We establish lower bounds for (i) the numbers of positive and negative terms and (ii) the number of sign changes in the sequence of Fourier coefficients at squarefree integers of a half-integral weight modular Hecke eigenform.

In terms of class field theory we give a necessary and sufficient condition for an integer to be representable by the quadratic form $x^{2}+xy+ny^{2}$ ($n\in \mathbb{N}$ arbitrary) under extra conditions $x\equiv 1\;\text{mod}\;m$, $y\equiv 0\;\text{mod}\;m$ on the variables. We also give some examples where their extended ring class numbers are less than or equal to $3$.

Using results from Ramanujan's lost notebook, Zudilin recently gave an insightful proof of a radial limit result of Folsom et al. for mock theta functions. Here we see that Mortenson's previous work on the dual nature of Appell–Lerch sums and partial theta functions and on constructing bilateral q-series with mixed mock modular behaviour is well suited for such radial limits. We present five more radial limit results, which follow from mixed mock modular bilateral q-hypergeometric series. We also obtain the mixed mock modular bilateral series for a universal mock theta function of Gordon and McIntosh. The later bilateral series can be used to compute radial limits for many classical second-, sixth-, eighth- and tenth-order mock theta functions.

Following Jacquet, Lapid and Rogawski, we regularize trilinear periods. We use the regularized trilinear periods to compute Fourier–Jacobi periods of residues of Eisenstein series on metaplectic groups, which has an application to the Gan–Gross–Prasad conjecture.

Let $E$ be an elliptic curve over $\mathbb{Q}$, and let ${\it\varrho}_{\flat }$ and ${\it\varrho}_{\sharp }$ be odd two-dimensional Artin representations for which ${\it\varrho}_{\flat }\otimes {\it\varrho}_{\sharp }$ is self-dual. The progress on modularity achieved in recent decades ensures the existence of normalized eigenforms $f$, $g$, and $h$ of respective weights two, one, and one, giving rise to $E$, ${\it\varrho}_{\flat }$, and ${\it\varrho}_{\sharp }$ via the constructions of Eichler and Shimura, and of Deligne and Serre. This article examines certain $p$-adic iterated integrals attached to the triple $(f,g,h)$, which are $p$-adic avatars of the leading term of the Hasse–Weil–Artin $L$-series $L(E,{\it\varrho}_{\flat }\otimes {\it\varrho}_{\sharp },s)$ when it has a double zero at the centre. A formula is proposed for these iterated integrals, involving the formal group logarithms of global points on $E$—referred to as Stark points—which are defined over the number field cut out by ${\it\varrho}_{\flat }\otimes {\it\varrho}_{\sharp }$. This formula can be viewed as an elliptic curve analogue of Stark’s conjecture on units attached to weight-one forms. It is proved when $g$ and $h$ are binary theta series attached to a common imaginary quadratic field in which $p$ splits, by relating the arithmetic quantities that arise in it to elliptic units and Heegner points. Fast algorithms for computing $p$-adic iterated integrals based on Katz expansions of overconvergent modular forms are then exploited to gather numerical evidence in more exotic scenarios, encompassing Mordell–Weil groups over cyclotomic fields, ring class fields of real quadratic fields (a setting which may shed light on the theory of Stark–Heegner points attached to Shintani-type cycles on ${\mathcal{H}}_{p}\times {\mathcal{H}}$), and extensions of $\mathbb{Q}$ with Galois group a central extension of the dihedral group $D_{2n}$ or of one of the exceptional subgroups $A_{4}$, $S_{4}$, and $A_{5}$ of $\mathbf{PGL}_{2}(\mathbb{C})$.

We prove modularity of formal series of Jacobi forms that satisfy a natural symmetry condition. They are formal analogs of Fourier–Jacobi expansions of Siegel modular forms. From our result and a theorem of Wei Zhang, we deduce Kudla’s conjecture on the modularity of generating series of special cycles of arbitrary codimension and for all orthogonal Shimura varieties.

We prove certain depth bounds for Arthur’s endoscopic transfer of representations from classical groups to the corresponding general linear groups over local fields of characteristic 0, with some restrictions on the residue characteristic. We then use these results and the method of Deligne and Kazhdan of studying representation theory over close local fields to obtain, under some restrictions on the characteristic, the local Langlands correspondence for split classical groups over local function fields from the corresponding result of Arthur in characteristic 0.

In this paper we establish a Chowla–Selberg formula for abelian CM fields. This is an identity which relates values of a Hilbert modular function at CM points to values of Euler’s gamma function ${\rm\Gamma}$ and an analogous function ${\rm\Gamma}_{2}$ at rational numbers. We combine this identity with work of Colmez to relate the CM values of the Hilbert modular function to Faltings heights of CM abelian varieties. We also give explicit formulas for products of exponentials of Faltings heights, allowing us to study some of their arithmetic properties using the Lang–Rohrlich conjecture.

The second author has recently introduced a new class of $L$-series in the arithmetic theory of function fields over finite fields. We show that the values at one of these $L$-series encode arithmetic information of a generalization of Drinfeld modules defined over Tate algebras that we introduce (the coefficients can be chosen in a Tate algebra). This enables us to generalize Anderson’s log-algebraicity theorem and an analogue of the Herbrand–Ribet theorem recently obtained by Taelman.

