We study a kernel function of the twisted symmetric square $L$ -function of elliptic modular forms. As an application, several exact special values of the $L$ -function are computed.

Let $\unicode[STIX]{x1D6E4}\subseteq \operatorname{PSL}(2,\mathbf{R})$ be a finite-volume Fuchsian group. The hyperbolic circle problem is the estimation of the number of elements of the $\unicode[STIX]{x1D6E4}$ -orbit of $z$ in a hyperbolic circle around $w$ of radius $R$ , where $z$ and $w$ are given points of the upper half plane and $R$ is a large number. An estimate with error term $\text{e}^{(2/3)R}$ is known, and this has not been improved for any group. Recently, Risager and Petridis proved that in the special case $\unicode[STIX]{x1D6E4}=\operatorname{PSL}(2,\mathbf{Z})$ taking $z=w$ and averaging over $z$ in a certain way the error term can be improved to $\text{e}^{(7/12+\unicode[STIX]{x1D716})R}$ . Here we show such an improvement for a general $\unicode[STIX]{x1D6E4}$ ; our error term is $\text{e}^{(5/8+\unicode[STIX]{x1D716})R}$ (which is better than $\text{e}^{(2/3)R}$ but weaker than the estimate of Risager and Petridis in the case $\unicode[STIX]{x1D6E4}=\operatorname{PSL}(2,\mathbf{Z})$ ). Our main tool is our generalization of the Selberg trace formula proved earlier.

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.

Let $p$ be a prime number and $F$ a totally real number field. For each prime $\mathfrak{p}$ of $F$ above $p$ we construct a Hecke operator $T_{\mathfrak{p}}$ acting on $(\text{mod}\,p^{m})$ Katz Hilbert modular classes which agrees with the classical Hecke operator at $\mathfrak{p}$ for global sections that lift to characteristic zero. Using these operators and the techniques of patching complexes of Calegari and Geraghty we prove that the Galois representations arising from torsion Hilbert modular classes of parallel weight $\mathbf{1}$ are unramified at $p$ when $[F:\mathbb{Q}]=2$ . Some partial and some conjectural results are obtained when $[F:\mathbb{Q}]>2$ .

Modular curves like X0(N) and X1(N) appear very frequently in arithmetic geometry. While their complex points are obtained as a quotient of the upper half plane by some subgroups of SL2(ℤ), they allow for a more arithmetic description as a solution to a moduli problem. We wish to give such a moduli description for two other modular curves, denoted here by Xnsp(p) and Xnsp+(p) associated to non-split Cartan subgroups and their normaliser in GL2(𝔽p). These modular curves appear for instance in Serre's problem of classifying all possible Galois structures of p-torsion points on elliptic curves over number fields. We give then a moduli-theoretic interpretation and a new proof of a result of Chen (Proc. London Math. Soc. (3) 77(1) (1998), 1–38; J. Algebra 231(1) (2000), 414–448).

For the modular variety attached to an arithmetic subgroup of an indefinite unitary group of signature $(1,n+1)$, with $n\geqslant 1$, we study Heegner divisors in the local Picard group over a boundary component of a compactification. For this purpose, we introduce local Borcherds products. We obtain a precise criterion for local Heegner divisors to be torsion elements in the Picard group, and further, as an application, we show that the obstructions to a local Heegner divisor being a torsion element can be described by certain spaces of vector-valued elliptic cusp forms, transforming under a Weil representation.

We show that the automorphic étale cohomology of a (possibly noncompact) PEL-type or Hodge-type Shimura variety in characteristic zero is canonically isomorphic to the cohomology of the associated nearby cycles over most of their mixed characteristics models constructed in the literature.

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].

For the groups $\operatorname{SO}(2n+1,F)$, where $F$ is a $p$-adic field, we consider the tempered irreducible representations of unipotent reduction. Lusztig has constructed and parametrized these representations. We prove that they satisfy the expected endoscopic identities which determine the parametrization.

Let $K$ be the field of fractions of a local Henselian discrete valuation ring ${\mathcal{O}}_{K}$ of characteristic zero with perfect residue field $k$ . Assuming potential semi-stable reduction, we show that an unramified Galois action on the second $\ell$ -adic cohomology group of a K3 surface over $K$ implies that the surface has good reduction after a finite and unramified extension. We give examples where this unramified extension is really needed. Moreover, we give applications to good reduction after tame extensions and Kuga–Satake Abelian varieties. On our way, we settle existence and termination of certain flops in mixed characteristic, and study group actions and their quotients on models of varieties.

