Jornada

Time (UTC-3)	Speaker	Title
10:00-11:00	Yves Martin (Universidad de Chile)	Every Siegel cusp form $F$ of weight $k$ over a Hecke congruence subgroup $\Gamma _0^n(N)$ of the symplectic group $Sp_n(\mathbb {Z}) \subseteq \mathbb {Z}^{2n, 2n}$ has a Fourier series representation \[ F(Z) = \sum _{T \in J_n} c(T) \exp {(2\pi i \ trace (TZ))}. \] Here $Z$ belongs to the Siegel space of degree $n$ and $J_n$ denotes the set of positive-definite, half-integral, symmetric, $n$ by $n$ matrices. In this talk we discuss the existence of simple looking matrices $T$ such that $c_F(T) \neq 0$ whenever $F \neq 0$. In this manner one gets proper, uncomplicated subsets $\Delta $ of $J_n$ such that \[ F = G \ \text{if, and only if}, \ c_F(T) = c_G(T) \ \text{for all $T \in \Delta $}. \] In particular, for $n=3$ we exhibit a set $\Delta $ parametrized by triples of primes numbers which characterize all Siegel cusp forms in $S_k^3(N, \chi )$ whenever $N$ is odd and conductor of $\chi = N$.
11:00-11:30	Coffee and Cookies	☕
11:30-12:30	Luis Santiago Palacios (USACH)	In recent years, $p$-adic $L$-functions have been a subject of considerable study. An important problem in modern number theory is to construct $p$-adic $L$-functions of automorphic forms. Let $K$ be an imaginary quadratic field and consider the automorphic forms of $\mathrm{GL}_2$ over $K$, better known as Bianchi modular forms. In this talk, we present the ideas of our construction of the $p$-adic $L$-function of non-cuspidal Bianchi modular forms given by the base change to $K$ of modular forms with complex multiplication by $K$. We also show a factorization of our $p$-adic $L$-functions as the product of two Katz $p$-adic $L$-functions, which are, the $p$-adic $L$-functions of Hecke characters of $K$.
14:30-15:30	Xenia Dimitrakopoulou (Warwick University)	In this talk we will explain what branching laws are and their link to critical $L$-values of certain automorphic representations of $U_{n+1} \times U_n$. We will show how to $p$-adically interpolate these branching laws and explain how this construction yields a $p$-adic $L$-function. Time permitting, we will discuss how to extend such constructions to more general spherical pairs.
15:30-16:00	Coffee and Cookies	☕
16:00-17:00	Jerson Caro (PUC)	Let $E$ be an elliptic curve defined over a function field $K$ of positive characteristic. When $E$ has split multiplicative reduction at the infinite place, there is an analogue of the modularity theorem of Wiles over $\mathbb {Q}$. In this talk, we will introduce Drinfeld curves and Drinfeld modular forms to give a general idea of this modularity theorem over function fields. In addition, we will introduce a conjecture of Watkins in 2002 that relates the Mordell-Weil rank with the modular degree of an elliptic curves over $\mathbb {Q}$, and then we will present an analogue of this conjecture over function fields and show some cases where this conjecture is satisfied. Those results are part of my PhD thesis under the supervision of Hector Pasten (PUC).

Time (UTC-3)	Speaker	Title
10:00-11:00	Yves Martin (Universidad de Chile)	Every Siegel cusp form \(F\) of weight \(k\) over a Hecke congruence subgroup \(\Gamma _0^n(N)\) of the symplectic group \(Sp_n(\mathbb {Z}) \subseteq \mathbb {Z}^{2n, 2n}\) has a Fourier series representation \[ F(Z) = \sum _{T \in J_n} c(T) \exp {(2\pi i \ trace (TZ))}. \] Here \(Z\) belongs to the Siegel space of degree \(n\) and \(J_n\) denotes the set of positive-definite, half-integral, symmetric, \(n\) by \(n\) matrices. In this talk we discuss the existence of simple looking matrices \(T\) such that \(c_F(T) \neq 0\) whenever \(F \neq 0\). In this manner one gets proper, uncomplicated subsets \(\Delta \) of \(J_n\) such that \[ F = G \ \text{if, and only if}, \ c_F(T) = c_G(T) \ \text{for all $T \in \Delta $}. \] In particular, for \(n=3\) we exhibit a set \(\Delta \) parametrized by triples of primes numbers which characterize all Siegel cusp forms in \(S_k^3(N, \chi )\) whenever \(N\) is odd and conductor of \(\chi = N\).
11:00-11:30	Coffee and Cookies	☕
11:30-12:30	Luis Santiago Palacios (USACH)	In recent years, \(p\)-adic \(L\)-functions have been a subject of considerable study. An important problem in modern number theory is to construct \(p\)-adic \(L\)-functions of automorphic forms. Let \(K\) be an imaginary quadratic field and consider the automorphic forms of \(\mathrm{GL}_2\) over \(K\), better known as Bianchi modular forms. In this talk, we present the ideas of our construction of the \(p\)-adic \(L\)-function of non-cuspidal Bianchi modular forms given by the base change to \(K\) of modular forms with complex multiplication by \(K\). We also show a factorization of our \(p\)-adic \(L\)-functions as the product of two Katz \(p\)-adic \(L\)-functions, which are, the \(p\)-adic \(L\)-functions of Hecke characters of \(K\).
14:30-15:30	Xenia Dimitrakopoulou (Warwick University)	In this talk we will explain what branching laws are and their link to critical \(L\)-values of certain automorphic representations of \(U_{n+1} \times U_n\). We will show how to \(p\)-adically interpolate these branching laws and explain how this construction yields a \(p\)-adic \(L\)-function. Time permitting, we will discuss how to extend such constructions to more general spherical pairs.
15:30-16:00	Coffee and Cookies	☕
16:00-17:00	Jerson Caro (PUC)	Let \(E\) be an elliptic curve defined over a function field \(K\) of positive characteristic. When \(E\) has split multiplicative reduction at the infinite place, there is an analogue of the modularity theorem of Wiles over \(\mathbb {Q}\). In this talk, we will introduce Drinfeld curves and Drinfeld modular forms to give a general idea of this modularity theorem over function fields. In addition, we will introduce a conjecture of Watkins in 2002 that relates the Mordell-Weil rank with the modular degree of an elliptic curves over \(\mathbb {Q}\), and then we will present an analogue of this conjecture over function fields and show some cases where this conjecture is satisfied. Those results are part of my PhD thesis under the supervision of Hector Pasten (PUC).