William J. The Theory of Numbers Higher Mathematics. Robert Carmichael. Number Theory. Analytic Number Theory. Carl Pomerance. Periods and Special Functions in Transcendence. Paula Tretkoff. Women in Numbers Europe. Mathematics for the Liberal Arts. Donald Bindner. Advances in the Theory of Numbers. Mathematics in the 21st Century. Pierre Cartier. Fundamentals of Hopf Algebras. Robert G. Open Problems in Mathematics. John Forbes Nash. Discriminant Equations in Diophantine Number Theory. Jan-Hendrik Evertse. Recent Advances in Hodge Theory. Matt Kerr.

How Numbers Work. New Scientist. Elsa Abbena. New Directions in Locally Compact Groups. Pierre-Emmanuel Caprace. A Panorama of Discrepancy Theory. Anand Srivastav. Hershel M. Christian Elsholtz. Number Treasury3. Margaret J Kenney. Calyampudi R. A Course on Basic Model Theory. Haimanti Sarbadhikari. Fundamentals of the theory of operator algebras. Michael Th. Drinfeld Moduli Schemes and Automorphic Forms. Yuval Z Flicker. Nat Reed. Singularities of the Minimal Model Program. Theory of H[superscript p] spaces. Codes on Euclidean Spheres. Beauville Surfaces and Groups.

Ingrid Bauer. Contributions in Analytic and Algebraic Number Theory. Valentin Blomer. Teo Mora. Jay Jorgenson. Iwasawa Theory Otmar Venjakob. Abstract We consider a system of k diagonal polynomials of degrees 1, 2, Using methods developed by W. Paul Pollack. Abstract Let s n denote the sum of the proper divisors of n, so, e. For example, 6 is a perfect number, and is an amicable number. Questions about perfect and amicable numbers constitute some of the oldest unsolved problems in mathematics. I will talk about old and new theorems concerning these numbers and their generalizations.

Abstract We consider two classical number theoretic problems that may seem quite. This observation suggest we might want to consider:. In , Masser and Zannier proved a result similar in the spirit of Lang's result for torsion points on a family of elliptic curves. In our talk we explain how both results come from the same general principle in arithmetic geometry, and at the same time we present a partial result to a more general conjecture which subsumes both Lang and Masser-Zannier theorems. Sujatha Ramdorai. Iwasawa theory studies certain arithmetic modules arising from such representations.

Hida theory provides a technique to package some of these representations in a family and study them simultaneously. Nick Harland. Abstract A primitive set is a set of positive integers with the property that no element of the set divides another. We go on to describe two new theorems on primitive sets. Karl Rubin. Abstract I will discuss some recent results joint with Zev Klagsbrun and Barry Mazur on the distribution of 2-Selmer ranks in families of quadratic twists of elliptic curves over arbitrary number fields.

We study the density of twists with a given 2-Selmer rank, and obtain some surprising results on the fraction of twists with 2-Selmer rank of given parity. Since the 2-Selmer rank is an upper bound for the Mordell-Weil rank, this work has consequences for Mordell-Weil ranks in families of quadratic twists. Jeanine Van Order. Paul Mezo. Abstract Part of the Langlands Program is to find a meaningful correspondence between representations of Galois groups and representations of reductive algebraic groups.

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I will attempt to motivate this through an example and then concentrate on what happens at a real Archimedean place of the global picture. In this context the idea of endoscopy arises in a natural fashion and suggests identities between representations of different Lie groups. These identities have been proven by Shelstad. I will sketch the theory of endoscopy under twisting by a group automorphism and describe character identities between discrete series representations.

We consider primitive prime divisors of zero orbits of polynomials and restrict to polynomials with zero linear term.

### Citas duplicadas

We then consider a generalized notion of primitive prime divisors as it applies to families of commutative polynomials. We consider families of commutative polynomials which all have zero linear term and give an effective bound on the number of terms in the forward orbit of 0 which do not have primitive prime divisors. This talk is based on joint work with Jason Bell and builds on previous joint work with Anna Haensch. Karen Yeats. Let p be prime, take the Kirchhoff polynomial of a graph, and count points on the variety of this polynomial over the finite field with p elements.

This invariant has important things to say about the Feynman integrals of scalar Feynman graphs, and links together the combinatorial and algebro-geometric approaches to understanding Feynman integrals. Tom Scanlon. Other anomalous intersections may arise as sums of such orbits. Our proof which was long but elementary yields bounds which explicitly depend on the characteristic. In these lectures, I shall explain how to deduce characteristic independent bounds from a differential algebraic argument.

Let A be a finite, nonempty set of positive integers. Imin Chen. Bennett, S. Dahmen, S. Himadri Ganguli. Adrian Belshaw. This is joint work with Peter Borwein. Jonathan Bober. Abstract I'll give a historical overview of computations of the zeta function on the critical line and then describe some recent computations from the past year or so.

David Roe. In this talk I will outline some number theoretic questions in which these problems arise, and describe joint work with Xavier Caruso in which we propose a general methodology for approaching them in practice. Gregory Margulis. Anna Haensch. Abstract A fundamental question in the study of integral quadratic forms is the representation problem which asks for an effective determination of the set of integers represented by a given quadratic form. A slightly different, but equally interesting problem, is the representation problem for inhomogeneous quadratic forms. In this talk, we will discuss a characterization of positive definite almost universal ternary inhomogeneous quadratic forms which satisfy some mild arithmetic conditions.

