Algebraic and analytic aspects of automorphic forms

Multiple Dirichlet series (MDS) are a tool for studying moments of quadratic Dirichlet L-functions, whereas the analytic properties of the MDS (meromorphic continuation and residues at poles) give information about the asymptotics of moments, an important problem in analytic number theory.  Part of this proposal is concerned with the fourth moment in the function field setting, when the group of functional equations satisfied by the MDS is an affine Weyl group. We propose to study MDS associated to all affine Weyl groups,  and to prove explicit formulas for residues of such MDS, leading to asymptotics for the fourth moment in the function field case that include secondary terms in the asymptotic expansion. In virtue of the local-global principle and of the dictionary between the function field and the rational numbers settings, we expect that our work will shed light on the fourth moment asymptotics over the rationals as well.

In a different direction, the PD and collaborators will investigate the trace formula for Hecke operators on the symplectic group Sp(4), using cohomological methods. This is a continuation of the PD's work on the trace formula for the modular group (joint with Don Zagier) and on congruence subgroups.

Seriile multiple Dirichlet (SMD) sunt un instrument pentru studiul momentelor L-functiilor Dirichlet patratice, in sensul ca informatii analitice despre SMD (continuare meromorfa si reziduuri la poli) implica informatii despre asimptota momentelor, o problema importanta in teoria analitica a numerelor. O parte din aceasta propunere se refera la al patrulea moment peste corpuri de functii, cand grupul de ecuatii functionale satisfacut de SMD este un grup Weyl afin. Ne propunem sa studiem SMD asociate grupurilor Weyl afine si sa demonstram formule explicite pentru reziduurile unor astfel de serii, ceea ce conduce la asimptote pentru al patrulea moment peste corpuri de functii, incluzand termeni secundari in dezvoltarea asimptotica. In virtutea principiului local-global si a dictionarului intre situatia peste corpuri de functii si corpuri de numere, ne asteptam ca acest proiect sa arunce lumina si asupra asimptotelor celui de-al patrulea moment peste numerele rationale. 

Intr-o alta directie, directorul de proiect impreuna cu colaboratori vor investiga formula de urma pentru operatorii Hecke peste grupul simplectic Sp(4), folosind metode coomologice. Aceasta reprezinta o continuare a rezultatatelor noastre despre formula de urma pentru grupul modular (impreuna cu Don Zagier) si pentru subgrupuri de congruenta. 

A well-known question in analytic number theory is the determination of the average size of the moments of central L-values of a family of automorphic objects. One family of interest to this project is the family of quadratic Dirichlet characters χD  associated to square-free integers D. In this case, having an asymptotic formula for all moments for the central values L(1/2,χD) implies the famous Lindelöf hypothesis on the growth of L(1/2,χD) with D. This is a very hard problem, and insight into it can be gained from studying the corresponding moments over function fields. That is, the characters χD are replaced with the quadratic characters associated to monic square-free polynomials D over a finite field. 

Weyl group multiple Dirichlet series (MDS) were introduced precisely to study the asymptotics of such moments, both over number fields and function fields. In the past three decades the subject has grown tremendously, and there are now several constructions of MDS over a global field, having a group of functional equations isomorphic to the Weyl group of a finite root system. For the MDS attached to the above moment problem, the group of functionals is infinite starting with the fourth moment. One of the results of this project, due to Diaconu and Twiss (Journal of Number Theory 2023), is a precise conjectural asymptotic formula for all moments, with an arbitrary number of second terms in the asymptotics starting with the fourth moment. The conjecture would follow from the analytic continuation of the associated MDS and its twists, and the knowledge of residues at its poles. This conjecture extends the well-known conjecture of Conrey-Keating-Farmer-Rubinstein-Snaith, who predicted the first polynomial term in the conjecture.

In the case of the fourth moment of quadratic Dirichlet L-functions over the rational function field, Diaconu, Pasol and the PD proved unconditionally such an asymptotic formula, for a modified moment sum, by studying the associated MDS (to appear in American Journal of Mathematics). This is the first such formula in the literature where an arbitrary number of secondary terms appear in the asymptotics. The group of functional equations satisfied by the MDS in this case is the (infinite) Weyl group of affine type D4. The desired analytic properties of the MDS follow from a new kind of functional equation that is specific to affine systems, which will likely play an important role in the combinatorics of affine roots systems.

