Building Bridges

Book of abstracts

3rd EU/US Workshop onAutomorphic Forms and Related Topics

18-22 July, 2016, University of Sarajevo

Organizers:

Jay Jorgenson (New York)

Lejla Smajlovic (Sarajevo)

Lynne Walling (Bristol)

Abelian varieties and the inverse Galois problem

Samuele Anni

Abstract

Let Q be an algebraic closure of Q,let n be a positive integer and let ` aprime number. Given a curve C overQ of genus g, it is possible to define aGalois representation ρ : Gal(Q/Q) →GSp2g(F`), where F` is the finite field of` elements and GSp2g is the general sym-plectic group in GL2g, corresponding tothe action of the absolute Galois groupGal(Q/Q) on the `-torsion points of itsJacobian variety J(C). If ρ is surjec-tive, then we realize GSp2g(F`) as a Ga-lois group over Q. In this talk I will firstdescribe a joint work with Pedro Lemosand Samir Siksek, concerning the realiza-tion of GSp6(F`) as a Galois group for in-finitely many odd primes ` and then thegeneralizations to arbitrary dimensions,joint work with Vladimir Dokchitser.

Dedekind sums for cofinite Fuchsian groups

Claire Burrin

Abstract

Dedekind sums are very well-studiedarithmetic sums, which determine themodular transformation of the eta-function. We present the constructionof the general analogue of the Dedekindsums, which we call Dedekind symbol,for cofinite Fuchsian groups. If time per-mits, we will discuss certain aspects of theDedekind symbol, such as its relation toSelberg–Kloosterman sums, the equidis-tribution mod 1 of its values, and the ex-istence of a reciprocity formula.

1

Period integrals and special values of L-functions

Andrew Corbett

Abstract

In the theory of L-functions, specialvalues are frequently shown to equal mys-terious arithmetic objects. I will talkabout some recent results on special val-ues of automorphic L-functions in thespirit of the Gan–Gross–Prasad conjec-tures. Our mysterious arithmetic objectsare period integrals of automorphic formson both GL(2) and GSp(4). The ap-plications are surprising and range fromnon-vanishing problems to bounding Lp-norms of modular forms.

Refinements of Boecherer’s conjecture

Martin Dickson

Abstract

In the 1980s Boecherer made a re-markable conjecture relating the Fouriercoefficients of a Siegel modular form ofdegree two to special values of the twistsof its spin L-function. I will talk aboutrefinements of this conjecture, and at-tempts to computationally verify these,which follow from an explicit form of theGan–Gross–Prasad conjectures. Part ofthis is joint work with A. Pitale, A. Saha,and R. Schmidt.

2

Torsion subgroups of elliptic curves over elemen-tary Abelian extensions

Ozlem Ejder

Abstract

Let K denote the quadratic fieldQ(√d) where d = −1 or −3. Let E be

an elliptic curve defined over K. In thispaper, we analyze the torsion subgroupsof E in the maximal elementary abelian2-extension of K.

Selmer groups and Beilinson-Bloch-Kato conjec-tures

Yara Elias

Abstract

Kolyvagin’s method of Euler systemsis used to bound Selmer groups associ-ated to elliptic curves defined over Q andcertain imaginary quadratic fields assum-ing the non-vanishing of a suitable Heeg-ner point. Its adaptations due to Neko-var, and Bertolini and Darmon allow usto bound Selmer groups associated tocertain modular forms twisted by alge-braic Hecke characters assuming the non-vanishing of a suitable Heegner cycle. Inthis talk, we will discuss the contributionsof these results to the Birch-Swinnerton-Dyer conjecture and the Beilinson-Bloch-Kato conjecture, respectively.

3

Aspects of the theta correspondence

Jens Funke

Abstract

Theta series associated to quadraticforms are a very classical way for con-structing automorphic forms. In thisoverview talk, we discuss some of the un-derlying representation-theoretic aspects.

Integral representations by sums of polygonal num-bers

Anna Haensch

Abstract

In this talk we will consider repre-sentations by ternary sums of polygo-nal numbers. A sum of three polygo-nal numbers is a ternary inhomogeneousquadratic polynomial, and by completingthe square the representation problem forsuch a ternary sum can be easily recastas a representation problem for a cosetof some quadratic lattice. Using the ma-chinery of quadratic lattices, we will con-sider when a sum of three polygonal num-bers represents all but finitely many pos-itive integers.