Using these general results, we will then characterize almost universal ternary sums of polygonal numbers. Alia Hamieh. In this talk, we report on a work in progress to establish some generalizations of such results. Lola Thompson. Abstract A polynomial is a product of distinct cyclotomic polynomials if and only if it is a divisor over Z [ x ] of x n —1 for some positive integer n. Asif Zaman.

Abstract Quantum Unique Ergodicity has been a widely studied conjecture of Rudnick and Sarnak , concerning the distribution of large frequency eigenstates on a negatively curved manifold. Work of Lindenstrauss combined with the elimination of escape of mass proved by Soundararajan confirmed AQUE for the classical modular surface. This talk is concerned with AQUE for Hilbert modular surfaces, and in particular, my thesis work involving the elimination of escape of mass in this case. A recursive algorithm for computing the dimensions of these spaces of newforms follows from the combination of these two results, but it is desirable to have a formula in closed form for these dimensions.

This formula is much more amenable to analysis and to computation. For example, we derive asymptotically sharp upper and lower bounds for these dimensions, and we compute their average orders. Brian Conrad. Abstract The theory of reductive groups has many applications in number theory, geometry, and representation theory. We will explain the motivation for this with examples , and discuss the structure theory that has been established in recent years, and mention some applications. If time permits, we'll discuss some more recent developments. This is joint work with O.

Gabber and G. Note for Attendees Please note the nonstandard starting time for this seminar. Eva Bayer Fluckiger. Abstract The Euclidean division is a basic tool when dealing with the ordinary integers. It does not extend to rings of integers of algebraic number fields in general. It is natural to ask how to measure the "deviation" from the Euclidean property, and this leads to the notion of Euclidean minimum. The case of totally real number fields is of special interest, in particular because of a conjectured upper bound conjecture attributed to Minkowski.

The talk will present some recent results, obtained jointly with Piotr Maciak. Abstract Berkovich's rigid analytic spaces are path-connected, Hausdorff, locally compact spaces that generalize non-archimedean fields in a way that allows conducting analysis. We use them to prove non-archimedean analogs of results in complex dynamics. We show that a similar, but not identical, result holds over non-archimedean fields, with applications to both global and local non-archimedean dynamics.

Abstract Let K be a field of characteristic zero and let k and l be two multiplicatively independent positive integers. We prove the following result: a power series F x in K [[ x ]] satisfies both a k - and a l -Mahler type functional equation if and only if it is a rational function. This proves a conjecture of Loxton and van der Poorten. This is joint work with Boris Adamczewski. Romyar Sharifi.

Abstract I will discuss a conjecture which provides a relationship between Manin symbols in the homology of modular curves and cup products of cyclotomic units in Galois cohomology with restricted ramification. I hope to be able to give something of an idea of its meaning, with an eye towards generalization. I will also relate recent progress by Fukaya and Kato that essentially proves the conjecture under a mild but difficult to remove hypothesis.

We also introduce a notion of the descending chain condition DCC for sequences of curves, and prove that there are sequences of generalized Mordell curves and of generalized Fermat curves satisfying DCC. Peter Schneider. I will sketch a solution for the Lie algebra of matrices. Daniel Katz. This is equivalent to the problem of finding binary sequences with small mean-squared aperiodic autocorrelation, important in engineering and physics.

The best known examples are polynomials whose coefficients are supplied by complex-valued characters of finite fields. In , Borwein, Choi, and Jedwab analyzed a construction that reliably produces polynomials with a low ratio of norms, and they conjectured that these polynomials do indeed break the record. We prove that their conjecture is true, and the character sum methods we have devised settle further conjectures. Ben Lundell. Abstract This informal talk will provide background material for graduate students and others who will be attending the seminar talk "Level-lowering for Galois representations" later today.

Modular forms have a weight and a level. We can attach a Galois representation to an eigenform. In this talk we will examine these three items in more detail. Note for Attendees Attendees are welcome to bring their lunch with them to this informal talk. Abstract About 25 years ago, Ribet proved his famous level lowering result, which is an existence statement about congruences between modular forms of different levels.

In this talk, I'll survey some recent progress towards giving a new proof of Ribet's result without any modularity assumptions. In place of a modular form, we start with a p -adic Galois representation, and in place of the level, we consider the conductor of this representation. Yiannis Sakellaridis. Abstract The study of periods of automorphic forms, and their relations with L -functions, has for a long time been regarded as a field separate from the mainstream of the Langlands program, a collection of fortunate coincidences allowing us to get our hands on difficult arithmetic objects.

The work of Jacquet and others, however, has continuously emphasized the relation of periods to functoriality: the nonvanishing of H -period integrals of G -automorphic forms where H is a spherical subgroup of G should detect functorial lifts from some other group to G. I will give an overview of this, mostly conjectural, program. Click here for abstract. Rachel Ollivier. Abstract Let F be a locally compact non archimedean field with residue characteristic p and G the group of F -points of a split connected reductive group.

Let k be an algebraically closed field of characteristic p. We are interested in the link between the k -representations of the spherical and affine Hecke algebras associated to G and the smooth k -representations of G. In particular, the so-called supersingular representations of G and the corresponding supersingular Hecke modules are still poorly understood.

However, these are the representations which are expected to play a prominent role in a potential mod p local Langlands correspondence for G. In the pro- p Iwahori-Hecke k -algebra H , we define a family of commutative subalgebras, each containing the center of H. We use these subalgebras to do the following: 1. Construct an inverse Satake isomorphism one can subsequently show that this is the inverse of the Satake isomorphism defined by Herzig. Prove that the center of H contains an affine semigroup algebra which is naturally isomorphic to the spherical Hecke algebra attached to an irreducible smooth k -representation of a given hyperspecial maximal compact subgroup of G.

