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1. Abelian surfaces over totally real fields are potentially modular

   https://doi.org/10.1007/s10240-021-00128-2  

   Boxer, George; Calegari, Frank; Gee, Toby; Pilloni, Vincent (December 2021, Publications mathématiques de l'IHÉS)

   Abstract We show that abelian surfaces (and consequently curves of genus 2) over totally real fields are potentially modular. As a consequence, we obtain the expected meromorphic continuation and functional equations of their Hasse–Weil zeta functions. We furthermore show the modularity of infinitely many abelian surfaces  $$A$$ A over  $${\mathbf {Q}}$$ Q with  $$\operatorname{End}_{ {\mathbf {C}}}A={\mathbf {Z}}$$ End C A = Z . We also deduce modularity and potential modularity results for genus one curves over (not necessarily CM) quadratic extensions of totally real fields. 

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2. Traces, high powers and one level density for families of curves over finite fields

   https://doi.org/10.1017/S030500411700041X  

   BUCUR, ALINA; COSTA, EDGAR; DAVID, CHANTAL; GUERREIRO, JOÃO; LOWRY–DUDA, DAVID (September 2018, Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract The zeta function of a curve C over a finite field may be expressed in terms of the characteristic polynomial of a unitary matrix Θ C . We develop and present a new technique to compute the expected value of tr(Θ C n ) for various moduli spaces of curves of genus g over a fixed finite field in the limit as g is large, generalising and extending the work of Rudnick [ Rud10 ] and Chinis [ Chi16 ]. This is achieved by using function field zeta functions, explicit formulae, and the densities of prime polynomials with prescribed ramification types at certain places as given in [ BDF + 16 ] and [ Zha ]. We extend [ BDF + 16 ] by describing explicit dependence on the place and give an explicit proof of the Lindelöf bound for function field Dirichlet L -functions L (1/2 + it , χ). As applications, we compute the one-level density for hyperelliptic curves, cyclic ℓ-covers, and cubic non-Galois covers. 

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3. Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields

   https://doi.org/10.1017/fmp.2022.14  

   Shankar, Ananth N.; Shankar, Arul; Tang, Yunqing; Tayou, Salim (January 2022, Forum of Mathematics, Pi)

   Abstract Given a K3 surface X over a number field K with potentially good reduction everywhere, we prove that the set of primes of K where the geometric Picard rank jumps is infinite. As a corollary, we prove that either $$X_{\overline {K}}$$ has infinitely many rational curves or X has infinitely many unirational specialisations. Our result on Picard ranks is a special case of more general results on exceptional classes for K3 type motives associated to GSpin Shimura varieties. These general results have several other applications. For instance, we prove that an abelian surface over a number field K with potentially good reduction everywhere is isogenous to a product of elliptic curves modulo infinitely many primes of K . 

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4. Torsion phenomena for zero-cycles on a product of curves over a number field

   https://doi.org/10.1007/s40993-024-00519-4  

   Gazaki, Evangelia; Love, Jonathan (March 2024, Research in Number Theory)

   Abstract For a smooth projective varietyXover an algebraic number fieldka conjecture of Bloch and Beilinson predicts that the kernel of the Albanese map ofXis a torsion group. In this article we consider a product$$X=C_1\times \cdots \times C_d$$ $X=C_1\times \cdots \times C_d$of smooth projective curves and show that if the conjecture is true for any subproduct of two curves, then it is true forX. For a product$$X=C_1\times C_2$$ $X=C_1\times C_2$of two curves over$$\mathbb {Q} $$ $Q$with positive genus we construct many nontrivial examples that satisfy the weaker property that the image of the natural map$$J_1(\mathbb {Q})\otimes J_2(\mathbb {Q})\xrightarrow {\varepsilon }{{\,\textrm{CH}\,}}_0(C_1\times C_2)$$ $J_1\left(Q\right)\otimes J_2\left(Q\right)\stackrel{\epsilon }{\to }{\text{CH}}_0(C_1\times C_2)$is finite, where$$J_i$$ $J_i$is the Jacobian variety of$$C_i$$ $C_i$. Our constructions include many new examples of non-isogenous pairs of elliptic curves$$E_1, E_2$$ $E_1,E_2$with positive rank, including the first known examples of rank greater than 1. Combining these constructions with our previous result, we obtain infinitely many nontrivial products$$X=C_1\times \cdots \times C_d$$ $X=C_1\times \cdots \times C_d$for which the analogous map$$\varepsilon $$ $\epsilon$has finite image. 

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5. Reductions of abelian surfaces over global function fields

   https://doi.org/10.1112/S0010437X22007473  

   Maulik, Davesh; Shankar, Ananth N.; Tang, Yunqing (April 2022, Compositio Mathematica)

   Let $$A$$ be a non-isotrivial ordinary abelian surface over a global function field of characteristic $p>0$ with good reduction everywhere. Suppose that $$A$$ does not have real multiplication by any real quadratic field with discriminant a multiple of $$p$$ . We prove that there are infinitely many places modulo which $$A$$ is isogenous to the product of two elliptic curves. 

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