish to exhibit the corresponding result involving Selmer groups that is consistent with the main conjecture.

Ashwath Rabindranath, University of Michigan
Some surfaces with non-polyhedral nef cones
We present a criterion for the cone of curves of a complex, smooth, projective surface to be non-polyhedral. In particular, we use this to prove that the cone of curves of $$C \times C$$ where $$C$$ is a curve of genus $$\ge2$$ is not polyhedral. The proof technique uses a (modified) construction of nef classes due to Vojta.

Vishal Saraswat, C.R.Rao Advanced Institute of Mathematics Statistics and Computer Science
Strengthening NTRU against message recovery attacks
There are two basic attacks on the NTRU cryptosystem: 1. ciphertext decryption attack (using lattice reduction to recover the plaintext from the ciphertext), and 2. key recovery attack (using lattice reduction to recover the secret key from the public key). In the most basic form, the complexity of the second attack is about the square of the first attack. We propose a twist in NTRU to increase the complexity of the message recovery attacks to be the same as that of the key recovery attack.

Joel Specter, Northwestern University
Commuting Endomorphisms of the $$p$$-adic Unit Disk
When can a pair of endomorphisms of $$\mathbf{Z}_p[[X]]/\mathbf{Z}_p$$ commute? Approaching this problem from the vantage point of dynamics on the $$p$$-adic unit disk, Lubin proved that whenever a non-invertible endomorphism $$f$$ commutes with a non-torsion automorphism $$u$$, the pair $$f$$ and $$u$$ exhibit many of the same properties as endomorphisms of a formal group over $$\mathbf{Z}_p.$$ Because of this, he posited that for such a pair of endomorphisms to exist, there in fact had to be a formal group 'somehow in the background.' My poster discusses how some of the dynamical systems of Lubin occur naturally as the restriction of the Galois action on certain Fontaine period rings and how one can use this observation to construct, in some cases, the formal groups conjectured by Lubin.

Conductors and discriminants for hyperelliptic curves
The Ogg-Saito formula relates the conductor of the minimal proper regular model of an elliptic curve over a local field (which is a certain numerical invariant that can be computed from the Galois action on étale cohomology groups) with the valuation of the discriminant of a minimal Weierstrass equation for the curve. The definition of the conductor naturally extends to models of curves of higher genus as well. Deligne defined a certain discriminant attached to regular models of curves, and Saito showed that the conductor equals the discriminant in this setting as well. Upper bounds on the conductor of a regular model can be used to give upper bounds on the number of components of the special fiber of the model, and this has applications to the study of rational points (Chabauty's method). We show that Deligne's discriminant for the minimal proper regular model of a hyperelliptic curve with rational Weierstrass points, over a local field with perfect residue field (and residue characteristic either zero or large enough compared to the genus) is bounded above by the valuation of the discriminant of an integral Weierstrass equation for the curve.

On the distribution of zeros of the derivatives of the Riemann zeta function and Dirichlet $$L$$-functions
The number of zeros and the distribution of the real part of non-real zeros of the derivatives of the Riemann zeta function have been investigated by Berndt, Levinson, Montgomery, and Akatsuka. Berndt, Levinson, and Montgomery investigated the general case, meanwhile Akatsuka gave sharper estimates for the first derivative of the Riemann zeta function under the truth of the Riemann hypothesis. Many properties of the zeros of the derivatives of the Dirichlet $$L$$-functions associated with primitive Dirichlet characters were studied by Yildirim. Among, Yildirim also studied the number of zeros. In this talk, we introduce a generalization of the results of Akatsuka to the $$k$$-th derivative (for positive integer $$k$$) of the Riemann zeta function. We also give a sharper estimate to the result of Yildirim on the number of zeros for the first derivative of the Dirichlet $$L$$-functions associated with primitive Dirichlet characters under the assumption of the generalized Riemann hypothesis.

Mckenzie Rachel West, Emory University
The Brauer Manin Obstruction for Cubic Surfaces
"The Hasse principle asks whether solutions to an equation in a local field extend to those in a global field. This does not always happen, with the Brauer-Manin obstruction being a common explanation. It is conjectured by Colliot-Thelene and Sansuc that for cubic surfaces the Brauer-Manin obstruction explains every instance where the Hasse principle fails. In 1973, Birch and Swinnerton-Dyer gave some of the first examples of the failure of the Hasse princple for cubic surfaces $m\operatorname{Norm}_{L/\mathbb{Q}}(ax+by+\phi z+\psi w) = (cx+dy)\operatorname{Norm}_{k/\mathbb{Q}}(x+\theta y),$ for given fields $$L$$ and $$k$$ over $$\mathbb{Q}$$ and fixed constants $$m,\ a,\ b,\ c,\ d,\ \phi,\ \psi,$$ and $$\theta$$. They make a rough number theoretic argument for the Brauer Manin obstruction in the case that the Hasse principle fails, focusing on the particular fields and constants. We make use of advancements in geometry and class field theory, taking a geometric look at this object and utilizing the correspondence between the Brauer group and the Picard group of a surface. "