## Theory of Numbers

Apply this to study the "supersingular block" of the category of finite length H -modules and relate it to supersingular representations of G. Abstract I'll give a report on joint work with Greenberg on the Iwasawa theory for modular forms of weight 1. Abstract In this talk, we'll survey the state-of-the-art on this and related topics, including recent techniques for solving such equations in more than two variables. Abstract I will first review some ideas of algebraic number theory from a computational point of view, and then address the computational complexity of the problems of determining whether an ideal A in the ring of integers of a fixed algebraic number field K is prime, and of finding the prime factorization of A.

Specifically, I will answer questions of Gil Kalai by giving polynomial-time reductions for the problems of determining whether A is prime and finding the prime factorization of A to the corresponding problems over the rational integers. I will then discuss the problem of factoring an algebraic integer into irreducibles and conclude with the problem of irreducibility testing. Sudhanshu Shekhar. Some of these results have been proved. Laurent Berger. Abstract I will recall the important features of the p -adic local Langlands correspondence for GL 2 Q p.

Extending this correspondence to other groups seems to require doing p -adic Hodge theory in a slightly different way. I will explain the new features that arise when one does this, in the simplest setting. Otmar Venjakob. Vorrapan Chandee. We obtain the constant 24, as a factor in the leading-order term of the eighth moment, which is as predicted for the eighth moment of the Riemann zeta function. This talk is based on joint work with Xiannan Li. Xiannan Li. Abstract The local Shimura correspondence relates representations of PGL 2 to those of the metaplectic cover of SL 2 , where both groups are p -adic and the representations are on C -vector spaces.

Many pieces of the usual construction break down, due to non-semisimplicity, when the representations are instead taken over a field of positive characteristic. The mod p case is especially problematic. The focus will be on the nonsupercuspical representations; I'll classify these for the cover of SL 2 , and relate them to the genuine spherical and Iwahori Hecke algebras. Abstract Let k be a positive integer that is congruent to 3 mod 4 , and let N be a positive square-free integer. This is accomplished by using a result of Waldspurger, which allows one to produce a basis for the forms that correspond to a given F via local considerations, once a form in the Kohnen space has been determined.

Ben Howard. Abstract For a weight two modular form f , the Gross-Zagier theorem is a formula relating two things: the central derivative of the convolution L -function of f with a weight-one theta series, and the Neron-Tate pairing of a Heegner point with itself. I'll discuss a generalization to higher weight modular forms, where the Heegner point is replaced by certain special cycles on a unitary Shimura variety. This is joint work with Jan Bruinier and Tonghai Yang.

Eric Naslund. Please see attached abstract file. Akos Magyar. Abstract We study the number of solutions of diophantine equations f x 1 , It has been established by Birch and Schmidt that one has the expected number of integer solutions if f is a homogeneous integral polymomial of sufficiently large rank with respect to its degree. We show that the same phenomenon holds when the variables are restricted to primes, extending the results of Hua for diagonal forms. We illustrate some of the ideas on quadratic forms and discuss some elements of the proof of the general case.

Jeff Achter. Abstract It occasionally happens that, for a certain type of complete intersection, the associated complex moduli space is actually open in an arithmetic quotient of a complex ball. I will describe recent work, particularly involving the case of cubic surfaces, which suggests that this unexpected structure is the complex realization of a morphism of integral moduli spaces. A modest payoff of the general theory is the calculation, in characteristic at least five, of the Galois group of the 27 lines on a sufficiently general cubic surface.

Michael Coons. Abstract In this talk, we survey past, present, and possible future results concerning the arithmetic nature of low complexity sequences. For example, what properties can be exhibited by numbers whose base expansion can be determined by a finite automaton? In the current context, this line of questioning was unknowingly initiated by Mahler, and later championed by Loxton and van der Poorten following the work of Cobham and Mendes France.

In addition to describing some historical work, this talk will describe some of the the current advancements and generalisations concerning Mahler's method. Karol Koziol. Using the interplay between these two algebras, we deduce a correspondence between "packets" of Hecke modules and mod- p projective Galois representations. Abstract Let f be an irreducible polynomial over a number field. Under what conditions is it true that all iterates of f are also irreducible? We call polynomials with this property stable. Eventual stability is the weaker property that the number of factors of the n th iterate of f is bounded uniformly in n.

We can extend this definition to rational functions. We conjecture that all rational functions are eventually stable when 0 is not periodic, and show that this is a case for a large class of functions using Newton polygon techniques. Abstract We present analogues of the classical conjectures of Manin-Mumford, Bogomolov and Pink-Zilber for function fields of arbitrary characteristic.

We also present a function field analogue of the Bounded Height Conjecture which appears in the study of the Pink-Zilber Conjecture. Abstract In the early 70's Davenport and Heilbronn derived the leading term in the asymptotic formula for the number of cubic number fields with bounded discriminant.

However, as algorithmic data became available, a large "gap" became evident between the actual number of cubic number fields of small discriminant and the asymptotic prediction. We will discuss this and the analogous situation in the function field setting. We will present methods for constructing and tabulating dihedral function fields which includes non-Galois cubics and prove the existence of a similar "gap" for cubic function fields of small discriminant and the leading term of the corresponding asymptotic.

Parimala Raman. Milnor conjecture is a theorem due to Voevodsky, Orlov and Vishik. I will survey some parts of his program and point out the key difficulties which remain. Recently, numerous irreducibility criteria for the mod p representations attached to elliptic curves over totally real fields have been developed David, Billerey, Freitas-Dieulefait, Freitas-Siksek. These are based on a technique which first appeared in Serre's Inventiones paper.

