quadratic extensions
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Author(s):  
George Boxer ◽  
Frank Calegari ◽  
Toby Gee ◽  
Vincent Pilloni

AbstractWe 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.


2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Asbjørn Christian Nordentoft

AbstractIn this paper, we study hybrid subconvexity bounds for class group 𝐿-functions associated to quadratic extensions K/\mathbb{Q} (real or imaginary). Our proof relies on relating the class group 𝐿-functions to Eisenstein series evaluated at Heegner points using formulas due to Hecke. The main technical contribution is the uniform sup norm bound for Eisenstein series E(z,1/2+it)\ll_{\varepsilon}y^{1/2}(\lvert t\rvert+1)^{1/3+\varepsilon}, y\gg 1, extending work of Blomer and Titchmarsh. Finally, we propose a uniform version of the sup norm conjecture for Eisenstein series.


2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Yao Cheng

AbstractWe establish explicit Ichino’s formulae for the central values of the triple product L-functions with emphasis on the calculations for the real place. The key ingredient for our computations is Proposition 8 which generalizes a result in [P. Michel and A. Venkatesh, The subconvexity problem for {\rm GL}_{2}, Publ. Math. Inst. Hautes Études Sci. 111 2010, 171–271]. As an application we prove the optimal upper bound of a sum of restricted {L^{2}}-norms of the {L^{2}}-normalized newforms on certain quadratic extensions with prime level and bounded spectral parameter following the methods in [V. Blomer, On the 4-norm of an automorphic form, J. Eur. Math. Soc. (JEMS) 15 2013, 5, 1825–1852].


Author(s):  
Eslam Badr ◽  
Francesc Bars

Let [Formula: see text] be a smooth plane curve of degree [Formula: see text] defined over a global field [Formula: see text] of characteristic [Formula: see text] or [Formula: see text] (up to an extra condition on [Formula: see text]). Unless the curve is bielliptic of degree four, we observe that it always admits finitely many quadratic points. We further show that there are only finitely many quadratic extensions [Formula: see text] when [Formula: see text] is a number field, in which we may have more points of [Formula: see text] than these over [Formula: see text]. In particular, we have this asymptotic phenomenon valid for Fermat’s and Klein’s equations. Second, we conjecture that there are two infinite sets [Formula: see text] and [Formula: see text] of isomorphism classes of smooth projective plane quartic curves over [Formula: see text] with a prescribed automorphism group, such that all members of [Formula: see text] (respectively [Formula: see text]) are bielliptic and have finitely (respectively infinitely) many quadratic points over a number field [Formula: see text]. We verify the conjecture over [Formula: see text] for [Formula: see text] and [Formula: see text]. The analog of the conjecture over global fields with [Formula: see text] is also considered.


2020 ◽  
Vol 327 ◽  
pp. 75-86
Author(s):  
Ruben Gamboa ◽  
John Cowles ◽  
Woodrow Gamboa
Keyword(s):  

2020 ◽  
Vol 16 (09) ◽  
pp. 1907-1922
Author(s):  
Stephan Baier ◽  
Rajneesh Kumar Singh

In this paper, we establish a version of the large sieve inequality with square moduli for imaginary quadratic extensions of rational function fields of odd characteristics.


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