4

Degenerate Eisenstein series for GLn and some ap-plications in number theory

Marcela Hanzer

Abstract

In this talk we will discuss degener-ate Eisenstein series for adelic general lin-ear groups; its poles and images. Wewill apply these results to some classicalEisenstein series. We will also briefly dis-cuss similar problems for classical groups.Most of the talk is joint work with GoranMuic.

Construction of Jacobi and Siegel cusp forms

Abhash Jha

Abstract

Given a fixed Jacobi cusp form, weconsider a family of linear maps con-structed using Rankin-Cohen bracketsand compute the adjoint of these linearmaps with respect to Petersson scalarproduct. The Fourier coefficients of theJacobi cusp form constructed using thismethod involve special values of certainDirichlet series of Rankin type associatedto Jacobi forms. We also discuss the simi-lar construction in the case of Siegel mod-ular forms of genus two. This is jointwork with B. Sahu.

5

Regularized determinant of the Lax-Phillips op-erator and central value of the scattering deter-minant

Jay Jorgenson

Abstract

Using the superzeta function ap-proach due to Voros, we define a Hurwitz-type zeta function ζ±B (s, z) constructedfrom the resonances associated to zI −[(1/2)I ±B], where B is the Lax-Phillipsscattering operator. We prove the mero-morphic continuation in s of ζ±B (s, z) and,define a determinant of the operatorszI − [(1/2)I ±B]. We obtain expressionsfor Selberg’s zeta function and the deter-minant φ(s) of the scattering matrix interms of the operator determinants. Us-ing this result, we evaluate φ(1/2) for anyfinite volume surface.

Weyl group multiple Dirichlet series and a meta-plectic Eisenstein series on GL(3,R)

Edmund Karasiewicz

Abstract

The theory of Weyl group multipleDirichlet series (Dirichlet series in sev-eral variables with a group of functionalequations isomorphic to a Weyl group)has been developed by Brubaker, Bump,Chinta, et al. These multiple Dirichlet se-ries are conjectured to be the Fourier co-efficients of metaplectic Eisenstein series.The work of Brubaker, Bump, Chinta, etal. includes a hypothesis that forces thebase field to be totally imaginary. We willconsider a specific metaplectic Eisensteinseries over R and see how this fits into theexisting theory.

6

Diophantine quadruples over finite fields

Matija Kazalicki

Abstract

A Diophantine m-tuple is a set ofm positive integers with the propertythat the product of any two of its dis-tinct elements increased by 1 is a per-fect square, e.g.. {1, 3, 8, 120} is a Dio-phantine quadruple. While the problemsrelated to these sets and their rationalcounterparts have a long history start-ing from Diophantus, a Diophantine m-tuples with elements in finite fields havenot been studied a lot.

In this talk, we will present theformula for the number of Diophan-tine quadruples in the finite field withp elements (p is a prime). Surpris-ingly, Fourier coefficients of some mod-ular forms appear in the formula. This isa joint work with Andrej Dujella.

Equidistribution of shears and applications

Dubi Kelmer

Abstract

A ”shear” is a unipotent translate ofa cuspidal geodesic ray in the quotientof the hyperbolic plane by a non-uniformdiscrete group (possibly of infinite co-volume). In joint work with Alex Kon-torovich, we prove the equidistribution ofshears under large translates. We give ap-plications including to moments of GL(2)automorphic L-functions, and to count-ing integer points on affine homogeneousvarieties. No prior knowledge of thesetopics will be assumed.

7

Periods of modular forms and identities betweenEisenstein series

Kamal Khuri-Makdisi

Abstract

Borisov and Gunnells observed in2001 that certain linear relations betweenproducts of two holomorphic weight 1Eisenstein series had the same structureas the relations between periods of mod-ular forms; a similar phenomenon existsin higher weights. We give a concep-tual reason for this observation in arbi-trary weight. This involves an uncon-ventional way of expanding the Rankin-Selberg convolution of a cusp form withan Eisenstein series.

On arbitrary products of eigenforms

Arvind Kumar

Abstract

We characterize all the cases in whichproducts of arbitrary numbers of nearlyholomorphic eigenforms and products ofarbitrary numbers of quasimodular eigen-forms for the full modular group SL2(Z)are again eigenforms.

8

Models for modular curves X0(N)

Iva Kodrnja

Abstract

The classical modular curve X0(N)is defined to be the quotient ofthe extended upper half-plane H∗ ={z ∈ C : Im(z) > 0}∪Q∪{∞} by the ac-tion of congruence subgroup Γ0(N). Ithas the structure of a compact Riemannsurface and thus can be interpreted assmooth irreducible projective algebraiccurve.