Such an element in Sha is said to be made visible by E'. In joint work with Tom Fisher, we have been able to finally find equations for these K3 surfaces, which allows us to determine visibility computationally in specific cases. Tonghai Yang. In this talk, I will explain this story. We study the variation of the canonical heights for this particular family. This is joint work with Dragos Ghioca. Dong Quan Nguyen. Abstract We discuss and show how to construct certain curves C of small genus satisfying the following conditions: 1 C is everywhere locally solvable; 2 the Jacobian of C is isogenous to the product of elliptic curves; 3 each elliptic curve factor in the Jacobian of C is of positive rank; 4 there is a Brauer-Manin obstruction for the failure of the Hasse principle for C.

This is the joint work with Mike Bennett. Conan Wong. Abstract Wolfgang Schmidt's Strong Subspace Theorem is a less well-known generalisation of his Subspace Theorem, and has not been studied much since its formulation in It is a result about integer points in parallelepipeds whose successive minima satisfy a certain condition. Thus, unlike the Subspace Theorem and its other generalisations, it falls within the field of the geometry of numbers. This self-contained talk reintroduces the Strong Subspace Theorem and describes some new preliminary results, including a stronger reformulation of the theorem in dimension two.

Guillermo Mantilla-Soler. Abstract Inspired by the invariant of a number field given by its Dedekind zeta function we define the notion of weak arithmetic equivalence , and we show that under certain ramification hypothesis this equivalence determines the local root numbers of the number field.

This is analogous to a result Rohrlich on the local root numbers of a rational elliptic curve. Additionally we prove that for tame non-totally real number fields the integral trace form is invariant under weak arithmetic equivalence. We show that the set containing all positive integers n such that the n -th iterate of x under f lands in Y is a union of at most finitely many arithmetic progressions along with a set of Banach density 0.

This is joint work with Jason Bell and Tom Tucker. Sharon Frechette. Abstract We construct Weyl group multiple Dirichlet series associated to root systems of type C , through a combinatorial recipe involving Gelfand-Tsetlin patterns. We also prove that our description matches the so called "stable case," as described for general root systems by Brubaker, Bump and Friedberg.

This is joing work with Jennifer Beineke and Ben Brubaker. Abstract This talk will retrace the main steps of the modern theory of prime numbers and in particular how the combinatorial sieve combined with the Dirichlet series theory to give birth to the modern representation of the primes via a linear combination of terms, some of which being "linear", while the other ones are "bilinear". Ashay Burungale. We outline the proof of the non-triviality of the p -adic formal group logarithm of Heegner points modulo p associated to the Rankin-Selberg convolution of an elliptic modular form of weight two and a theta series over the Z l -anticyclotomic extension of K.

We also make remarks regarding the analogous non-triviality of generalised Heegner cycles. Abstract Let F be a totally real number field. In particular, I will describe a sheaf-function dictionary for quasicharacters of tori over local fields, and early progress toward a definition for the affine Grassmannian for reductive groups over K.

Bianca Viray. Alex Lubotzky. Abstract Ramanujan graphs are optimal expanders from spectral point of view. Explicit constructions of such graphs were given in the 80's as quotients of the Bruhat-Tits tree associated with GL 2 over a local field F , by suitable congruence subgroups. The spectral bounds were proved using works of Hecke, Deligne and Drinfeld on the "Ramanujan conjecture" in the theory of automorphic forms.

This gives finite simplical complxes which on one hand are "random like" and at the same time have strong symmetries. Recently various applications have been found in combinatorics, coding theory and in relation to Gromov's overlapping properties. We will describe these developments and give some details on recent applications. Khoa Nguyen. Please see attached abstract. Sinnou David. We shall explore the function field analogue of this problem. Dick Gross. I will then discuss results of Bhargava, Poonen, Stoll, Shankar and Wang, which show that most curves in these families have the minimal number of rational points.

Miljan Brakocevic. Abstract An Euler system is a collection of global arithmetic objects, most notably global cohomology classes arising from geometry, which are related to L -functions and can be made to vary in p -adic families. The talk will mainly focus on the Euler system of Hegneer points that played a key role in the seminal work of Gross—Zagier and Kolyvagin on the Birch and Swinnerton—Dyer Conjecture. We will present a construction of anticyclotomic p -adic Rankin—Selberg L -functions and discuss some related reciprocity laws in the spirit of Kato.

Abstract Let X be a genus 2 g curve defined over an arbitrary field of characteristic not equal to 2 and let J X the Jacobian variety of X.

## WHEN SETS CAN AND CANNOT HAVE SUM-DOMINANT SUBSETS

We say that a Jacobian variety is decomposable if it is isogenous to a product of abelian varieties. The type of decomposition can by characterized by the type of kernel of the isogeny and the dimensions of the varieties in the product. Additionally, we insist that the isogeny is polarized. In this talk we describe a family of non-hyperelliptic genus 2 g curves whose Jacobians are decomposable in this way.

We prove that all genus 4 curves whose Jacobian has this decomposition type are either in this family or arise from a different construction considered by Legendre. Joint work with Nils Bruin. Raman Parimala. There are several positive results in this direction for connected linear algebraic groups which are rational, thanks to the patching techniques developed by Harbater-Hartmann-Krashen.

Abstract Let R be a discrete valuation ring with fraction field F. Two algebraic objects say, quadratic forms defined over R are said to be rationally isomorphic if they become isomorphic after extending scalars to F. In the case of unimodular quadratic forms, it is a classical result that rational isomorphism is equivalent to isomorphism. This has been recently extended to "almost umimodular" forms by Auel, Parimala and Suresh. I will present further generalizations to related objects: hermitian forms over involutary R -algebras, quadratic spaces equipped with a group action "G-forms" , and systems of quadratic forms.