Let $\mathbf{G}$ be the connected reductive group of type $E_{7,3}$ over $\mathbb{Q}$ and $\mathfrak{T}$ be the corresponding symmetric domain in $\mathbb{C}^{27}$. Let ${\rm\Gamma}=\mathbf{G}(\mathbb{Z})$ be the arithmetic subgroup defined by Baily. In this paper, for any positive integer $k\geqslant 10$, we will construct a (non-zero) holomorphic cusp form on $\mathfrak{T}$ of weight $2k$ with respect to ${\rm\Gamma}$ from a Hecke cusp form in $S_{2k-8}(\text{SL}_{2}(\mathbb{Z}))$. We follow Ikeda’s idea of using Siegel’s Eisenstein series, their Fourier–Jacobi expansions, and the compatible family of Eisenstein series.

We propose to generalize the work of Régis Dupont for computing modular polynomials in dimension $2$ to new invariants. We describe an algorithm to compute modular polynomials for invariants derived from theta constants and prove heuristically that this algorithm is quasi-linear in its output size. Some properties of the modular polynomials defined from quotients of theta constants are analyzed. We report on experiments with our implementation.

It is well known that every elliptic curve over the rationals admits a parametrization by means of modular functions. In this short note, we show that only finitely many elliptic curves over $\mathbf{Q}$ can be parametrized by modular units. This answers a question raised by W. Zudilin in a recent work on Mahler measures. Further, we give the list of all elliptic curves $E$ of conductor up to 1000 parametrized by modular units supported in the rational torsion subgroup of $E$. Finally, we raise several open questions.

We construct local and global metaplectic double covers of odd general spin groups, using the cover of Matsumoto of spin groups. Following Kazhdan and Patterson, a local exceptional representation is the unique irreducible quotient of a principal series representation, induced from a certain exceptional character. The global exceptional representation is obtained as the multi-residue of an Eisenstein series: it is an automorphic representation, and it decomposes as the restricted tensor product of local exceptional representations. As in the case of the small representation of $\mathit{SO}_{2n+1}$ of Bump, Friedberg, and Ginzburg, exceptional representations enjoy the vanishing of a large class of twisted Jacquet modules (locally), or Fourier coefficients (globally). Consequently they are useful in many settings, including lifting problems and Rankin–Selberg integrals. We describe one application, to a calculation of a co-period integral.

We prove that formal Fourier Jacobi expansions of degree two are Siegel modular forms. As a corollary, we deduce modularity of the generating function of special cycles of codimension two, which were defined by Kudla. A second application is the proof of termination of an algorithm to compute Fourier expansions of arbitrary Siegel modular forms of degree two. Combining both results enables us to determine relations of special cycles in the second Chow group.

In 2006, F. Luca and I. E. Shparlinski (Proc. Indian Acad. Sci. (Math. Sci.)116(1) (2006), 1–8) proved that there are only finitely many pairs (n, m) of positive integers which satisfy the Diophantine equation |τ(n!)|=m!, where τ is the Ramanujan function. In this paper, we follow the same approach of Luca and Shparlinski (Proc. Indian Acad. Sci. (Math. Sci.)116(1) (2006), 1–8) to determine all solutions of the above equation. The proof of our main theorem uses linear forms in two logarithms and arithmetic properties of the Ramanujan function.

Suppose E is an elliptic curve over $\Bbb Q$, and p>3 is a split multiplicative prime for E. Let q ≠ p be an auxiliary prime, and fix an integer m coprime to pq. We prove the generalised Mazur–Tate–Teitelbaum conjecture for E at the prime p, over number fields $K\subset \Bbb Q\big(\mu_{{q^{\infty}}},\;\!^{q^{\infty}\!\!\!\!}\sqrt{m}\big)$ such that p remains inert in $K\cap\Bbb Q(\mu_{{q^{\infty}}})^+$. The proof makes use of an improved p-adic L-function, which can be associated to the Rankin convolution of two Hilbert modular forms of unequal parallel weight.

We use a relative trace formula on $\text{GL}(2)$ to compute a sum of twisted modular $L$-functions anywhere in the critical strip, weighted by a Fourier coefficient and a Hecke eigenvalue. When the weight $k$ or level $N$ is sufficiently large, the sum is nonzero. Specializing to the central point, we show in some cases that the resulting bound for the average is as good as that predicted by the Lindelöf hypothesis in the $k$ and $N$ aspects.

In a recent important paper, Hoffstein and Hulse [Multiple Dirichlet series and shifted convolutions, arXiv:1110.4868v2] generalized the notion of Rankin–Selberg convolution $L$-functions by defining shifted convolution$L$-functions. We investigate symmetrized versions of their functions, and we prove that the generating functions of certain special values are linear combinations of weakly holomorphic quasimodular forms and “mixed mock modular” forms.

We exhibit an action of Conway’s group – the automorphism group of the Leech lattice – on a distinguished super vertex operator algebra, and we prove that the associated graded trace functions are normalized principal moduli, all having vanishing constant terms in their Fourier expansion. Thus we construct the natural analogue of the Frenkel–Lepowsky–Meurman moonshine module for Conway’s group. The super vertex operator algebra we consider admits a natural characterization, in direct analogy with that conjectured to hold for the moonshine module vertex operator algebra. It also admits a unique canonically twisted module, and the action of the Conway group naturally extends. We prove a special case of generalized moonshine for the Conway group, by showing that the graded trace functions arising from its action on the canonically twisted module are constant in the case of Leech lattice automorphisms with fixed points, and are principal moduli for genus-zero groups otherwise.

We show that a weakly holomorphic modular function can be written as a sum of modular units of higher level. Furthermore, we find a necessary and sufficient condition for a meromorphic Siegel modular function of degree g to have neither a zero nor a pole on a certain subset of the Siegel upper half-space .