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Suppose that
$G$
is a simple reductive group over
$\mathbf{Q}$
, with an exceptional Dynkin type and with
$G(\mathbf{R})$
quaternionic (in the sense of Gross–Wallach). In a previous paper, we gave an explicit form of the Fourier expansion of modular forms on
$G$
along the unipotent radical of the Heisenberg parabolic. In this paper, we give the Fourier expansion of the minimal modular form
$\unicode[STIX]{x1D703}_{Gan}$
on quaternionic
$E_{8}$
and some applications. The
$Sym^{8}(V_{2})$
-valued automorphic function
$\unicode[STIX]{x1D703}_{Gan}$
is a weight 4, level one modular form on
$E_{8}$
, which has been studied by Gan. The applications we give are the construction of special modular forms on quaternionic
$E_{7},E_{6}$
and
$G_{2}$
. We also discuss a family of degenerate Heisenberg Eisenstein series on the groups
$G$
, which may be thought of as an analogue to the quaternionic exceptional groups of the holomorphic Siegel Eisenstein series on the groups
$\operatorname{GSp}_{2n}$
.
Let
$G$
be an anisotropic semisimple group over a totally real number field
$F$
. Suppose that
$G$
is compact at all but one infinite place
$v_{0}$
. In addition, suppose that
$G_{v_{0}}$
is
$\mathbb{R}$
-almost simple, not split, and has a Cartan involution defined over
$F$
. If
$Y$
is a congruence arithmetic manifold of non-positive curvature associated with
$G$
, we prove that there exists a sequence of Laplace eigenfunctions on
$Y$
whose sup norms grow like a power of the eigenvalue.
Let
$\unicode[STIX]{x1D707}(m,n)$
(respectively,
$\unicode[STIX]{x1D702}(m,n)$
) denote the number of odd-balanced unimodal sequences of size
$2n$
and rank
$m$
with even parts congruent to
$2\!\!\hspace{0.6em}{\rm mod}\hspace{0.2em}4$
(respectively,
$0\!\!\hspace{0.6em}{\rm mod}\hspace{0.2em}4$
) and odd parts at most half the peak. We prove that two-variable generating functions for
$\unicode[STIX]{x1D707}(m,n)$
and
$\unicode[STIX]{x1D702}(m,n)$
are simultaneously quantum Jacobi forms and mock Jacobi forms. These odd-balanced unimodal rank generating functions are also duals to partial theta functions originally studied by Ramanujan. Our results also show that there is a single
$C^{\infty }$
function in
$\mathbb{R}\times \mathbb{R}$
to which the errors to modularity of these two different functions extend. We also exploit the quantum Jacobi properties of these generating functions to show, when viewed as functions of the two variables
$w$
and
$q$
, how they can be expressed as the same simple Laurent polynomial when evaluated at pairs of roots of unity. Finally, we make a conjecture which fully characterizes the parity of the number of odd-balanced unimodal sequences of size
$2n$
with even parts congruent to
$0\!\!\hspace{0.6em}{\rm mod}\hspace{0.2em}4$
and odd parts at most half the peak.
We introduce a new family of real-analytic modular forms on the upper-half plane. They are arguably the simplest class of ‘mixed’ versions of modular forms of level one and are constructed out of real and imaginary parts of iterated integrals of holomorphic Eisenstein series. They form an algebra of functions satisfying many properties analogous to classical holomorphic modular forms. In particular, they admit expansions in
$q,\overline{q}$
and
$\log |q|$
involving only rational numbers and single-valued multiple zeta values. The first nontrivial functions in this class are real-analytic Eisenstein series.
This paper completes the construction of
$p$
-adic
$L$
-functions for unitary groups. More precisely, in Harris, Li and Skinner [‘
$p$
-adic
$L$
-functions for unitary Shimura varieties. I. Construction of the Eisenstein measure’, Doc. Math.Extra Vol. (2006), 393–464 (electronic)], three of the authors proposed an approach to constructing such
$p$
-adic
$L$
-functions (Part I). Building on more recent results, including the first named author’s construction of Eisenstein measures and
$p$
-adic differential operators [Eischen, ‘A
$p$
-adic Eisenstein measure for unitary groups’, J. Reine Angew. Math.699 (2015), 111–142; ‘
$p$
-adic differential operators on automorphic forms on unitary groups’, Ann. Inst. Fourier (Grenoble)62(1) (2012), 177–243], Part II of the present paper provides the calculations of local
$\unicode[STIX]{x1D701}$
-integrals occurring in the Euler product (including at
$p$
). Part III of the present paper develops the formalism needed to pair Eisenstein measures with Hida families in the setting of the doubling method.
We prove an upper bound for the fifth moment of Hecke L-functions associated to holomorphic Hecke cusp forms of full level and weight k in a dyadic interval K ≤ k ≤2K, as K → ∞. The bound is sharp on Selberg’s eigenvalue conjecture.
We consider an extension of the Ramanujan series with a variable
$x$
. If we let
$x=x_{0}$
, we call the resulting series ‘Ramanujan series with the shift
$x_{0}$
’. Then we relate these shifted series to some
$q$
-series and solve the case of level
$4$
with the shift
$x_{0}=1/2$
. Finally, we indicate a possible way towards proving some patterns observed by the author corresponding to the levels
$\ell =1,2,3$
and the shift
$x_{0}=1/2$
.