The results can be regarded as versions of the Grothendieck—Serre conjecture for certain non-reductive groups. Joint work with Eva Bayer—Fluckiger. Chung Pang Mok. Abstract Inter-universal Teichmuller theory, as developed by Mochizuki in the past decade, is an analogue for number fields of the classical Teichmuller theory, and also of the p -adic Teichmuller theory of Mochizuki. In this theory, the ring structure of a number field is subject to non-ring theoretic deformation.

Absolute anabelian geometry, a refinement of anabelian geometry, plays a crucial role in inter-universal Teichmuller theory. In this talk, we will try to give an introduction to these ideas. Nicolas Billerey. Abstract In this talk I will discuss modularity of reducible mod l Galois representations. By analogy with the irreducible case, I will state several questions regarding characterization and optimization of the weights and levels of the various cuspidal forms attached to such representations. Finally I'll give an application of these results to the determination of an explicit lower bound for the highest degree of the coefficient fields of newforms of prime level and trivial Nebentypus.

This is a joint work with Ricardo Menares. Julio Andrade. Abstract In this seminar I will discuss a function field analogue of a classical problem in analytic number theory, concerning the auto-correlations of divisor functions, in the limit of a large finite field. Cheng-Chiang Tsai. Abstract An admissible representation of a reductive p -adic group has its character as a distribution on the group, invariant under conjugation.

## Some conferences I've heard about

The asymptotic behavior of the character is given by so-called Harish-Chandra—Howe local character expansion, which expressed the character near the identity in terms of a finite linear combination of Fourier transforms of nilpotent orbital integrals. In this talk, we show examples about how the coefficients in this expansion arise as the numbers of rational points on varieties over the residue field, which will be certain covers of hyperelliptic curves in our example. We also talk about how the endoscopy transfer identity appears as geometric identities regarding the first cohomology of these curves.

Colin Weir. Abstract An elliptic curve in characteristic p is either ordinary or supersingular, depending on whether or not it has points of order p. It is known that elliptic curves are typically ordinary, and also exactly how many are super-singular for each prime p. However, for higher genus curves little is known. In this talk, we will discuss several higher genus generalizations of supersingular elliptic curves, focussing on the hyperelliptic case.

In particular we discuss recent heuristics, computational results, and theorems on the proportion of hyperelliptic curves that are non-ordinary. This allows us to perform 3, 3 -isogeny descent on various simple principally polarized abelian surfaces and exhibit non-trivial 3-part in their Tate-Shafarevich groups. This is joint work with Victor Flynn and Damiano Testa. Abstract Darmon outlined a program which is suited to potentially resolving one parameter families of generalized Fermat equations.

He gave explicit descriptions of Frey representations and conductor calculations for Fermat equations of signature p , p , r. Somewhat less explicit results are stated for signature r , r , p , and even less for signature q , r , p. For the equation r , r , p , there are at least three competing Frey curve constructions: superelliptic curves of hypergeometric type due to Darmon, hyperelliptic curves due to Kraus, and elliptic curves with models over totally real fields due to Freitas. I will survey these Frey curve constructions and end by giving explicit Frey hyperelliptic curves for signatures 2, r , p and 3,5, p.

Abstract I will describe a general technique to extend various results about non-degenerate quadratic and hermitian forms to systems of possibly degenerate hermitian forms. The idea is based on a certain categorical equivalence. Abstract Apollonius's Theorem states that given three mutually tangent circles, there are exactly two circles which are tangent to all three.

We derive an asymptotic formula for the number of Diophantine quadruples whose elements are bounded by x. In doing so, we extend two existing tools in ways that might be of independent interest. We also adapt an argument of Hooley on the equidistribution of solutions of polynomial congruences to handle reducible quadratic polynomials. Abstract Understanding the least quadratic nonresidue mod p is a classical problem, with a history stretching back to Gauss. The approach which has led to the strongest results uses character sums, objects which are ubiquitous in analytic number theory.

I will discuss character sums, their connection to the least nonresidue, and some recent work of myself and Jonathan Bober University of Bristol on a promising new approach to the problem. We give a partial result in the case of hyperbolic manifolds of dimension 4 and 5. This is a joint work with my supervisor Lior Silberman. Haruzo Hida. Dijana Kreso. Any polynomial of degree greater than 1 can clearly be represented as a composition of indecomposable polynomials. Such a representation, called a complete decomposition, does not need to be unique.

Ritt in the 's described the extent of the non-uniqueness for complex polynomials.

In so doing, Ritt exhibited some invariants of complete decompositions of complex polynomials. Building on the methods developed by Ritt, Fried and others, and by shifting to the setting of maps between curves, we extend and generalize known results on invariants of complete decompositions.

These results are obtained jointly with Michael Zieve. We further present some methods for showing indecomposability and discuss applications of such results to Diophantine equations. These methods are described in a joint survey paper with Robert F. Samir Siksek. Abstract In , Carl Jacobi made the experimental observation that all integers are sums of seven non negative cubes, with precisely 17 exceptions, the largest of which is Michael Mossinghoff.

This is joint work with Tim Trudgian. Abstract I will give an overview of things we know about c 2 invariant of a graph. This is an invariant investigated principally by Brown and Schnetz which comes from counting points on the hypersurface defined by the Kirchhoff polynomial of a graph. This invariant predicts many properties of the Feynman integral of the graph. It connects with deep things like modular forms and motives.