The canonical equation of X0(N)(also called modular equation) is hard tocompute and has several practical draw-backs. In his doctoral thesis, S. Galbraithhas presented a method for finding defin-ing equations of X0(N) based on canon-ical embedding. He used cusp forms ofweight 2 to define maps from X0(N) toprojective space. Recently, G. Muic hasextended this method and maps X0(N)to projective plane using modular formsof higher weight.

In this talk I will present results frommy doctoral thesis which concerns withMuic’s method. Some of the main topicsare the choice of appropriate modularforms and computation of the degree ofthe model. I will give a few examplesof models obtained in this way usingη-quotients.

9

Selberg trace formula as a Dirichlet series

Min Lee

Abstract

We explore the idea of Conrey and Liof presenting the Selberg trace formulafor Hecke operators, as a Dirichlet se-ries. We explore the idea of Conrey andLi presenting the Selberg trace formulafor Hecke operators, as a Dirichlet series.We enhance their work in few ways andpresent several applications of our for-mula. This is a joint work with AndrewBooker.

A lower bound for the least prime in an arithmeticprogression

Junxian Li

Abstract

Fix k a positive integer, and let `be coprime to k. Let p(k, `) denote thesmallest prime equivalent to ` (mod k),and set P (k) to be the maximum ofall the p(k, `). We seek lower boundsfor P (k). In particular, we show thatfor almost every k one has P (k) �φ(k) log k log2 k log4 k/ log3 k,. This isa joint work with K. Pratt and G.Shakan based on the recent work on largegaps between primes of of Ford, Green,Konyangin, Maynard, and Tao.

10

Bounds on the arithmetic genus of Hilbert modu-lar varieties

Benjamin Linowitz

Abstract

Associated to every totally real num-ber field k of degree n is a Hilbert mod-ular variety which arises as a desingu-larization of the compactification of thequotient space of the Hilbert modulargroup acting on an n-fold product of com-plex upper half planes. In this talk wewill discuss some recent results concern-ing the arithmetic genus of Hilbert modu-lar varieties. In particular we will exhibitupper and lower bounds for the arith-metic genus of Hilbert modular varietiesin terms of the degree and discriminantof k. Additionally, we will discuss somerecent progress on the related problem ofclassifying those totally real fields k forwhich the associated Hilbert modular va-riety is not of general type.

Using theta lifts to understand Eisenstein denom-inators

Robert Little

Abstract

We use the (extended) Shimura-Shintani correspondence (in the cohomo-logical setting of Funke-Millson) to inves-tigate Harders theory of denominators ofEisenstein cohomology classes on modu-lar curves.

11

Fourier coefficients of automorphic forms for higherrank groups

Guangshi Lyu

Abstract

Fourier coefficients of automorphicforms are interesting and important ob-jects in modern number theory. Inthis talk, I shall introduce some recentprogress on orthogonality between addi-tive characters and Fourier coefficients ofcusp forms over primes. If time permits,I shall also talk about shifted convolutionsums for higher rank groups. This talk isbased on my recent work joint with FeiHou and Yujiao Jiang.

Zeros of L-functions of certain half integral weightcusp forms

Jaban Meher

Abstract

We will discuss about the zeros of L-functions attached to half integral weightcusp forms for the group Γ0(4).

12

Computing the p-adic periods of abelian surfacesfrom automorphic forms

Marc Masdeu

Abstract

Let F be a number field, and let f bea normalized eigenform modular form ofweight 2 and level N for GL(2,F ). It isconjectured that there is an abelian vari-ety Af attached to such a modular form.This abelian variety should have dimen-sion equal to the degree of the field ofHecke eigenvalues, and should have goodreduction outside N . In those instanceswhere the Eichler-Shimura constructionis not available (for example when F isnot totally-real) little is known about howto find Af .

In joint work with Xavier Guitart, wepresent a p-adic conjectural construction(subject to several restrictions, in partic-ular p should divide N) of Af , and il-lustrate how in favourable situations itcan be used to find equations for abeliansurfaces Af as jacobians of hyperellipticcurves.

Properties of Sturm’s operator

Kathrin Maurischat

Abstract

For non-holomorphic modular forms,the projection to their holomorphic com-ponents is described by Sturm’s operatorin all cases known to us. We show thatthere are interesting situations in whichits image fails to be holomorphic, for ex-ample the Siegel case of genus two andthe low weight three.

