The aim of this paper is to study class number relations over function fields and the intersections of Hirzebruch-Zagier type divisors on the Drinfeld-Stuhler modular surfaces. The main bridge is a particular "harmonic" theta series with nebentypus. Using the strong approximation theorem, the Fourier coefficients of this series are expressed in two ways; one comes from modified Hurwitz class numbers and another gives the intersection numbers in question.

We prove that the dimension of the p -Selmer group of an elliptic curve is controlled by certain analytic quantities associated with modular symbols as conjectured by Kurihara.

Let k be a field of characteristic 0. Let G be a reductive group over the ring of Laurent polynomials R=k[x_1^{\pm 1},\ldots,x_n^{\pm 1}] . We prove that G is isotropic over R if and only if it is isotropic over the field of fractions k(x_1,\ldots,x_n) of R , and if this is the case, then the natural map H^1_{\acute{e}t}(R,G)\to H^1_{\acute{e}t}(k(x_1,\ldots,x_n),G) has trivial kernel and G is loop reductive. In particular, we settle in positive the conjecture of V. Chernousov, P. Gille, and A. Pianzola that H^1_{Zar}(R,G)=\ast for such groups G . We also deduce that if G is a reductive group over R of isotropic rank \ge 2 , then the natural map of non-stable K_1 -functors K_1^G(R)\to K_1^G\bigl( k((x_1))\ldots((x_n)) \bigr) is injective, and an isomorphism if G is moreover semisimple.

In the present paper, we first construct a pairing on the space of analytic distributions associated with \text{GSp}_{2g} . By considering the overconvergent parabolic cohomology groups and following the work of Johansson-Newton, we construct the cuspidal eigenvariety for \text{GSp}_{2g} . The pairing on the analytic distributions then induces a pairing on some coherent sheaves of the cuspidal eigenvariety. As an application, we follow the strategy of Bellaïche to study the ramification locus of the cuspidal eigenvariety over the corresponding weight space.

Motivated by the weight part of Serre's conjecture we consider the following question. Let K/\mathbb{Q}_p be a finite extension and suppose \overline{\rho} : G_K \rightarrow \operatorname{GL}_n(\overline{\mathbb{F}}_p) admits a crystalline lift with Hodge-Tate weights contained in the range [0,p] . Does \overline{\rho} admit a potentially diagonalisable crystalline lift of the same Hodge-Tate weights? We answer this question in the affirmative when K = \mathbb{Q}_p and n \leq 5 , and \overline{\rho} satisfies a mild 'cyclotomic-free' condition. We also prove partial results when K/\mathbb{Q}_p is unramified and n is arbitrary.

We point out some mistakes in our paper [ibid. 23, 2043–2077 (2018; Zbl 1430.46050)]. The most severe one is a gap in our proof that the smallest exact crossed product functor, as constructed in [loc. cit.], is automatically Morita compatible. We were not able to close this gap so far. So, at this time, we cannot conclude that the C^* -group algebra C_{\epsilon_{M}}^*(G)=\mathbb{C}\rtimes_{\epsilon_{M}} G corresponding to the smallest exact Morita compatible crossed-product functor \rtimes_{\epsilon_{M}} is equal to the reduced group C^* -algebra C_r^*(G) .

For a G -scheme X with a given equivariant perfect obstruction theory, we prove a virtual equivariant Grothendieck-Riemann-Roch formula, this is an extension of a result of B. Fantechi and L. Göttsche [Geom. Topol. 14, No. 1, 83–115 (2010; Zbl 1194.14017)] to the equivariant context. We also prove a virtual non-abelian localization theorem for schemes over \mathbb{C} with proper actions.

Given a Galois cover of curves over F_p , we relate the p-adic valuation of epsilon constants appearing in functional equations of Artin L-functions to an equivariant Euler characteristic. Our main theorem generalises a result of Chinburg from the tamely to the weakly ramified case. We furthermore apply Chinburg’s result to obtain a ‘weak’ relation in the general case. In the Appendix, we study, in this arbitrarily wildly ramified case, the integrality of p-adic valuations of epsilon constants.

We construct and estimate the fundamental solution of highly anisotropic space-inhomogeneous integro-differential operators. We use the Levy method. We give applications to the Cauchy problem for such operators.

Let $Q \to B$ be a quadric fibration and $T \to B$ a family of sextic du Val del Pezzo surfaces. Making use of the theory of noncommutative mixed motives, we establish a precise relation between the Schur-finiteness conjecture for $Q$, resp. for $T$, and the Schur-finiteness conjecture for $B$. As an application, we prove the Schur-finiteness conjecture for $Q$, resp. for $T$, when $B$ is low-dimensional. Along the way, we obtain a proof of the Schur-finiteness conjecture for smooth complete intersections of two or three quadric hypersurfaces. Finally, we prove similar results for the Bass-finiteness conjecture.