Many computations involving it come down to playing around with polynomials defined from the graph and so its also combinatorial. The fun and power of it come from the interplay of all three of these things. Abstract Let G be a connected reductive algebraic group defined over a number field. The harmonic analysis of the adelic points of G leads to a decomposition of the regular representation into automorphic representations.

The irreducible subrepresentations in this decomposition form the so-called discrete spectrum. Arthur has recently provided such a description for symplectic and special orthogonal groups in terms of sets of representations called A rthur -packets. The structure of A-packets is not well understood, and relies in part on their local analogues. We will outline a method for computing A-packets for real groups. Manin defined an obstruction in terms of the Brauer group to detect the failure of Hasse principle for varieties over number fields which is referred to as the Brauer-Manin obstruction.

This obstruction is the only obstruction to Hasse principle for torsors under connected linear algebraic groups over k. This obstruction can be used to produce examples of principal homogeneous spaces under tori which fail Hasse principle over function fields of p -adic curves. Abstract This will be largely a survey of classical results: I will recall the analytic class number formula, and the Minkowski—Siegel mass formula for the "number" of quadratic forms in a genus, as well as Tamagawa's reformulation of these results as a volume computation.

Then I will discuss a similar formula for the number of elliptic curves in an isogeny class, and we will see that it can again appear in two versions: one is due to Gekeler and comes from probabilistic and equidistribution considerations, and the other is due to Langlands and Kottwitz and is based on a volume computation. This talk is motivated by the joint project with Jeff Achter, Ali Altug and Luis Garcia, where we explore the connection between these two formulas and generalize Gekeler's result to counting principally polarized Abelian varieties.

The talk will be completely non-technical and is not aimed at the experts who would find most of it very familiar. Eknath Ghate. Bhattacharya and S. This reduces the computation to the automorphic side. Matilde Lalin. Abstract The Mahler measure of a multivariable polynomial P is given by the integral of log P where each of the variables moves on the unit circle and with respect to the Haar measure. In Boyd made a systematic numerical study of the Mahler measure of many polynomial families and found interesting conjectural relationships to special values of L -functions of elliptic curves.

I will discuss some of those results and present new ones in collaboration with D. Samart and W. Sheng-Chi Liu. This is joint work with Riad Masri and Matt Young. John Binder. Abstract Consider the set of Hecke eigenforms of a fixed weight k and very large level N , and pick a prime p coprime to N. How are the eigenvalues of the operator T p distributed? We will relate this question to a more abstract question about the distribution of local components of cuspidal automorphic representations. For a reductive algebraic group G defined over a p -adic field L , there is a measure, called the 'Plancherel measure', which is expected to serve as the limiting distribution under many circumstances.

We'll define this measure, give a precise formulation of the problem, and discuss progress in this area of research. No prior knowledge of reductive groups or automorphic forms will be assumed. Myrto Mavraki. Mohammad Bardestani. Abstract A recent result of Karpenko and Merkurjev states that the essential dimension of a p-group G over a field K containing a primitive p'th root of unity is equal to the minimal dimension of faithful representations of G over K.

Motivated by this result, it is then interesting to compute the minimal dimension of complex faithful representations of a given finite p-group. In this talk we will show how Lie algebraic method, namely Howe-Kirillov's orbit method, can be applied to answer this question for some classes of p-groups. In this talk, we will discuss a variety of approaches to this equation that share the common thread of failing to solve it. This is joint work with Aaron Levin. Abstract The recent work of Minhyong Kim provides us with a very general framework for addressing questions of Diophantine finiteness for hyperbolic curves in an effective way.

We will discuss some first steps towards making his strategy explicit for certain classes of higher genus curves, resulting in an effective version of Faltings' finiteness theorem. Together with the recent work of Jennifer Balakrishnan and Netan Dogra on bielliptic curves, these form the first examples of higher genus curves for which Kim's programme may be carried out. So far, it remains an open problem to exhibit a natural example of such a number. We present a number of algorithms that compute absolutely normal numbers in the sense that they successively output the digits to a given base of a real number that can be shown to be absolutely normal.

We analyze these algorithms with respect to computational complexity and speed of convergence to normality. Furthermore, we adapt one of the algorithms to the more general setting of Pisot numbers as bases and conclude with some open problems. Bin Xu. Abstract The irreducible smooth representations of Arthur class are the local components of automorphic representations. They are conjectured to be parametrized by the Arthur parameters, which form a subset of the usual Langlands parameters. The set of irreducible representations associated with a single Arthur parameter is called an Arthur packet.

Following Arthur's classification theory of automorphic representations of symplectic and orthogonal groups, the Arthur packets are now known in these cases. On the other hand, Moeglin independently constructed these packets in the p -adic case by using very different methods. In this talk, I would like to describe a combinatorial procedure to study the structure of the Arthur packets following the works of Moeglin.

As an application, we show the size of Arthur packets in these cases can be given by counting integral or half-integral points in certain polytopes. Abstract We will start by introducing Langlands parameters and their enhanced versions. We will illustrate our construction with simple examples. It is joint work with Ahmed Moussaoui and Maarten Solleveld. Dan Collins. Abstract "Which numbers and in particular, which primes are sums of two rational cubes" is a classical and still-not-entirely-solved Diophantine problem. I'll talk about how it turns into a problem about rational points on elliptic curves, and how it can then be attacked using the modern machinery of the arithmetic of elliptic curves.