In a recent preprint, Bergström, Diaconu, Petersen and Westerland have proved the moment conjecture over the rational function field with the first polynomial term, subject to a purely topological statement: a stability result for the cohomology of a certain moduli space. The latter is close to being proved in a work in progress of Petersen, Randal-Williams and Westerland. These very recent results open new venues of research both in number theory and topology.

In another direction, the PD studied the trace formula for Hecke operators on Fuchsian groups. We consider the Lefschetz trace of the Hecke operator on the cohomology of such a group with values in an arbitrary module, and we show that this trace is encoded by a certain algebraic operator independent of the module. This generalizes a proof of the Eichler-Selberg trace formula given by the PD together with Don Zagier, and the resulting formula has the potential of generalizing to other groups.

O problema binecunoscuta in teoria analitica a numerelor este determinarea marimii medii a momentelor valorilor centrale de L-functii intr-o familie de obiecte automorfe. O familie care ne-a interesat in acest proiect este cea a caracterelor Dirichlet χD  asosciate intregilor liberi de patrate D. In acest caz, o formula asimptotica pentru toate momentele valorilor centrale L(1/2,χD) implica faimoasa ipoteza a lui Lindelöf despre cresterea  valorii L(1/2,χD) cu D. Aceasta este o problema dificila, dar informatii asupra ei pot fi obtinute studiind momentele corespunzatoare peste corpuri de functii. Aceasta presupune inlocuirea caracterelor χD cu caractere patratice asociate polinoamelor monice libere de patrate D peste un corp finit.

Seriile multiple Dirichlet de tip Weyl (SMD) au fost introduse in scopul studierii asimptotelor pentru astfel de momente, atat peste corpuri de numere cat si peste corpuri de functii. In ultimele trei decade, subiectul s-a dezvoltat considerabil si acum sunt mai multe constructii de SMD peste un corp global, avand un grup de ecuatii functionale izomorf cu grupul Weyl al unui sistem de radacini finit. Pentru SMD atasate momentelor de mai sus, grupul de ecuatii functionale devine infinit incepand cu momentul al patrulea. Unul din rezultatele proiectului, datorat lui Diaconu si Twiss (Journal of Number Theory 2023), este o formula asimptotica conjecturala precisa pentru toate momentele, cu un numar arbitrar de termeni secundari incepand cu al patrulea moment. Conjectura este o consecinta a continuarii analitice a SMD asociate si ale rasucirilor acestora, si din determinarea reziduurilor acestora la poli. Aceasta conjectura extinde binecunoscuta conjectura  a lui Conrey-Keating-Farmer-Rubinstein-Snaith, care au conjecturat primul termen polinomial in formula asimptotica. 

In cazul momentului al patrulea al seriilor patratice Dirichlet peste corpul rational de functii, Diaconu, Pasol si PD au demonstrat neconditionat o astfel de formula asimptotica, pentru o suma modificata, ca urmare a studiului SMD asociate (acceptat la publicare in American Journal of Mathematics). Aceasta este prima formula in literatura in care un numar arbitrar de termeni secundari apar in asimptota. Grupul de ecuatii functionale satisfacut de SMD in acest caz este grupul Weyl (infinit) de tip D4  afin. Proprietatile analitice ale SMD de care este nevoie sunt o consecinta a unui nou tip de ecuatie functionala care este specifica sistemelor afine de radacini, si care este probabil ca va juca un rol important in combinatorica sistemelor de radacini afine.

Intr-un preprint recent, Bergström, Diaconu, Petersen si Westerland au demonstrat conjectura momentelor peste corpul rational de functii, cu primul termen polinomial in asimptota, folosind o proprietate pur topologica: un rezultat de stabilitate pentru coomologia unui spatiu de moduli anume. Aceasta proprietate este aproape de a fi demonstrata intr-o lucrare in curs de redactare de catre Petersen, Randal-Williams si Westerland. Aceste rezultate foarte recente deschid noi cai de studiu atat in teoria numerelor, cat si in topologie. 

Intr-o alta directie, PD a studiat formula de urma Hecke pentru grupuri Fuchsiene. Am considerat urma Lefschetz al operatorului Hecke pe coomologia unui astfel de grup cu coeficienti intr-un modul arbitrar, si am aratat ca aceasta urma este codata de un operator algebric independent de modul. Aceasta generalizeaza o demonstratie a formulei de urma Eichler-Selberg data de PD impreuna cu Don Zagier, si formula care rezulta are potentialul de a fi generalizata la alte grupuri.