13

Lower bounds on dimensions of mod-p Hecke al-gebras

Anna Medvedovsky

Abstract

In 2012, Nicolas and Serre revived in-terest in the study of mod-p Hecke al-gebras when they proved that, for p =2, the Hecke algebra acting on level-oneforms of all weights is the power-seriesring F2[[T3, T5]]: a surprising and sur-prisingly explicit result. Their elemen-tary arguments do not appear to gener-alize directly to other primes, but theirtools — the Hecke recursion, the nilpo-tence filtration — serve as the backboneof a new method, uniform and entirelyin characteristic p, for obtaining lowerbounds on dimensions of mod-p Hecke al-gebras. I will present this new method(currently implemented in the genus-zerocase), compare it with others, and discussvarious future directions. The talk shouldbe accessible to anyone with some famil-iarity with modular forms and Hecke op-erators. The key technical result is purealgebra, combinatorial in flavor; and maybe of independent interest.

14

Numerical computation of generalized τ-Li coeffi-cients

Almasa Odzak

Abstract

The importance of generalized Li andτ -Li coefficients comes from the crite-ria relating generalized Riemann or tau-Riemann hypothesis for an L-functionwith certain properties of these coeffi-cients attached to that L-function. Anexample is the Li criterion for the Rie-mann hypothesis stating that the Rie-mann hypothesis is equivalent to non-negativity of corresponding Li coeffi-cients. Some later results include asymp-totic behavior of these coefficients.

There are a number of approachesto the problem of numerically comput-ing the Li and τ -Li coefficients of a givenfunction. Computationally speaking, theproblem is quite demanding. The mainissue is the accumulation of the errorterms and the time required for the com-putation. Arb, a C library for arbitrary-precision floating-point ball arithmeticwith automatic error control, turns outto be very useful tool.

Our investigations are focused on twovery broad classes of functions S][(σ0;σ1)and S]

R. Both classes are interested sincethey contain functions that are assumedto violate Riemann hypothesis in whichcase numerical evaluation of τ -Li coeffi-cients may serve as a tool for proving exis-tence of eventual zeros off the critical lineand eventual zeros off the critical strip.

15

Rational points on twisted modular curves

Ekin Ozman

Abstract

Studying rational points on an alge-braic curve is one of the main problems innumber theory. In this talk, I will presentresults about points on certain twists ofthe classical modular curve. Many ofthese twisted curves violate the Hasseprinciple. In the cases that genus is big-ger than one, some of these violationsare explained by the Mordell-Weil sievemethod which will be defined. In genusone cases, it is possible to study Hasseprinciple violations via local-global traceobstructions.

An analytic class number type formula for PSL2(Z)

Anna von Pippich

Abstract

For any Fuchsian subgroup Γ ⊂PSL2(R) of the first kind, Selberg intro-duced the Selberg zeta function in anal-ogy to the Riemann zeta function us-ing the lengths of simple closed geodesicson Γ\H instead of prime numbers. Inthis talk, we report on a generalizedRiemann–Roch isometry in Arakelov ge-ometry. As arithmetic application, we de-termine the special value at s = 1 of thederivative of the Selberg zeta function inthe case Γ = PSL2(Z). This is joint workwith Gerard Freixas.

16

Elliptic curves with maximally disjoint divisionfields

James Ricci

Abstract

One of the many interesting alge-braic objects associated to a given el-liptic curve, E, defined over the ratio-nal numbers, is its full-torsion represen-tation ρE : Gal(Q/Q) → GL2(Z). Gen-eralizing this idea, one can create an-other full-torsion Galois representation,

ρ(E1,E2) : Gal(Q/Q) →(

GL2(Z))2

asso-

ciated to a pair (E1, E2) of elliptic curvesdefined over Q. The goal of this talk isto provide an infinite number of concreteexamples of pairs of elliptic curves whoseassociated full-torsion Galois representa-tion ρ(E1,E2) has maximal image.