In particular, proving that certain primes can be written as a sum of two cubes can be accomplished by constructing a Heegner-type point and proving it's nonzero. This is a subtle question and has been carried out in different ways by Elkies and by Dasgupta-Voight. My work in process gives a new method to carry this out, based on a new construction I've given of a certain type of anticyclotomic p-adic L-function.

Following the new format for the number theory seminar, this talk will consist of two minute parts. The first 45 minutes will be expository and is intended to be accessible for graduate students in number theory. There will be a short break when people are welcome to leave , and the second 45 minutes will be at a higher level. Michel Raibaut. This concept gives a better understanding of operations on distributions such as product or pullback and it plays an important role in the theory of partial differential equations.

In , Howe introduced a notion of wave front set for some Lie group representations and in , Heifetz gave an analogous version in the p-adic context. In this talk, in the t-adic context in characteristic zero, using Cluckers-Loeser motivic integration we will present analogous constructions of test functions, distributions and wave front sets.

In particular, we will explain how definability can be used as a substitute for topological compactness of the sphere in the real and p-adic contexts to obtain finiteness. This a joint work with R. Cluckers, and F. Lee Troupe. This talk concerns two instances where the answer is yes.

Abstract In the theory of automorphic representations the study of L-functions plays a key role. A common method to study the analytic behavior of such functions and, in fact, proving that they are meromorphic functions is the Rankin-Selberg method. In this method an integral representation, with good analytic properties, is attached to the L-function. Many examples of Rankin-Selberg integrals were studied along the years.

In a pioneering paper "A new-way to get Euler products", Krelle, I.

Piatetski-Shapiro and S. Rallis suggested a remarkable mechanism that makes it possible to use integrals containing a "non-unique model" by a slight strengthening of the unramified computations. In the second part of my talk I will present a joint work with N. In its unfolded form a term which will be explained in the talk , the integrals contain a non-unique model and we apply the new-way mechanism.

The unramified computation gives rise to two interesting objects: the generating function of the L-function and its approximations. Nuno Freitas. It's proof was completed in by the groundbreaking work of Andrew Wiles on the modularity of semistable elliptic curves over Q. From its proof a new revolutionary method to attack Diophantine equations was born.

Geo Kam-Fai Tam. Abstract Part I: Representations of reductive groups over local fields In this talk, we describe the representations of certain reductive groups over local fields and the representations of Weil groups. Then we review the class field theory for local fields and the Langlands correspondence for these reductive groups. The talk will be a brief overview of the representation theory of reductive groups over local fields, largely based on examples of low-rank groups. This work is partly joint with Corinne Blondel. Uri Shapira. Abstract I will discuss some aspects of the tight connection between the study of certain dynamical systems and number theory most notably to questions in Diophantine approximation and the geometry of numbers.

Kiran Kedlaya. Abstract As its name is meant to suggest, the subject of p-adic Hodge theory was historically concerned with the relationship between different cohomology theories attached to p-adic algebraic varieties. Within p-adic Hodge theory, the concept of a perfectoid space discussed in my PIMS lectures arose quite naturally and has led to improvements in the subject which were in some sense "expected".

However, it also had several "unexpected" applications rather far afield. We'll survey three of these: Deligne's weight-monodromy conjecture Scholze ; Galois representations associated to torsion cohomology of arithmetic groups Scholze, Caraiani-Scholze ; and the direct summand conjecture of commutative algebra Andre, Bhatt. Abstract Our work on modulo p smooth representations of reductive p-adic groups classification of irreducible admissible representations, functoriality properties,Satake homomorphisms, with Abe, Henniart, Herzig, Ollivier , raises questions for which a kind of answer is required if we want to venture further.

Niccolo' Ronchetti. Abstract Let E be an elliptic curve over the rationals. Hida and Y. Abstract Let K be a field and let P be a polynomial. What can we say about the field generated by the roots of P and of all its iterates? I will discuss some questions motivated by this general problem when K is a p-adic field. Along the way, we'll see Coleman power series, p-adic dynamical systems and a little bit of p-adic Hodge theory.

Sandra Rozensztajn. In particular, I will be interested in the following situation: fix Hodge-Tate weights and a residual representation, and consider the locus parametrizing crystalline representations with the given weights and reduction modulo p. What can be said about this locus in general? Ian Petrow. Abstract We consider the set of isomorphism classes of elliptic curves over a finite field k from a probabilistic point of view. In joint work with N. The formulas are necessarily complicated but quite usable in practice as one knows a lot about these spaces of modular forms.

Dima Jakobson. Abstract This is joint work with Frederic Naud Avignon. After reviewing general results about resonances on asymptotically hyperbolic manifolds, we discuss some recent results on the distribution of resonances for infinite index congruence subgroups of SL 2,Z , as well as some conjectures.

Pierre Colmez. Abstract We will present different ways to compute the p-adi etale cohomology of layers of the Drinfeld tower, and give applications to the p-adic local Langlands correspondence. Lecture 1 of 4. Jared Weinstein. Abstract On a complex variety, you can integrate a differential form over a cycle to get a period. For instance, an elliptic curve has two periods, whose quotient gives an element of the upper half plane. There is a family of concepts Hodge decomposition, variation of Hodge structures, Shimura varieties arising from the study of periods on families of complex varieties.

What if the complex variety is replaced with a rigid-analytic variety over a p-adic field? We will review work of Tate, Fontaine, Kedlaya-Liu, Scholze and others that falls under the domain of p-adic Hodge theory. One goal will be to understand the surprising Hodge-Tate period map, defined by Scholze, attached to the modular curve at infinite level. Lecture 2 of 4. Wieslawa Niziol. Lecture 3 of 4. Lecture 4 of 4. Abstract The classical theory of phi, Gamma -modules relates continuous p-adic representations of the Galois group of a p-adic field with modules over a certain mildly noncommutative ring.