We show that two distinct singular moduli
$j(\unicode[STIX]{x1D70F}),j(\unicode[STIX]{x1D70F}^{\prime })$
, such that for some positive integers
$m$
and
$n$
the numbers
$1,j(\unicode[STIX]{x1D70F})^{m}$
and
$j(\unicode[STIX]{x1D70F}^{\prime })^{n}$
are linearly dependent over
$\mathbb{Q}$
, generate the same number field of degree at most two. This completes a result of Riffaut [‘Equations with powers of singular moduli’, Int. J. Number Theory, to appear], who proved the above theorem except for two explicit pairs of exceptions consisting of numbers of degree three. The purpose of this article is to treat these two remaining cases.
We prove an exact formula for the second moment of Rankin–Selberg
$L$
-functions
$L(\frac{1}{2},f\times g)$
twisted by
$\unicode[STIX]{x1D706}_{f}(p)$
, where
$g$
is a fixed holomorphic cusp form and
$f$
is summed over automorphic forms of a given level
$q$
. The formula is a reciprocity relation that exchanges the twist parameter
$p$
and the level
$q$
. The method involves the Bruggeman–Kuznetsov trace formula on both ends; finally the reciprocity relation is established by an identity of sums of Kloosterman sums.
We describe the Schwarzian equations for the 328 completely replicable functions with integral
$q$
-coefficients [Ford et al., ‘More on replicable functions’, Comm. Algebra 22 (1994) no. 13, 5175–5193].
We prove some new structure results for automorphic products of singular weight. First, we give a simple characterisation of the Borcherds function
$\unicode[STIX]{x1D6F7}_{12}$
. Second, we show that holomorphic automorphic products of singular weight on lattices of prime level exist only in small signatures and we derive an explicit bound. Finally, we give a complete classification of reflective automorphic products of singular weight on lattices of prime level.
We give a Rankin–Selberg integral representation for the Spin (degree eight)
$L$
-function on
$\operatorname{PGSp}_{6}$
that applies to the cuspidal automorphic representations associated to Siegel modular forms. If
$\unicode[STIX]{x1D70B}$
corresponds to a level-one Siegel modular form
$f$
of even weight, and if
$f$
has a nonvanishing maximal Fourier coefficient (defined below), then we deduce the functional equation and finiteness of poles of the completed Spin
$L$
-function
$\unicode[STIX]{x1D6EC}(\unicode[STIX]{x1D70B},\text{Spin},s)$
of
$\unicode[STIX]{x1D70B}$
.
We prove analogs of the Bezout and the Bernstein–Kushnirenko–Khovanskii theorems for systems of algebraic differential conditions over differentially closed fields. Namely, given a system of algebraic conditions on the first
$l$
derivatives of an
$n$
-tuple of functions, which admits finitely many solutions, we show that the number of solutions is bounded by an appropriate constant (depending singly-exponentially on
$n$
and
$l$
) times the volume of the Newton polytope of the set of conditions. This improves a doubly-exponential estimate due to Hrushovski and Pillay. We illustrate the application of our estimates in two diophantine contexts: to counting transcendental lattice points on algebraic subvarieties of semi-abelian varieties, following Hrushovski and Pillay; and to counting the number of intersections between isogeny classes of elliptic curves and algebraic varieties, following Freitag and Scanlon. In both cases we obtain bounds which are singly-exponential (improving the known doubly-exponential bounds) and which exhibit the natural asymptotic growth with respect to the degrees of the equations involved.
We provide a concrete example of a normal basis for a finite Galois extension which is not abelian. More precisely, let
$\mathbb{C}(X(N))$
be the field of meromorphic functions on the modular curve
$X(N)$
of level
$N$
. We construct a completely free element in the extension
$\mathbb{C}(X(N))/\mathbb{C}(X(1))$
by means of Siegel functions.
By constructing suitable Borcherds forms on Shimura curves and using Schofer’s formula for norms of values of Borcherds forms at CM points, we determine all of the equations of hyperelliptic Shimura curves
$X_{0}^{D}(N)$
. As a byproduct, we also address the problem of whether a modular form on Shimura curves
$X_{0}^{D}(N)/W_{D,N}$
with a divisor supported on CM divisors can be realized as a Borcherds form, where
$X_{0}^{D}(N)/W_{D,N}$
denotes the quotient of
$X_{0}^{D}(N)$
by all of the Atkin–Lehner involutions. The construction of Borcherds forms is done by solving certain integer programming problems.
We investigate two kinds of Fricke families, those consisting of Fricke functions and those consisting of Siegel functions. In terms of their special values we then generate ray class fields of imaginary quadratic fields over the Hilbert class fields, which are related to the Lang–Schertz conjecture.
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 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 .
We construct explicit bases for spaces of overconvergent
$p$
-adic modular forms when
$p=2,3$
and study their interaction with the Atkin operator. This results in an extension of Lauder’s algorithms for overconvergent modular forms. We illustrate these algorithms with computations of slope sequences of some
$2$
-adic eigencurves and the construction of Chow–Heegner points on elliptic curves via special values of Rankin triple product L-functions.