Integers that can be written as the sum of twocubes

Eugenia Rosu

Abstract

The Birch and Swinnerton-Dyer con-jecture predicts that we have non-torsionrational points on an elliptic curve iff theL-function corresponding to the ellipticcurve vanishes at 1. Thus BSD predictsthat a positive integer N is the sum of twocubes if L(EN , 1) = 0, where L(EN , s) isthe L-function corresponding to the el-liptic curve EN : x3 + y3 = N . Us-ing methods from automorphic forms, wehave computed several formulas that re-late L(EN , 1) to the trace of a ratio ofspecific theta functions at a CM point.This offers a criterion for when the inte-ger N is the sum of two cubes. Further-more, when L(EN , 1) is nonzero we get aformula for the number of elements in theTate-Shafarevich group and show that itis a square in certain cases.

17

Convolution sums of the divisor functions and thenumber of representations of certain quadratic forms

Brundaban Sahu

Abstract

We report about our recent results onthe evaluation of some convolution sumsof the divisor functions and their use indetermining the number of representa-tions of certain quadratic forms. By com-paring with similar results, we also makesome observations about the Fourier co-efficients of certain cusp forms. This is ajoint work with B. Ramakrishnan.

On the Weyl law for certain quantum graphs

Lamija Sceta

Abstract

We apply a special case of Tauberiantheorem for the Laplace transform to thesuitably transformed trace formula in thesetting of quantum graphs with generalself adjoint boundary conditions and ob-tain new bounds for the remainder termin the Weyl law.

Dimension formulas for Hilbert modular forms andShimizu L-series

Fredrik Stromberg

Abstract

I will discuss some recent results onjoint work with N.-P. Skoruppa, regard-ing explicit methods, in particular dimen-sion formulas, for vector-valued Hilbertmodular forms.

18

The Hauptmodul at elliptic points of certain arith-metic groups

Holger Then

Abstract

Let N be a square-free integer suchthat the arithmetic group Γ0(N)+ hasgenus zero; there are 44 such groups. LetjN denote the associated Hauptmodulnormalized to have residue equal to oneand constant term equal to zero in its q-expansion. In joint work together withLejla Smajlovic and Jay Jorgenson, weprove that the Hauptmodul at any ellipticpoint of the surface associated to Γ0(N)+

is an algebraic integer. Moreover, for eachsuch N and elliptic point e, we explic-itly evaluate jN (e) in radicals and con-struct generating polynomials of the cor-responding class fields of the orders asso-ciated to the elliptic point under consid-eration.

The sign changes of Fourier coefficients of Eisen-stein series

Lola Thompson

Abstract

We study a variety of statistical ques-tions concerning the signs of the Fouriercoefficients of Eisenstein series, provinganalogues of several well-known theoremsfor cusp forms. This talk is based on jointwork with Benjamin Linowitz.

19

Elliptic and hyperbolic Eisenstein series as thetalifts

Fabian Volz

Abstract

Generalising the concept of classicalnon-holomorphic Eisenstein series asso-ciated to cusps, one can define ellipticEisenstein series associated to points inthe upper-half plane H, and hyperbolicEisenstein series associated to geodesicsin H. In my talk I will show that av-eraged versions of these elliptic and hy-perbolic Eisenstein series can be obtainedas Theta lifts of signature (2, 1) of someweighted Poincare series. These averagedEisenstein series can also be regarded astraces of elliptic and hyperbolic kernelfunctions.

On the distribution of Hecke eigenvalues for GL(2)

Nahid Walji

Abstract

Given a self-dual cuspidal automor-phic representation for GL(2) over a num-ber field, we establish the existence of aninfinite number of Hecke eigenvalues thatare greater than an explicit positive con-stant, and an infinite number of Heckeeigenvalues that are less than an explicitnegative constant. This provides an an-swer to a question of Serre. We also con-sider analogous problems for cuspidal au-tomorphic representations that are notself-dual.

20

On the lifting of Hilbert cusp forms

Shunsuke Yamana

Abstract

Starting from a Hilbert cusp form,I will construct holomorphic cusp formson certain Hermitian symmetric domainsof higher degree. In the case of Siegelcusp forms, this is a generalization of theSaito-Kurokawa lifting from degree twoto higher degrees and the Ikeda liftingfrom the rational number field to totallyreal number fields.

This is a joint work with TamotsuIkeda.

21

Partners:

EUROPEAN MATHEMATICAL

SOCIETY

HEILBRONN INSTITUTE FOR MATHEMATICAL

RESEARCH

SCHOOL OF ECONOMICS AND

BUSINESS SARAJEVO

NUMBER THEORY FOUNDATION

NATIONAL SCIENCE

FOUNDATION FOUNDATION

COMPOSITIO MATHEMATICA

AMERICAN INSTITUTE OF MATHEMATICS