We describe a method for applying a key property of perfectoid spaces, the analytic analogue of Drinfeld's lemma, to the construction of "multivariate phi, Gamma -modules" corresponding to p-adic Galois representations in more exotic ways. Based on joint work with Annie Carter and Gergely Zabradi. Abstract Suppose that rho is an irreducible automorphic n-dimensional global p-adic Galois representation that is upper-triangular locally at p. In work in progress we prove many new cases of this conjecture, assuming that rho is moreover crystalline with distinct Hodge-Tate weights. Bharathwaj Palvannan.

We will discuss certain non-primitive Iwasawa modules that have a free resolution of length one over appropriate Iwasawa algebras. This is joint work with Alexandra Nichifor. The possible ways of writing a polynomial as a composition of lower degree polynomials were studied by many authors. There are applications to several areas of mathematics. In my talk I will address about some Diophantine applications. This talk will highlight some of these results and the techniques used to obtain them, including recent work of the speaker and Greg Martin UBC.

Ken Ono. Borcherds won the Fields medal in for his proof of the Monstrous Moonshine Conjecture. Loosely speaking, the conjecture asserts that the representation theory of the Monster, the largest sporadic finite simple group, is dictated by the Fourier expansions of a distinguished set of modular functions. This conjecture arose from astonishing coincidences noticed by finite group theorists and arithmetic geometers in the s.

Recently, mathematical physicists have revisited moonshine, and they discovered evidence of undiscovered moonshine which some believe will have applications to string theory and 3d quantum gravity. The speaker and his collaborators have been developing the mathematical facets of this theory, and have proved the conjectures which have been formulated. These results include a proof of the Umbral Moonshine Conjecture, and Moonshine for the first sporadic finite simple group which does not occur as a subgroup or subquotient of the Monster. The most recent Moonshine announced here yields unexpected applications to the arithmetic elliptic curves thanks to theorems related to the Birch and Swinnerton-Dyer Conjecture and the Main Conjectures of Iwasawa theory for modular forms.

Abstract We study a generalization of the quadratic sieve to a geometric setting. We apply this to counting points of bounded height on an l-cyclic cover over the rational function field and we consider a question of Serre. In addition to the geometric quadratic sieve, we use Fourier analysis over function fields, deep results of Deligne and Katz about cancellation of mixed character sums over finite fields, and a bound on the number of points of bounded height due to Browning and Vishe.

This is joint work with A. Bucur, A. Cojocaru, and L. Nathan Ng. Abstract Hardy and Littlewood initiated the study of the 2k-th moments of the Riemann zeta function on the critical line. In Hardy and Littlewood established an asymptotic formula for the second moment and in Ingham established an asymptotic formula for the fourth moment. In this talk we consider the sixth moment of the zeta function on the critical line. We show that a conjectural formula for a certain family of ternary additive divisor sums implies an asymptotic formula for the sixth moment.

This builds on earlier work of Ivic and of Conrey-Gonek. Abstract Using p-adic local Langlands correspondence for GL2 Qp , we prove that the support of patched modules constructed by Caraiani, Emerton, Gee, Geraghty, Paskunas, and Shin meet every irreducible component of the potentially semistable deformation ring. As a consequence, a local restriction in the proof of Fontaine-Mazur conjecture by Kisin is removed. In Iwasawa Theory, we study the behaviours of E over a tower of number fields. For example, it is known that the Mordell Weil ranks of E over all p-power cyclotomic extensions of Q are bounded when p does not divide the conductor of E.

Surprisingly, the techniques required to show this are very different depending on the number of points on the finite curve when we consider E reduced modulo p. The easier case is when E has "ordinary" reduction at p and the more difficult case is when E has "supersingular" reduction at p. I will review the Iwasawa-theoretic tools used to study the behaviours of E over cyclotomic fields in these two cases.

I will also discuss some recent developments on the Iwasawa theory of elliptic curves over quadratic extensions of Q. Jasmin Matz. Abstract Analytic torsion is a classical invariant for compact Riemannian manifolds. The Cheeger-Mueller Theorem relates it to its combinatorial equivalent, the Reidemeister torsion. This can be exploited to study the torsion homology of certain arithmetic lattices as in recent work of Bergeron and Venkatesh. Joint work with W. Adrian Iovita. Abstract p-Adic modular forms have first been defined by J.

Serre as q-expansions and have later been interpreted geometrically by N. Katz as sections of certain modular line bundles over the ordinary locus of the relevant modular curves. Katz also defined overconvergent modular forms of integer weights as overconvergent sections of the modular line bundles of that weight. Many years later H. Hida and respectively R. Coleman defined ordinary, respectively finite slope overconvergent modular forms of arbitrary, p-adic weight as q-expansions and using these Coleman and Mazur constructed at the end of the 90's the famous eigencurve.

Recently, together with Andreatta, Pilloni and Stevens we have been able to geometrically redefine the overconvergent modular forms of Hida and Coleman and so we were able to generalize these constructions to Hilbert and Siegel modular forms. Abstract Consider a smooth projective variety over a number field. The image of the associated complex Abel--Jacobi map inside the transcendental intermediate Jacobian is an abelian variety. We show that this abelian variety admits a distinguished model over the number field. Among other applications, this tool allows us to answer a recent question of Mazur; recover an old result of Deligne; and give new constructions of period maps over arithmetic bases.

Vaidehee Thatte. Abstract In classical ramification theory, we consider extensions of complete discrete valuation rings with perfect residue fields.