Let $\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2}$ be a pair of cuspidal complex, or $\ell$-adic, representations of the general linear group of rank $n$ over a nonarchimedean local field $F$ of residual characteristic $p$, different to $\ell$. Whenever the local Rankin–Selberg $L$-factor $L(X,\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2})$ is nontrivial, we exhibit explicit test vectors in the Whittaker models of $\unicode[STIX]{x1D70B}_{1}$ and $\unicode[STIX]{x1D70B}_{2}$ such that the local Rankin–Selberg integral associated to these vectors and to the characteristic function of $\mathfrak{o}_{F}^{n}$ is equal to $L(X,\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2})$. As an application we prove that the $L$-factor of a pair of banal $\ell$-modular cuspidal representations is the reduction modulo $\ell$ of the $L$-factor of any pair of $\ell$-adic lifts.

Suppose that $F/F^{+}$ is a CM extension of number fields in which the prime $p$ splits completely and every other prime is unramified. Fix a place $w|p$ of $F$ . Suppose that $\overline{r}:\operatorname{Gal}(\overline{F}/F)\rightarrow \text{GL}_{3}(\overline{\mathbb{F}}_{p})$ is a continuous irreducible Galois representation such that $\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$ is upper-triangular, maximally non-split, and generic. If $\overline{r}$ is automorphic, and some suitable technical conditions hold, we show that $\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$ can be recovered from the $\text{GL}_{3}(F_{w})$ -action on a space of mod $p$ automorphic forms on a compact unitary group. On the way we prove results about weights in Serre’s conjecture for $\overline{r}$ , show the existence of an ordinary lifting of $\overline{r}$ , and prove the freeness of certain Taylor–Wiles patched modules in this context. We also show the existence of many Galois representations $\overline{r}$ to which our main theorem applies.

Let $A\rightarrow B$ be a morphism of Artin local rings with the same embedding dimension. We prove that any $A$ -flat $B$ -module is $B$ -flat. This freeness criterion was conjectured by de Smit in 1997 and improves Diamond’s criterion [The Taylor–Wiles construction and multiplicity one, Invent. Math. 128 (1997), 379–391, Theorem 2.1]. We also prove that if there is a nonzero $A$ -flat $B$ -module, then $A\rightarrow B$ is flat and is a relative complete intersection. Then we explain how this result allows one to simplify Wiles’s proof of Fermat’s last theorem: we do not need the so-called ‘Taylor–Wiles systems’ any more.

We prove a general formula for the $p$ -adic heights of Heegner points on modular abelian varieties with potentially ordinary (good or semistable) reduction at the primes above $p$ . The formula is in terms of the cyclotomic derivative of a Rankin–Selberg $p$ -adic $L$ -function, which we construct. It generalises previous work of Perrin-Riou, Howard, and the author to the context of the work of Yuan–Zhang–Zhang on the archimedean Gross–Zagier formula and of Waldspurger on toric periods. We further construct analytic functions interpolating Heegner points in the anticyclotomic variables, and obtain a version of our formula for them. It is complemented, when the relevant root number is $+1$ rather than $-1$ , by an anticyclotomic version of the Waldspurger formula. When combined with work of Fouquet, the anticyclotomic Gross–Zagier formula implies one divisibility in a $p$ -adic Birch and Swinnerton-Dyer conjecture in anticyclotomic families. Other applications described in the text will appear separately.

In this paper, we consider how to express an Iwahori–Whittaker function through Demazure characters. Under some interesting combinatorial conditions, we obtain an explicit formula and thereby a generalization of the Casselman–Shalika formula. Under the same conditions, we compute the transition matrix between two natural bases for the space of Iwahori fixed vectors of an induced representation of a $p$-adic group; this corrects a result of Bump–Nakasuji.

We prove the explicit version of the Buzzard–Diamond–Jarvis conjecture formulated by Dembele et al. (Serre weights and wild ramification in two-dimensional Galois representations, Preprint (2016), arXiv:1603.07708 [math.NT]). More precisely, we prove that it is equivalent to the original Buzzard–Diamond–Jarvis conjecture, which was proved for odd primes (under a mild Taylor–Wiles hypothesis) in earlier work of the third author and coauthors.

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 consider mod  $p$ Hilbert modular forms associated to a totally real field of degree  $d$ in which $p$ is unramified. We prove that every such form arises by multiplication by partial Hasse invariants from one whose weight (a $d$ -tuple of integers) lies in a certain cone contained in the set of non-negative weights, answering a question of Andreatta and Goren. The proof is based on properties of the Goren–Oort stratification on mod  $p$ Hilbert modular varieties established by Goren and Oort, and Tian and Xiao.

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 study the multiple Eisenstein series introduced by Gangl, Kaneko and Zagier. We give a proof of (restricted) finite double shuffle relations for multiple Eisenstein series by revealing an explicit connection between the Fourier expansion of multiple Eisenstein series and the Goncharov co-product on Hopf algebras of iterated integrals.