Michael Wijaya, Dartmouth College
A function-field analogue of Conway's topograph
Conway's topograph is a visual method to display values of an integral binary quadratic form over $$\mathbb{Z}$$. This method leads to a simple and elegant method of classifying all integral binary quadratic forms and answering some basic questions about them. Let $$\mathbb{F}_{q}$$ be a finite field with odd characteristic, $$A = \mathbb{F}_{q}[T]$$, and $$\widehat{K} = \mathbb{F}_{q}((T^{-1}))$$. We develop an analogue of Conway's topograph for binary quadratic forms over $$A$$ by exploiting the connection between Conway's topograph and hyperbolic geometry. Following Paulin, we use the Bruhat–Tits tree $$\mathcal{T}_{q+1}$$ of $$\operatorname{SL}_{2}(\widehat{K})$$ as our function-field analogue of the hyperbolic plane. After we recast the underlying infrastructure of Conway's topograph in terms of constructions on $$\mathcal{T}_{q+1}$$, we formulate and prove an analogue of Conway's climbing lemma. We then show that just as in the classical setting, there is a unique "well" (respectively, "river") on the topograph of any definite (respectively, indefinite) binary quadratic form over $$A$$.

Ka Lun Wong, University of Hawaii at Manoa
Zagier's sums of powers of quadratic polynomials when the discriminants are negative
Zagier studied some functions defined as sums of powers of quadratic polynomials with integer coefficients and discovered that these functions have several surprising properties and are related to many other subjects, including modular forms of weight $$2k$$ and special values of zeta functions. Zagier mentions that his definition does not work well and/or becomes unnatural when $$k$$ is odd. We redefine Zagier's sums by changing the summation condition. That allows us to consider splittings of positive discriminants of the quadratic forms under summation into products of two (positive or negative) discriminants. Finally, our sums, while essentially coincide with those of Zagier in the case when $$k$$ is even, allow us to cover in a similar way the case when $$k$$ is odd.

Tian An Wong, CUNY Graduate Center
Sums of zeroes in the Arthur-Selberg Trace Formula
The Arthur-Selberg trace formula for a noncompact reductive group involves logarithmic derivatives of automorphic $$L$$-functions, which may be rewritten as certain sums of zeroes of $$L$$-functions. This suggests studying the distribution of zeroes using the trace formula. I will describe this in the simplest case, which is $$\operatorname{GL}(2)$$.

Fan Zhou, Ohio State University
Generalization of Voronoi Formula and a New Proof for $$\operatorname{GL}(3)$$
We discover new Voronoi formulae for automorphic forms on $$\operatorname{GL}(n)$$ for $$n\ge4$$. There are $$[n/2]$$ different Voronoi formulae on $$\operatorname{GL}(n)$$, which are Poisson summation formulae weighted by Fourier coefficients of the automorphic form with twists by some hyper-Kloosterman sums.

Huilin Zhu, Xiamen University and UBC
A generalized conjecture of ternary pure exponential Diophantine equation in ring of integers
Let $$a,b,c$$ be positive integers. In 1933, K. Mahler used his $$p$$-adic analogue of the Thue-Siegel method to prove that the ternary pure exponential Diophantine equation $$a^{x}+b^{y}=c^{z}$$ has only finitely many solutions $$(x,y,z)$$. His method is ineffective in the sense that it gives no indication on the number of possible solutions. In 1940, an effective result for the number of the solutions was given by A. O. Gel'fond. In 1956, L. Jesmanowicz conjectured that if $$a^{2}+b^{2}=c^{2}$$, then the Diophantine equation $$a^{x}+b^{y}=c^{z}$$ has only one integer solution $$(x,y,z)=(2,2,2)$$. In 1994, N. Terai conjectured that $$a^{x}+b^{y}=c^{z}$$ always has at most one positive integer solution $$(x,y,z)$$, especially when $$a^{p}+b^{q}=c^{r}$$, $$a,b,c,p,q,r\gt 1$$ are fixed integers, $$a^{x}+b^{y}=c^{z}$$ has exactly one positive integral solution $$(x,y,z)=(p,q,r)$$. L. J. Alex pointed that Terai's conjecture is clearly false. For example the equation $$2^{x}+2^{y}=2^{z}$$ has infinitely many solutions of the form $$(x,y,z)=(k,k,k+1)$$, $$k\gt1$$. The condition $$\gcd(a,b)=1$$ should be added to the hypotheses of the conjecture. In 1999, simple counterexamples to this statement have been found by Z.-F. Cao, who suggested that the condition $$\max\{a,b,c\}\gt7$$ should be added to the hypotheses of Terai's conjecture. However, it turns out that this condition is not sufficient to ensure the correctness. In 2003, a family of counterexamples has been found by M.-H. Le, that is, $$(2^{n}-1)^{x}+2^{y}=(2^{n}+1)^{z}$$ has two positive integer solutions $$(x,y,z)=(1,1,1)$$ and $$(2,n+1,2)$$. Now the accepted Terai conjecture is the following: for any fixed and coprime integers $$a$$ ,$$b$$ and $$c$$ greater than $$1$$, $$a^{x}+b^{y}=c^{z}$$ has at most one solution $$(x,y,z)$$ with $$\min\{x,y,z\}\gt1$$. In 2006, N. Hirata-Kohno proved $$a^{x}+b^{y}=c^{z}$$ has at most $$2^{32}$$ positive integer solutions. We will give a generalized conjecture on ternary pure exponential Diophantine equation $$a^{x}+b^{y}=c^{z}$$ in ring of integers, that is to say, if we allow $$x,y,z$$ to be negative or zero or positive integers and $$a,b,c$$ to be coprime or not coprime, we will describe an upper bound for the number of solutions of $$a^{x}+b^{y}=c^{z}$$ according to discussion in some cases.